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+page_content='04894v1  [math-ph]  12 Jan 2023  Ground state energy of the dilute spin-polarized Fermi  gas: Upper bound via cluster expansion  Asbjørn Bækgaard Lauritsen∗ and Robert Seiringer†  IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria  13 January 2023  Abstract  We prove an upper bound on the ground state energy of the dilute spin-polarized  Fermi gas capturing the leading correction to the kinetic energy resulting from repulsive  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' One of the main ingredients in the proof is a rigorous implementation of the  fermionic cluster expansion of Gaudin, Gillespie and Ripka (Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content=' A, 176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 (1971),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content='  Contents  1  Introduction and main results  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content='1  Calculation of the normalization constant .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content='2  Calculation of the 1-particle reduced density .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content='3  Calculation of the 2-particle reduced density .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content='  20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4  Calculation of the 3-particle reduced density .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content='4  Absolute convergence of the Γ3-sum .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content='  30  ∗alaurits@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='at  1  4  Energy of the trial state  31  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1  Thermodynamic limit via a box method  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content='  57  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3 Small diagrams in 1 dimension (proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='21) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content='  58  B Derivative Lebesgue constants (proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content='1 Reduction to simpler tetrahedron .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content='  62  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 Reduction from d = 3 to d = 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content='  63  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3 Reduction from d = 2 to d = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content='4 Bounding the one-dimensional integrals .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  70  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5 Bounding the j = 3 two-dimensional integral .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  72  1  Introduction and main results  We consider a Fermi gas of N particles in a box Λ = ΛL = [−L/2, L/2]d in d dimensions,  d = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We will mostly focus on the case d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The particles interact via a two-body  interaction v, which we assume to be positive, radial and of compact support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In particular we  allow for v to have a hard core, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' v(x) = ∞ for |x| ≤ r for some r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In natural units  where ℏ = 1 and the mass of the particles is m = 1/2 the Hamiltonian of the system takes the  form  HN =  N  �  i=1  −∆xi +  �  j<k  v(xj − xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We are interested in spin-polarized fermions, meaning that all the spins are aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We may  thus equivalently forget about the spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This means that the Hamiltonian should be realized  on the fermionic N-particle space of antisymmetric wavefunctions L2  a(ΛN) = �N L2(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We  consider the ground state energy density in the thermodynamic limit  ed(ρ) =  lim  L→∞  N/Ld→ρ  inf  ΨN∈L2  a(ΛN)  ∥ΨN∥2  L2=1  ⟨ΨN|HN|ΨN⟩  Ld  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  It is a result of Robinson [Rob71] that the thermodynamic limit exists, and that it is independent  of boundary conditions (say, Dirichlet, Neumann or periodic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We study the dilute limit, where the inter-particle spacing is large compared to the length  scale set by the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For spin-polarized fermions, the relevant lengthscale is the p-wave  scattering length a which we define below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Our main theorem is the upper bound  ed=3(ρ) ≤ 3  5(6π2)2/3ρ5/3 + 12π  5 (6π)2/3a3ρ8/3  �  1 − 9  35(6π2)2/3a2  0ρ2/3 + o  �  (a3ρ)2/3��  2  in the dilute limit a3ρ ≪ 1, where a0 is another length related to the scattering length, also  defined below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The leading term 3  5(6π2)2/3ρ5/3 is the kinetic energy density of the free Fermi  gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The next term 12π  5 (6π)2/3a3ρ8/3 naturally results from the two-body interactions using that  the two-body density vanishes quadratically at incident points, leading to the cubic behavior  in the scattering length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Finally, the correction term of order a3a2  0ρ10/3 is a consequence of the  fourth-order behaviour of the two-particle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This formula is expected to be sharp, at least to order a4ρ3 (in the sense that there is no term  of this order) [WDS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Indeed, to this order it appears as an equality in the physics literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For instance, the formula (to order a3ρ8/3) follows by truncating expansion formulas of Jastrow  [Jas55], Iwamoto and Yamada [IY57], Clark and Westhaus [CW68;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' WC68] or Gaudin, Gillespie  and Ripka [GGR71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Additionally the formula (to order a3ρ8/3) is claimed by Efimov and  Amus’ya [Efi66;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' EA65], see also [WDS20] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Our result thus verifies  this formula from the physics literature, at least as an upper bound: Both, that the leading  correction to the kinetic energy is as described, and that there are no contributions of order  a4ρ3, see [WDS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' An important ingredient in our proof is a rigorous implementation of the  cluster expansion introduced by Gaudin, Gillespie and Ripka [GGR71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For the dilute Fermi gas one can also study the setting where different spins are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This is studied in [FGHP21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Gia22a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' LSS05], see also [Gia22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This system is realized by  having the Hamiltonian HN act on a definite spin-sector L2  a(ΛN↑) ⊗ L2  a(ΛN↓), where one fixes  the number of spin-up and -down particles to be N↑ and N↓ = N −N↑ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The energy  density satisfies (in 3 dimensions)  ed=3(ρ↑, ρ↓) = 3  5(6π2)2/3 �  ρ5/3  ↑  + ρ5/3  ↓  �  + 8πasρ↓ρ↑ + o(asρ2),  where ρσ denotes the density of particles of spin σ ∈ {↑, ↓} and ρ = ρ↓ + ρ↑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Here as is the  s-wave scattering length of the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The leading term is again the kinetic energy density  of a free Fermi gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The next to leading order correction was first shown in [LSS05] and later  in [FGHP21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Gia22a] using different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The next correction is conjectured to be the  Huang–Yang term [HY57] of order a2  sρ7/3, see [Gia22a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Gia22b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note that even the Huang–  Yang term of order a2  sρ7/3 is much larger than the leading correction in the spin-polarized case  of order a3ρ8/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For the dilute Fermi gas with spin, effectively only fermions of different spins interact (to  leading order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For fermions of different spins, the Pauli exclusion principle does not give  any restriction, and the energy correction of the interaction is the same as for a dilute Bose  gas (to leading order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For fermions of the same spin, the Pauli exclusion principle gives an  inherent repulsion between the fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This gives the effect that the energy correction of the  interaction is much smaller for fermions all of the same spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  In addition to the dilute Fermi gas, much work has been done on dilute Bose gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Here  one realizes the Hamiltonian on the bosonic N-particle space of symmetric functions L2  s(ΛN) =  L2(Λ)⊗symN instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' One has the asymptotic formula (in 3 dimensions)  ed=3(ρ) = 4πasρ2  �  1 + 128  15√π(a3  sρ)1/2 + o  �  (a3  sρ)1/2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The leading term was shown by Dyson [Dys57] for an upper bound and Lieb and Yngvason  [LY98] for the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The next correction, known as the Lee–Huang–Yang correction  [LHY57], was shown as an upper bound in [BCS21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' YY09] and as a lower bound in [FS21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  FS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In some sense, the term 12π  5 (6π)2/3a3ρ8/3 for the same-spin fermions is the fermionic  analogue of the 4πasρ2 for the bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' It is the leading correction to the energy of the free  Fermi/Bose gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  3  Finally also some lower-dimensional problems have been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The 2-dimensional dilute  Fermi gas with different spins present is studied in [LSS05], where the leading correction to the  kinetic energy is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We show that for the spin-polarized setting in 2 dimensions we have  the upper bound  ed=2(ρ) ≤ π  8 ρ2 + π2  4 a2ρ3[1 + o(1)]  as a2ρ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Additionally, the 1-dimensional spin-polarized Fermi gas is studied in [ARS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Agerskov,  Reuvers and Solovej [ARS22] show that  ed=1(ρ) = π2  3 ρ3 + 2π2  3 aρ4 [1 + o(1)]  as aρ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We give a new proof of this as an upper bound with an improved error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1  Precise statement of results  We now give the precise statement of our main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We start with the 3-dimensional  setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' First, we define the p-wave scattering length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (See also [LY01, Appendix A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' SY20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The p-wave scattering length a of the interaction v is defined by the minimiza-  tion problem  12πa3 = inf  �ˆ  R3 |x|2  �  |∇f0(x)|2 + 1  2v(x)|f0(x)|2  �  dx : f0(x) → 1 as |x| → ∞  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The minimizer f0 is the (p-wave) scattering function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (In case v has a hard core, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' v(x) = ∞  for |x| ≤ r one has to interpret v(x) dx as a measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Necessarily then the minimizer has  f0(x) = 0 for |x| ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  We collect properties of the scattering function f0 in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We define the length a0 as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The length a0 is given by  3a2  0 =  1  12πa3  ˆ  R3 |x|4  �  |∇f0(x)|2 + 1  2v(x)|f0(x)|2  �  dx,  where f0 is the scattering function of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The normalization is chosen so that a  hard core interaction of radius R0 has a0 = a = R0, see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (If v has a hard core we  interpret v(x) dx as a measure as in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  We can now state our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Suppose that v ≥ 0 is radial and compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then, for sufficiently  small a3ρ, the ground-state energy density satisfies  ed=3(ρ) ≤ 3  5(6π2)2/3ρ5/3  + 12π  5 (6π)2/3a3ρ8/3  �  1 − 9  35(6π2)2/3a2  0ρ2/3 + O  �  (a3ρ)2/3+1/21| log(a3ρ)|6��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The essential steps in the proof are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  4  (1) Show the absolute convergence of the formal cluster expansion formulas of [GGR71] for the  reduced densities of a Jastrow-type trial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The criterion for absolute convergence will  not hold uniformly in the system size, and in order to allow for a larger particle number we  need to introduce the “Fermi polyhedron”, described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2, as an approximation  to the Fermi ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The formulas of [GGR71] are computed in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3  and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4 and stated in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The absolute convergence is proven in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (2) Bound the energy of the Jastrow-type trial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For this we shall in particular need  bounds on “derivative Lebesgue constants” given in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9 and proven in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The computation of the energy of such a Jastrow-type trial state is given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (3) Use a box method to glue together trial states in smaller boxes to obtain a bound in the  thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This is done in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' One may weaken the assumptions on the interaction v a bit at the cost of a longer  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The compact support and that v ≥ 0 are not strictly necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Essentially, we just need  sufficiently good bounds on integrals of the scattering function f0 as used in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 and 4  and that the “stability condition” of the tree-graph bound [PU09, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Uel18] used  in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' With the same method one should be able to improve the error bound slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  At best one could get the error to be Oε(ρ5/3(a3ρ)2−ε) for any ε > 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=', the error-term in the  theorem, O((a3ρ)2/3+1/21| log(a3ρ)|6) could be replaced by Oε((a3ρ)1−ε)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This is similar to the  recent work on the Bose gas [BCGOPS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We shall discuss this further in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We consider the lower-dimensional problems next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We start with 2 dimensions, where the  scattering length is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The (2-dimensional) p-wave scattering length a of the interaction v is defined  by the minimization problem  4πa2 = inf  �ˆ  R2 |x|2  �  |∇f0(x)|2 + 1  2v(x)|f0(x)|2  �  dx : f0(x) → 1 as |x| → ∞  �  The minimizer f0 is the (2-dimensional) (p-wave) scattering function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  With this, we may state the 2-dimensional analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7 (Two dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Suppose that v ≥ 0 is radial and compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then,  for sufficiently small a2ρ, the ground-state energy density satisfies  ed=2(ρ) ≤ π  8 ρ2 + π2  4 a2ρ3  �  1 + O  �  a2ρ| log(a2ρ)|2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We sketch in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 how to adapt the proof in the 3-dimensional setting to 2 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Finally, we consider the 1-dimensional problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The scattering length is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The (1-dimensional) p-wave scattering length a of the interaction v is defined  by the minimization problem  2a = inf  �ˆ  R  |x|2  �  |∂f0(x)|2 + 1  2v(x)|f0(x)|2  �  dx : f0(x) → 1 as |x| → ∞  �  The minimizer f0 is the (1-dimensional) (p-wave) scattering function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  5  We show in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12 that Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8 agrees with the (seemingly different) definition  of the scattering length in [ARS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' With this, we may state the 1-dimensional analogue of  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9 (One dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Suppose that v ≥ 0 is even and compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Suppose  moreover that  ´ � 1  2vf 2  0 + |∂f0|2�  dx < ∞, where f0 denotes the (p-wave) scattering function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Then, for sufficiently small aρ, the ground-state energy density satisfies  ed=1(ρ) ≤ π2  3 ρ3 + 2π2  3 aρ4  �  1 + O  �  (aρ)13/17��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We remark that Agerskov, Reuvers and Solovej [ARS22] recently showed (almost) the same  result with a matching lower bound ed=1(ρ) ≥  π2  3 ρ3 + 2π2  3 aρ4(1 + o(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Compared to their  result we treat a slightly different class of potentials and obtain an improved error bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The  conjectured next contribution is of order a2ρ5, see [ARS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10 (On the assumptions on v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Any smooth interaction or an interaction with a hard  core (meaning that v(x) = +∞ for |x| ≤ a0 for some a0 > 0) satisfies  ´ �1  2vf 2  0 + |∂f0|2�  dx < ∞,  see Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We sketch in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 how to adapt the proof in the 3-dimensional setting to 1 dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This turns out to be more involved than adapting the argument to 2 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In Section 2 we give some preliminary computations  and in particular we introduce the “Fermi polyhedron”, a polyhedral approximation to the  Fermi ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In Section 3 we introduce the fermionic cluster expansion of Gaudin, Gillespie  and Ripka [GGR71] and we find conditions on absolute convergence of the resulting formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  In the subsequent Section 4 we compute the energy of a Jastrow-type trial state and glue  many of them together using a box method to form trial states of arbitrary many particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Finally, in Section 5 we sketch how to adapt the argument to the lower-dimensional settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In  Appendix A we give computations of “small diagrams” needed for some bounds in Sections 4  and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 and in Appendix B we give the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9, an important lemma used in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  2  Preliminary computations  We will construct a trial state using a box method, and bound the energy of such trial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  To use such a box method we need to use Dirichlet boundary conditions in each smaller box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3 we show that we may construct trial states with Dirichlet boundary condition out of  trial states with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We will thus use periodic boundary conditions  in the box Λ = [−L/2, L/2]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For periodic boundary conditions, the Hamiltonian is given by  HN = Hper  N,L =  N  �  j=1  −∆j +  �  i<j  vper(xi − xj),  where ∆j denotes the Laplacian on the j’th coordinate and vper(x) = �  n∈Z3 v(x + nL), the  periodized interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By a slight abuse of notation we write v = vper, since we will choose L  bigger than the range of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The trial state in each smaller box is given by the Jastrow-type [Jas55] trial state (also  known as a Bijl-Dingle-Jastrow-type trial state)  ψN =  1  √CN  �  i<j  f(xi − xj)DN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xN),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1)  6  where f is a scaled and cut-off version of the scatting function f0, DN is an appropriately chosen  Slater determinant, and CN is a normalization constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' More precisely,  f(x) =  �  1  1−a3/b3f0(|x|)  |x| ≤ b,  1  |x| ≥ b,  DN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xN) = det [uk(xi)]1≤i≤N  k∈PF  ,  uk(x) =  1  L3/2eikx,  where f0 is the p-wave scattering function, |·| := minn∈Z3 |· − nL|R3 (with |·|R3 denoting the  norm on R3), b > R0, the range of v, is some cut-off to be chosen later, PF is a polyhedral  approximation to the Fermi ball BF of radius kF described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2, and the number of  particles is N = #PF, the number of points in PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We choose b to be larger than the range of  v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' in particular, then f is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (Note that the metric on the torus is d(x, y) = |x − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We will abuse notation slightly and denote by |·| also the absolute value of some number or the  norm on R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  Before going further with the proof we first fix some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We introduce the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For any function h and edge (of some graph) e = (i, j) we will write he = hij = h(xi − xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We denote by C a generic positive constant whose value may change line by line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For expressions A, B we write A ≲ B if there exists some constant C > 0 such that A ≤ CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  If both A ≲ B and B ≲ A we write A ∼ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For a vector x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xd) ∈ Rd we write x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xd for its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We will fix the Fermi momentum kF and then choose L, N large but finite depending on kF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The density of particles in the trial state ψN is ρ := N/L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The limit of small density a3ρ → 0  will be realized as kFa → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  To compute the energy of the trial state ψN note that for (real-valued) functions F, G we have  ˆ  |∇(FG)|2 =  ˆ  |∇F|2|G|2 −  ˆ  |F|2G∆G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Using this on F = �  i<j fij and G = DN we have  ⟨ψN|HN|ψN⟩ = E0 + 2  �  j<k  �  ψN  �����  ����  ∇f(xj − xk)  f(xj − xk)  ����  2  + 1  2v(xj − xk)  �����ψN  �  + 6  �  i<j<k  �  ψN  ����  ∇fij∇fjk  fijfjk  ����ψN  �  = E0 +  ¨  ρ(2)  Jas(x1, x2)  �����  ∇f(x1 − x2)  f(x1 − x2)  ����  2  + 1  2v(x1 − x2)  �  dx1 dx2  +  ˚  ρ(3)  Jas(x1, x2, x3)∇f12∇f23  f12f23  dx1 dx2 dx3,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2)  where E0 = �  k∈PF |k|2 is the kinetic energy of  1  √  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='DN and ρ(n)  Jas denotes the n-particle reduced  density of the trial state ψN, given by  ρ(n)  Jas(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xn) = N(N−1) · · · (N−n+1)  ˙  |ψN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xN)|2 dxn+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxN,  n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3)  The division by f is non-problematic even where f = 0, since it cancels with the corresponding  factors of f in ψN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We need to compute ρ(2)  Jas and bound ρ(3)  Jas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Before we start on this endeavour  we first recall some properties of the scattering function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1  The scattering function  The scattering function f0 is defined by the minimization problem in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1, see also  [LY01, Appendix A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' SY20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In particular f0 satisfies the corresponding Euler-Lagrange equation  −4x · ∇f0 − 2|x|2∆f0 + |x|2vf0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The minimizer f0 is radial and with a slight abuse of notation we sometimes write f0(|x|) =  f0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In radial coordinates the Euler-Lagrange equations reads  − ∂2  rf0 − 4  r∂rf0 + 1  2vf0 = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4)  where ∂r denotes the derivative in the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This is the same equation as for s-wave  scattering in 5 dimensions, see [LY01, Appendix A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, properties of this carry over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In  particular f0(x) = 1 −  a3  |x|3 for x outside the support of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Moreover  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 ([LY01, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The scattering function f0 satisfies  �  1 − a3  |x|3  �  + ≤ f0(x) ≤ 1  for all x and |∇f0(x)| ≤ 3a3  |x|4 for |x| > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We give a short proof here for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' From the radial Euler-Lagrange equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4) we have ∂r(r4∂rf0) = vr4f0/2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Denote by fhc =  �  1 − a3  |x|3  �  + the solution for a hard core potential of range a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then  r4∂rfhc =  �  3a3  r > a  0  r < a  In particular ∂r(r4∂rfhc) = 0 for r > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We thus see that ∂rf0 ≤ ∂rfhc = 3a3r−4 and f0 ≥ fhc  for r > a by integrating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Trivially f0 ≥ 0 = fhc for r ≤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' A hard core interaction of range R0 > 0,  vhc(x) =  �  +∞  |x| ≤ R0,  0  |x| > R0,  has f0(x) = fhc(x) =  �  1 − a3  |x|3  �  + and thus a0 = a = R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2  The “Fermi polyhedron”  We now introduce a polyhedral approximation to the Fermi ball BF = {k ∈ 2π  L Z3 : |k| ≤ kF}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We discuss why we need this in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The problem is that  ˆ  [0,L]3  1  L3  ������  �  k∈B(kF )∩ 2π  L Z3  eikx  ������  dx =  1  (2π)3  ˆ  [0,2π]3  ������  �  q∈B(cN1/3)∩Z3  eiqu  ������  du ∼ N1/3  for large N (see [GL19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Lif06] and references therein) is too big for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Note  that this behaviour is a consequence of taking the absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  In fact we have that  1  L3  ´ �  k∈B(kF )∩ 2π  L Z3 eikx dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  8  This type of quantity is referred to as the Lebesgue constant [GL19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Lif06] of some domain  Ω,  L(Ω) :=  1  (2π)3  ˆ  [0,2π]3  ������  �  q∈Ω∩Z3  eiqu  ������  du  These kinds of integrals appear in estimates in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For an overview of such  Lebesgue constants, see [GL19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Lif06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Of particular relevance for us is the fact that the  Lebesgue constants are much smaller for polyhedral domains than for balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Hence we introduce  the polyhedron P = P(N) as an approximation of the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Then the scaled version  PF = kFP ∩ 2π  L Z3 approximates the Fermi ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We will refer to PF as the Fermi polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  In [KL18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1] it is shown that for any fixed convex polyhedron P ′ of s vertices  L(RP ′) =  1  (2π)3  ˆ  [0,2π]3  ������  �  q∈RP ′∩Z3  eiqu  ������  du ≤ Cs(log R)3 + C(s)(log R)2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5)  for any R > 2, in particular for R ∼ N1/3, where C(s) is some unknown function of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We  will improve on this bound for the specific polyhedron P = P(N) to control the s-dependence  of the subleading (in R) terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' of C(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We first give an almost correct definition of the  polyhedron P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  “Definition” 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4 (Simple definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The polyhedron P is chosen to be the convex hull of  s = s(N) points κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κs on a sphere of radius 1+δ, where δ is chosen such that Vol(P) = 4π/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We moreover choose the set of points to have the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The points are evenly distributed, meaning that the distance d between any pair of points  satisfies d ≳ s−1/2, and that for any k on the sphere of radius 1 + δ the distance from k to  the closest point is ≲ s−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is, for some constants c, C > 0 we have d ≥ cs−1/2 and  infj |k − κj| ≤ Cs−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  P is invariant under any map (k1, k2, k3) �→ (±ka, ±kb, ±kc) for {a, b, c} = {1, 2, 3}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  reflection in or permutation of any of the axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The Fermi polyhedron is the rescaled version defined as PF := kFP ∩ 2π  L Z3, where L is chosen  large (depending on kF) such that kFL is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note that the symmetry constraint adds a restriction on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For instance, a  generic point away from any plane of symmetry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  k1, k2, k3 all different and non-zero)  has 48 images (including itself) when reflected by the maps (k1, k2, k3) �→ (±ka, ±kb, ±kc) for  {a, b, c} = {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For s points on a sphere of radius 1 + δ, the natural lengthscale is (1 + δ)s−1/2 ∼ s−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The  requirement that the points are evenly distributed then ensures that all pairs of close points  (for any reasonable definition of “close points”) have a pairwise distance of this order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For all purposes apart from the technical argument in Appendix B one may  take this as the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In particular, the convergence criterion of the cluster expansion  formulas of Gaudin, Gillespie and Ripka [GGR71], given in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4, holds also for this  simpler definition of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We provide this simpler definition to better give an intuition of the  construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We now give the actual definition of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We first give the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9 we  give a few comments and in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10 we give a short motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  9  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7 (Actual definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The polyhedron P with s corners and the “centre” z is  constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  First, choose a big number Q, the “size of the primes” satisfying  Q−1/4 ≤ Cs−1,  N4/3 ≪ Q ≤ CNC  in the limit N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Pick three large distinct primes Q1, Q2, Q3 with Qj ∼ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Place s evenly distributed points κR  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κR  s on the sphere of radius Q−3/4 and such that the  set of points {κR  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κR  s } is invariant under the symmetries (k1, k2, k3) �→ (±ka, ±kb, ±kc)  for {a, b, c} = {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Here, evenly distributed means that the distance between any pair of points is d ≳ s−1/2Q−3/4  and that for any k on the sphere of radius Q−3/4 the distance from k to the nearest point is  ≲ s−1/2Q−3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is, d ≥ cs−1/2Q−3/4 and infj  ��k − κR  j  �� ≤ Cs−1/2Q−3/4 for some constants  c, C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Find points κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κs of the form  κj =  � p1  j  Q1  , p2  j  Q2  , p3  j  Q3  �  ,  pµ  j ∈ Z,  µ = 1, 2, 3,  j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , s,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6)  such that  ��κj − κR  j  �� ≲ Q−1 for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , s and such that the set of points {κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κs} is  invariant under the symmetries (k1, k2, k3) �→ (±k1, ±k2, ±k3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Define ˜P as the convex hull of all the points κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is, ˜P = conv{κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κs}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Define P as σ ˜P, where σ is chosen such that Vol(P) = 4π/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We will refer to the scaled  points σκj = σ(p1  j/Q1, p2  j/Q2, p3  j/Q3) for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , s as corners of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Define P R = σ conv{κR  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κR  s } as the scaled convex hull of all the initial points κR  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κR  s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Define the centre as z = σ(1/Q1, 1/Q2, 1/Q3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The Fermi polyhedron is the rescaled version defined as PF := kFP ∩ 2π  L Z3, where L is chosen  large (depending on kF) such that kF L  2π is rational and large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We additionally define P R  F := kFP R ∩ 2π  L Z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We choose N := #PF, so that the Fermi polyhedron is filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The dependence  in N of, for instance, Q should therefore more precisely be given in terms of a dependence on  kFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note that N = ρL3 ∼ (kFL)3 and kF = (6π2ρ)1/3(1 + O(N−1/3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We will choose also s depending on N (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' on kFL) satisfying s → ∞ as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9 (Comments on and properties of the construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We collect here some prop-  erties of the Fermi polyhedron, some of which will only be needed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The points κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κs are evenly distributed on a thickened sphere of radius Q−3/4 – their  radial coordinates are |κj| = Q−3/4 + O(Q−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Indeed, the points κR  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κR  s are evenly  distributed and Q−1 ≪ s−1/2Q−3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For s points on a thickened sphere of radius Q−3/4, the  natural lengthscale between points is s−1/2Q−3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  10  There is some constraint on the number of points s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' A generic point κ ∈ S (with κ1, κ2, κ3  all different and non-zero) has 48 images, including itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The constraint on s is more or less  the same as for the simpler ‘‘Definition” 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  By choosing the points κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κs as in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6) we break the symmetries of permuting  the coordinates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (k1, k2, k3) �→ (ka, kb, kc) if (a, b, c) ̸= (1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' These symmetries are  however still almost satisfied, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We choose s, Q such that Q−1/4 ≪ s−1/2 in the limit of large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Hence, for N sufficiently  large, all the chosen points {κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κs} are extreme points of ˜P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' all corners are extreme  points of the polyhedron P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is, the name “corner” is well-chosen, and we do not have  any superfluous points in the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For any three points (xi, yi, zi) ∈ R3, i = 1, 2, 3 the plane through them is given by the  equation  \uf8eb  \uf8ed  (y2 − y1)(z3 − z1) − (y3 − y1)(z2 − z1)  (z2 − z1)(x3 − x1) − (z3 − z1)(x2 − x1)  (x2 − x1)(y3 − y1) − (x3 − x1)(y2 − y1)  \uf8f6  \uf8f8 ·  \uf8eb  \uf8ed  x  y  z  \uf8f6  \uf8f8 = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Hence, for three points K1, K2, K3 of the form Ki = (p1  i /Q1, p2  i /Q2, p3  i /Q3), pµ  i ∈ Z, i, µ =  1, 2, 3 the plane through them is given by  α1  Q2Q3  k1 +  α2  Q1Q3  k2 +  α3  Q1Q2  k3 = γ ∈ Q,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7)  where  α1 = (p2  2 − p2  1)(p3  3 − p3  1) − (p2  3 − p2  1)(p3  2 − p3  1) ∈ Z  and similarly for α2, α3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' From these formulas it is immediate that |αj| ≤ C√Q for j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For some planes we may have αj = 0 for some j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We claim that σ = Q3/4(1 + O(s−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In particular, that any point on the boundary ∂P has  radial coordinate 1 + O(s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' To see this, note that Q3/4 ˜P is a polyhedron whose corners are  evenly spaced and have radial coordinates r with r = 1 + O(Q−1/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, by scaling Q3/4 ˜P  by 1 − CQ−1/4 we get that (1 − CQ−1/4)Q3/4 ˜P ⊂ B1(0) so that this has volume ≤ 4π  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' It  follows that σ ≥ Q3/4(1 − CQ−1/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' On the other hand, scaling Q3/4 ˜P by 1 + Cs−1 we have  that (1 + Cs−1)Q3/4 ˜P ⊃ B1(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Indeed, since the distance from any point k on the sphere of  radius 1 to any corner of Q3/4 ˜P is ≲ s−1/2, and the sphere is locally quadratic, the smallest  radial coordinate r of a point on the boundary ∂(Q3/4 ˜P) is r ≥ 1 − Cs−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' It follows that  σ ≤ (1 + Cs−1)Q3/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Since Q−1/4 ≤ Cs−1 this shows the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Note moreover that σ is irrational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Indeed, the volume of a polyhedron with rational corners  is rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (This is easily seen for tetrahedra, of which any polyhedron is an essentially  disjoint union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Thus σ3 = πr for a rational r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Hence the equations of the planes defined by  corners of P (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' scaled points) are of the form Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7) with an irrational constant σγ  on the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Indeed, the corners of P (and the central point z) are all scaled by  σ compared to points of the form (p1/Q1, p2/Q2, p3/Q3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The equation of the plane through  three scaled points only differ by scaling the constant term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Since σ is irrational, and the  constant term was rational for the unscaled points, this shows the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We now construct a triangulation of ∂P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For all (2-dimensional) triangular faces of P simply  consider these as part of the triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is, we construct edges between any pair of  11  the three corners of such a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Some of the (2-dimensional) faces of P may be polygons  of more than 3 sides (1-dimensional faces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Construct edges between all pairs of corners  sharing a side (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' a 1-dimensional face) and choose one corner and construct edges from  this corner to all other corners of the polygon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Doing this constructs a triangulation of ∂P and we will refer to all pairs of corners with  an edge between them as close or neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Since the points {κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κs} are evenly dis-  tributed, that the distance between any pair of close corners is d ∼ s−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Additionally, one may note that the corners of P have ≤ C many neighbours since the points  are evenly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The reason we need  LkF  2π  rational will only become apparent in Appendix B and will be  explained there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10 (Motivation of construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The purpose of the construction is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Firstly  we avoid a casework argument as in the proof of [KL18, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5] of whether the coefficients  of the planes are rational or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The argument in Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7 is heavily inspired by  [KL18, Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9], where such casework is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Secondly we have good control over  how many (and which) lattice points (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' points in 2π  L Z3) can lie on each plane (or, rather, a  closely related plane, see Appendix B for the details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  These are technical details only needed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We reiterate, that apart from the  arguments in Appendix B, the reader may have the simpler ‘‘Definition” 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4 in mind instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  As mentioned in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9 the Fermi polyhedron is almost symmetric under permutation of  the axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This is formalized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For µ ̸= ν let Fµν be the map that permutes kµ and kν (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' F12(k1, k2, k3) =  (k2, k1, k3), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then for any function t ≥ 0 we have  �  k∈ 2π  L Z3  ��χ(k∈PF ) − χ(k∈Fµν(PF ))  �� t(k) ≲ Q−1/4N sup  |k|∼kF  t(k) ≲ N2/3 sup  |k|∼kF  t(k),  where Q is as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7 and χ denotes the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note that  �  k∈ 2π  L Z3  ��χ(k∈PF ) − χ(k∈Fµν(PF ))  �� t(k)  ≤  �  k∈ 2π  L Z3  ���χ(k∈PF ) − χ(k∈P R  F )  ��� t(k) +  �  k∈ 2π  L Z3  ���χ(k∈Fµν(PF )) − χ(k∈Fµν(P R  F ))  ��� t(k)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8)  since P R  F is invariant under permutation of the axes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Fµν(P R  F ) = P R  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The points {κj}j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',s  only differ from {κR  j }j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',s by at most ∼ Q−1 thus the points {σκj}j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',s (the corners of P)  only differ from the points {σκR  j }j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',s by ∼ Q−1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Hence, the support of χ(k∈PF ) − χ(k∈P R  F ) is  contained in a shell of width ∼ kFQ−1/4 around the surface ∂(kFP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is,  supp  �  χ(k∈PF ) − χ(k∈P R  F )  �  ⊂  �  k ∈ 2π  L Z3 : dist(k, ∂(kFP)) ≲ kFQ−1/4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The surface ∂(kF P) has area ∼ k2  F so  Vol  ��  k ∈ R3 : dist(k, ∂(kFP)) ≲ kFQ−1/4��  ≲ k3  FQ−1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  12  The spacing between the k’s in 2π  L Z3 is ∼ L−1 and any k with dist(k, ∂(kFP)) ≲ kFQ−1/4 has  |k| ∼ kF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus  �  k∈PF  ���χ(k∈PF ) − χ(k∈P R  F )  ��� t(k) ≲ L3k3  FQ−1/4 sup  |k|∼kF  t(k) ∼ Q−1/4N sup  |k|∼kF  t(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The same argument applies to the second summand in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We conclude the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We now improve on Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5) for our polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The Lebesgue constant of the Fermi polyhedron satisfies  ˆ  Λ  1  L3  �����  �  k∈PF  eikx  ����� dx =  1  (2π)3  ˆ  [0,2π]3  �������  �  q∈  � LkF  2π P  �  ∩Z3  eiqu  �������  du ≤ Cs(log N)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The proof is (almost) the same as given in [KL18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We need to be a bit more  careful in the decomposition into tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Define R = LkF  2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We decompose RP into tetrahedra using the “central” point z from  the construction of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We triangulate the surface of RP as in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For each triangle  in the triangulation add the point Rz to form a tetrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note that Rz /∈ Z3 since |Rz| ≤  CRQ−1/4 ≪ 1 and z ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This gives m = O(s) many (closed) tetrahedra {Tj} such that  RP = � Tj and that Tj ∩ Tj′ is a tetrahedron of lower dimension (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' the central point Rz,  a line segment or a triangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then, as in [KL18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1] by the inclusion–exclusion  principle we have  L(RP) =  1  (2π)3  ˆ ������  �  j  �  q∈Tj∩Z3  eiqu −  �  j<j′  �  q∈Tj∩Tj′∩Z3  eiqu + · · ·  ������  du  ≤  m  �  ℓ=1  �  j1<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='<jℓ  L(Tj1 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' ∩ Tjℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  In [KL18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1] it is shown that for a d-dimensional tetrahedron T with T ⊂ [0, n1] ×  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' × [0, nd] we have L(T) ≤ C(d) �d  i=1 log(ni + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' All the tetrahedra in our construction are  d-dimensional for d ≤ 3 and contained in boxes [0, CR]d (after translations by lattice vectors  κ ∈ Z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Hence for all tuples Tj1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , Tjℓ we have  L(Tj1 ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' ∩ Tjℓ) ≤ C(log R)d ≤ C(log N)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We need to count how many summands we have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The 3-dimensional tetrahedra each appear just  once, and there are m = O(s) many of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The 2-dimensional tetrahedra (triangles) appear  just once, namely in the term L(Tj ∩ Tj′) where the triangle is the intersection Tj ∩ Tj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Hence  there are O(s) many such terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The 1-dimensional tetrahedra (line segments) may appear  more times, with 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , C many Tj’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Indeed an edge may be shared by more tetrahedra,  but only a bounded number of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (This follows from the points being well-distributed,  so each corner of P has a bounded number of neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  Since there is also only O(s)  many 1-dimensional line segments this gives also just a contribution O(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The central point  appears many times, but all appearances contribute 0, since Rz /∈ Z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We conclude that  L(RP) ≤ Cs(log N)3 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  13  By replacing BF with PF we make an error in the kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (The Fermi ball BF is the  set of momenta of the Slater determinant with lowest kinetic energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') We now bound the error  made with this approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is, we consider  �  k∈PF  |k|2 −  �  k∈BF  |k|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Note that there might not be the same number of summands in both sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' To compute this  difference we interpret the sums as Riemann-sums and replace them with the corresponding  integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' It is a simple exercise to show that the error made in this replacement is Ck2  FN2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  That is,  �  k∈PF  |k|2 −  �  k∈BF  |k|2 =  L3  (2π)3  �ˆ  kF P  |k|2 dk −  ˆ  B(kF )  |k|2 dk  �  + O  �  k2  FN2/3�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The integrals can be computed in spherical coordinates,  ˆ  kF P  |k|2 dk −  ˆ  B(kF )  |k|2 dk =  ˆ  S2  �ˆ kF R(ω)  0  r4 dr −  ˆ kF  0  r4 dr  �  dω  For kFP the radial limit is kFR(ω) = kF(1 + ε(ω)), where ε(ω) = O(s−1) uniformly in ω by the  argument in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Expanding the powers of R we thus get  ˆ  kF P  |k|2 dk −  ˆ  B(kF )  |k|2 dk = k5  F  ˆ  S2  �  ε(ω) + O(s−2)  �  dω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  By construction, P has volume 4π/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is, kFP and B(kF) have the same volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This  means that  0 =  ˆ  S2  �ˆ kF R(ω)  0  r2 dr −  ˆ kF  0  r2 dr  �  dω = k3  F  ˆ  S2  �  ε(ω) + O(s−2)  �  dω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We thus get that  �  k∈PF  |k|2 −  �  k∈BF  |k|2 = O(k2  FNs−2) + O  �  k2  FN2/3�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We conclude the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The kinetic energy of the (Slater determinant with momenta in the) Fermi  polyhedron satisfies  �  k∈PF  |k|2 =  �  k∈BF  |k|2 �  1 + O(N−1/3) + O(s−2)  �  = 3  5(6π)2/3ρ2/3N  �  1 + O(N−1/3) + O(s−2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The computation above gives the first equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The second follows by noting that  �  k∈BF |k|2 is a Riemann sum for  L3  (2π)3  ˆ  |k|≤kF  |k|2 dk =  4π  5(2π)3k5  FL3 = 3  5(6π2)2/3ρ2/3N(1 + O(N−1/3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  14  Completely analogously one can show that  �  k∈PF  |k|4 = 18π2  7  (6π)1/3ρ4/3N  �  1 + O(N−1/3) + O(s−2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9)  We need this formula for Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='14 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Additionally we need a formula for �  k∈PF |k1|4,  where k1 refers to the first coordinate of k = (k1, k2, k3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Here we have  �  k∈PF  |k1|4 = 18π2  35 (6π)1/3ρ4/3N  �  1 + O(N−1/3) + O(s−1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10)  To see this we compare it to �  k∈BF |k1|4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The only difference from above is when doing the  spherical integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We have  �  k∈PF  |k1|4 −  �  k∈BF  |k1|4  =  L3  (2π)3  ˆ 2π  0  dθ  ˆ π  0  dφ cos(φ)4  �ˆ kF (1+ε(φ,θ))  0  r6 dr −  ˆ kF  0  r6 dr  �  + O(k4  FN2/3)  = O(k4  FNs−1) + O(k4  FN2/3),  since we can’t use the volume constraint that  ´  S2 ε(ω) dω = O(s−2) but only that ε(ω) = O(s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Thus we only get an error of (relative) size s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The sum over BF may be readily computed  by computing the corresponding integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3  Reduced densities of the Slater determinant  We now consider the 2-particle reduced density of the (normalized) Slater determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We  have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The 2-particle reduced density of the (normalized) Slater determinant  1  √  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='DN  satisfies  ρ(2)(x1, x2) = (6π2)2/3  5  ρ8/3|x1 − x2|2  �  1 − 3(6π2)2/3  35  ρ2/3|x1 − x2|2  + O(N−1/3) + O(s−2) + O(N−1/3ρ2/3|x1 − x2|2) + O(ρ4/3|x1 − x2|4)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This follows from a Taylor expansion akin to the argument in [ARS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The Slater determinant is in particular a quasi-free state, hence we get by Wick’s rule  that  ρ(2)(x1, x2) = ρ(1)(x1)ρ(1)(x2) − γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' x2)γ(1)  N (x2, x1),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11)  where γ(1)  N denotes the (kernel of the) reduced 1-particle density matrix of the Slater determi-  nant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We have  γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' x2) =  �  k∈PF  uk(x1)uk(x2) = 1  L3  �  k∈PF  eik(x1−x2),  ρ(1)(x1) = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  By translation invariance, γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' x2) is a function of x1 − x2 only, and we shall Taylor expand  γ(1)  N in x1 − x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By construction PF is reflection symmetric in the axes, see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This  15  means that all odd orders vanish and that all off-diagonal second order terms vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, by  defining x12 = (x1  12, x2  12, x3  12) = x1 − x2 and expanding all the exponentials we get  γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' x2) = 1  L3  �  k∈PF  �  1 − 1  2(k · (x1 − x2))2 + 1  24(k · (x1 − x2))4 + O(|k|6|x1 − x2|6)  �  = ρ −  1  2L3  �  k∈PF  �  |k1|2|x1  12|2 + |k2|2|x2  12|2 + |k3|2|x3  12|2�  +  1  24L3  � �  k∈PF  �  |k1|4|x1  12|4 + |k2|4|x2  12|4 + |k3|4|x3  12|4�  +  �  k∈PF  �  |k1|2|k2|2|x1  12|2|x2  12|2 + |k1|2|k3|2|x1  12|2|x3  12|2 + |k2|2|k3|2|x2  12|2|x3  12|2�  �  + O(ρ3|x1 − x2|6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11 we may write  �  k∈PF  |kµ|2 = 1  3  �  k∈PF  |k|2 + O  �  N2/3k2  F  �  and similar for the � |kµ|4 and � |kµ|2|kν|2-sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Using this the second order term is given  by  − 1  6L3  �  k∈PF  |k|2|x12|2 + O(ρ5/3|x12|2N−1/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Similarly by also rewriting everything in terms of |x12|4 and [|x1  12|4 + |x2  12|4 + |x3  12|4] the fourth  order term is given by  1  48L3  �� �  k∈PF  |k|4 − 3  �  k∈PF  |k1|4  �  |x12|4 +  �  5  �  k∈PF  |k1|4 −  �  k∈PF  |k|4  �  �  |x1  12|4 + |x2  12|4 + |x3  12|4�  �  + O(ρ7/3|x12|4N−1/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13 and Equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10) we get that  γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' x2) = ρ − (6π2)2/3  10  |x1 − x2|2 + 3π2(6π2)1/3  140  ρ7/3|x1 − x2|4 + O(ρ5/3N−1/3|x1 − x2|2)  + O(ρ5/3s−2|x1 − x2|2) + O(ρ7/3N−1/3|x1 − x2|4) + O(ρ7/3s−1|x1 − x2|4)  + O(ρ3|x1 − x2|6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Plugging this into Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11) we conclude the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Finally, we have the following bound on the 3-particle reduced density  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The 3-particle reduced density of the (normalized) Slater determinant  1  √  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='DN  satisfies  ρ(3)(x1, x2, x3) ≤ Cρ5|x1 − x2|2|x1 − x3|2|x2 − x3|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note that ρ(3) vanishes whenever any 2 of the 3 particles are incident and moreover that  ρ(3) is symmetric under exchange of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We may bound derivatives of ρ(3) as we did  for ρ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By Taylor’s theorem we conclude the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  16  3  Gaudin-Gillespie-Ripka-expansion  We now present the cluster expansion of Gaudin, Gillespie and Ripka [GGR71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The argument  given here is essentially the same as in [GGR71], only we give sufficient conditions for the  formulas [GGR71, Equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='19), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4)], given in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4, to hold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=', for  absolute convergence of the expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Recall the definition of the trial state ψN in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We calculate the the normal-  ization constant CN and the reduced densities ρ(1)  Jas, ρ(2)  Jas and ρ(3)  Jas defined in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3)  We remark that the computation given in the following is not just valid for the function f  and (square of a) Slater determinant |DN|2 we choose, but these can be replaced by a more  general function and determinant of a more general matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We comment on this further in  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1  Calculation of the normalization constant  First we give the calculation of CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Rewriting the f’s in terms of g = f 2 − 1 we have  CN =  ˙ �  i<j  f 2  ij|DN|2 dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxN =  ˙ �  i<j  (1 + gij)|DN|2 dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxN  We factor out the gij and group terms with the same number of values xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (For instance g12g23  and g45g46g56 both have 3 values xi appearing, the values x1, x2, x3 and x4, x5, x6, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  To state the result we define Gp as the set of all graphs on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , p} such that each vertex has  degree at least 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' is incident to at least one edge) and define  Wp(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xp) :=  �  G∈Gp  �  e∈G  ge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (Note that for p = 0, 1 we have Gp = ∅ and so Wp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') By the symmetry of permuting the  coordinates we have  CN =  ˙ �  1 + N(N − 1)  2  W2(x1, x2) + N(N − 1)(N − 2)  3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  W3(x1, x2, x3) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  × |DN|2 dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxN  = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  1 +  N  �  p=2  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ˙  Wp(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xp)∆p dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxp  �  ,  where we introduced (following the notation of [GGR71])  ∆p := ρ(p) = N(N − 1) · · · (N − p + 1)  ˙  1  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='|DN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xN)|2 dxp+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  A simple calculation using the Wick rule shows that  ∆p = det  �  γ(1)  N (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj)  �  1≤i,j≤p = det  � �  k∈PF  uk(xi)uk(xj)  �  1≤i,j≤p  = det[S∗  pSp],  where Sp is the PF × p “Slater”-matrix with entries uk(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This has rank min{N, p} = p and  so by taking this determinant as the definition of ∆p for p > N we have ∆p = 0 for p > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  17  Thus we may extend the summation to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We now expand out the determinant and the Wp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  That is  CN  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' = 1 +  ∞  �  p=2  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  G∈Gp  π∈Sp  (−1)π  ˙ �  e∈G  ge  p  �  j=1  γ(1)  N (xj, xπ(j)) dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxp,  where Sp denotes the symmetric group on p elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We will consider π and G together as  a diagram (π, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We give a slightly more general definition for what a diagram is, as we will  need such for the calculation of the reduced densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Define the set Gq  p as the set of all graphs on q “external” vertices {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , q}  and p “internal” vertices {q + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , q + p} such that all internal vertices have degree at least  1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' each internal vertex has at least one incident edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The external edges are allowed to  have degree zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' have no incident edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For q = 0 we recover G0  p = Gp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  A diagram (π, G) on q “external” and p “internal” vertices is a pair of a permutation  π ∈ Sq+p (viewed as a directed graph on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , q + p}) and a graph G ∈ Gq  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We denote the  set of all diagrams on q “external” vertices and p “internal” vertices by Dq  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We will sometimes refer to edges in G as g-edges, directed edges in π as γ(1)  N -edges and the  graph G as a g-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The value of a diagram (π, G) ∈ Dq  p is the function  Γq  π,G(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xq) := (−1)π  ˙ �  e∈G  ge  q+p  �  j=1  γ(1)  N (xj, xπ(j)) dxq+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxq+p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For q = 0 we write Γπ,G = Γ0  π,G and Dp = D0  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  A diagram (π, G) is said to be linked if the graph ˜G with edges the union of edges in G  and directed edges in π is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The set of all linked diagrams on q “external” and p  “internal” vertices is denoted Lq  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For q = 0 we write Lp = L0  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  By the translation invariance we have that Γ1  π,G is a constant for any diagram (π, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  1  2  3  4  5  6  7  8  9  10  11  (π1, G1)  (π2, G2)  (π3, G3)  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1: A diagram (π, G) decomposed into linked components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The dashed lines  denote g-edges and the arrows (i, j) denote that π(i) = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  In terms of diagrams we thus have  CN  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' = 1 +  ∞  �  p=2  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈Dp  Γπ,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  If (π, G) is not linked we may decompose it into its linked components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Here the integration  factorizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We split the sum according to the number of linked components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Each linked  18  component has at least 2 vertices, since each vertex must be connected to another vertex with  an edge in the corresponding graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We get  CN  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' = 1 +  ∞  �  p=2  ∞  �  k=1  ����  # lnk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' cps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  p1≥2  · ·  �  pk≥2  �  ��  �  sizes linked cps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  χ(� pℓ=p)  �  (π1,G1)∈Lp1  · ·  �  (πk,Gk)∈Lpk  �  ��  �  linked components  Γπ1,G1  p1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  · · Γπk,Gk  pk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1)  Here χ is the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The factor 1/(k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') comes from counting the possible labellings of  the k linked components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The factors 1/(p1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , 1/(pk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') come from counting how to distribute  the p vertices {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , p} between the linked components of prescribed sizes p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This  gives the factor  �  p  p1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',pk  �  =  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  p1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='.pk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=', which together with the factor 1/p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' already present gives the  claimed formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We want to pull the p-summation inside the p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , pk-summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This is allowed once we  check that �  p  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈Lp Γπ,G is absolutely convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' More precisely we need that the p-sum  is absolutely convergent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' �  p  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ���  �  (π,G)∈Lp Γπ,G  ��� < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This is the content of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We conclude that if the assumptions of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 are satisfied then  CN  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' = 1 +  ∞  �  k=1  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  p1≥2  · ·  �  pk≥2  �  (π1,G1)∈Lp1  · ·  �  (πk,Gk)∈Lpk  Γπ1,G1  p1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  · · Γπk,Gk  pk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  = 1 +  ∞  �  k=1  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  \uf8eb  \uf8ed  ∞  �  p=2  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈Lp  Γπ,G  \uf8f6  \uf8f8  k  = exp  \uf8eb  \uf8ed  ∞  �  p=2  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈Lp  Γπ,G  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2  Calculation of the 1-particle reduced density  We consider now the 1-particle reduced density of the Jastrow trial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We have by the  translation invariance that ρ(1)  Jas = ρ(1) = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We nonetheless compute it here, as we need the  formula in terms of (linked) diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We have similarly as before  ρ(1)  Jas(x1) = N  CN  ˙  �  2≤i≤N  f 2  1i  �  2≤i<j≤N  f 2  ij|DN|2 dx2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxN  = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  CN  �  ∆1 +  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ˙  X1  p∆p+1 dx2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxp+1  �  ,  where  X1  p =  �  G∈G1p  �  e∈G  ge,  with G1  p as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Again this is what one gets by just expanding all the products in  the first line and grouping them in terms of how many xi’s appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The sum is again extended  to ∞, since ∆p = 0 for p > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We again expand out the determinant and the X1  p’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For each summand π ∈ Sp+1 and  G ∈ G1  p we again think of them together as a diagram (π, G) ∈ D1  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The formula for ρ(1)  Jas in  terms of diagrams is  ρ(1)  Jas = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  CN  ∞  �  p=0  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈D1p  Γ1  π,G = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  CN  \uf8ee  \uf8f0ρ(1) +  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈D1p  Γ1  π,G  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  19  As for the normalization we write out the diagrams in terms of their linked components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' There  is a distinguished linked component, namely the one containing the vertex {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We will write  its size as p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' It is convenient to take “size” to mean number of internal vertices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' p∗ = 0  if {1} is not connected to any other vertex by either an edge in G or an edge in π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Similarly  “number of linked components” means disregarding the distinguished one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Analogously to the computation in Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1) we thus get for any p ≥ 0  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈D1p  Γ1  π,G =  �  (π∗,G∗)∈L1p  Γ1  π∗,G∗  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  +  \uf8ee  \uf8f0  ∞  �  k=1  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  p∗≥0  �  p1≥2  · ·  �  pk≥2  χ(�  ℓ∈{∗,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k} pℓ=p)  �  (π∗,G∗)∈L1p∗  Γ1  π∗,G∗  p∗!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ×  �  (π1,G1)∈Lp1  · ·  �  (πk,Gk)∈Lpk  Γπ1,G1  p1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  · · Γπk,Gk  pk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  \uf8f9  \uf8fb ,  where the superscript 1 refers to the slightly modified structure as described in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1,  where there may be no g-edges connecting to {1}, and there is no integration over x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note  that (π1, G1) ∈ Lp1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , (πk, Gk) ∈ Lpk only deal with internal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Again here we take the sum over p’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We are allowed to permute the p-sum inside of the  p∗- and p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , pk-sums if the sums over linked diagrams are absolutely summable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is, if  �  p≥0  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ������  �  (π,G)∈L1p  Γ1  π,G  ������  < ∞,  �  p≥2  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ������  �  (π,G)∈Lp  Γπ,G  ������  < ∞,  then we have, as for the normalization in Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2), that  ρ(1)  Jas = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  CN  ∞  �  p=0  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈D1p  Γ1  π,G  = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  CN  \uf8ee  \uf8f0�  p∗≥0  �  (π∗,G∗)∈Lp∗  Γ1  π∗,G∗  p∗!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  \uf8f9  \uf8fb ×  \uf8ee  \uf8ef\uf8f0  ∞  �  k=0  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  \uf8eb  \uf8ed�  p≥2  �  (π,G)∈Lp  Γπ,G  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  \uf8f6  \uf8f8  k\uf8f9  \uf8fa\uf8fb  =  �  p≥0  �  (π,G)∈L1p  Γ1  π,G  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  = ρ(1) +  �  p≥1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L1p  Γ1  π,G,  where we used Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2) and that the p = 0 term just gives the 1-particle density of the  Slater determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, by translation invariance, we have  ρ = ρ(1) = ρ(1)  Jas =  �  p≥0  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L1p  Γ1  π,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3  Calculation of the 2-particle reduced density  Let us now compute the 2-particle reduced density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' As before by expanding all the f 2 = 1 + g  factors apart from the factor f12 we get  ρ(2)  Jas = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  CN  f 2  12  ∞  �  p=0  ˆ  X2  p∆p+2 dx3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxp+2,  X2  p =  �  G∈G2p  �  e∈G  ge,  20  where G2  p is as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We again decompose the diagrams into linked components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' However, we need to distinguish  between the cases where {1} and {2} are both in the same component or in two different  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The computation is analogous to the computation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We get  ��(2)  Jas(x1, x2) = f 2  12  \uf8eb  \uf8ec  \uf8ec  \uf8ed  �  p1,p2≥0  �  (π1,G1)∈L1  p1  (π2,G2)∈L1  p2  Γ1  π1,G1(x1)Γ1  π2,G2(x2)  p1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='p2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  ��  �  {1} and {2} in different linked components  +  �  p12≥0  �  (π12,G12)∈L2p12  Γ2  π12,G12(x1, x2)  p12!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  ��  �  {1} and {2} in same linked component  \uf8f6  \uf8f7  \uf8f7  \uf8f8  = f 2  12  \uf8eb  \uf8edρ(1)  Jas(x1)ρ(1)  Jas(x2) +  �  p12≥0  1  p12!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π12,G12)∈L2p12  Γ2  π12,G12(x1, x2)  \uf8f6  \uf8f8  after pulling in the sum over p ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The p12 = 0 term together with the term ρ(1)  Jas(x1)ρ(1)  Jas(x2) =  ρ(1)(x1)ρ(1)(x2) = ρ2 gives ρ(2) by Wick’s rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The condition of absolute convergence is  �  p≥2  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ������  �  (π,G)∈Lp  Γπ,G  ������  < ∞,  �  p≥0  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ������  �  (π,G)∈L1p  Γ1  π,G  ������  < ∞,  �  p≥0  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ������  �  (π,G)∈L2p  Γ2  π,G  ������  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4  Calculation of the 3-particle reduced density  The calculation of the 3-particle reduced density follows along the same arguments as for the  2-particle reduced density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We introduce the relevant diagrams and decompose these according  to their linked components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' As for the 2-particle reduced density we distinguish between the  cases according to whether the external vertices {1, 2, 3} are in the same or different linked  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' They are either in 1, 2 or 3 different components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, schematically  ρ(3)  Jas = f 2  12f 2  13f 2  23  �  �  all in different  Γ1Γ1Γ1 +  � �  2 in one  Γ1(x1)Γ2(x2, x3) + permutations  �  +  �  all in same  Γ3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Any case where one external vertex is in its own linked component, the contribution for such a  linked component is ρ(1)  Jas = ρ(1) = ρ (assuming absolute convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus,  ρ(3)  Jas(x1, x2, x3) = f 2  12f 2  13f 2  23  �  ρ3 + ρ  �  p≥0  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2p  �  Γ2  π,G(x1, x2) + Γ2  π,G(x1, x3) + Γ2  π,G(x2, x3)  �  +  �  p≥0  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L3p  Γ3  π,G(x1, x2, x3)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  All the p = 0-terms together give ρ(3) by Wick’s rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The condition for absolute convergence  is that for any q ≤ 3 we have �  p≥0  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ���  �  (π,G)∈Lq  p Γq  π,G  ��� < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5  Summarising the results  For the absolute convergence we have  21  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' There exists a constant c > 0 such that if sa3ρ log(b/a)(log N)3 < c, then  �  p≥0  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ������  �  (π,G)∈Lq  p  Γq  π,G  ������  < ∞  for any 0 ≤ q ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' As mentioned in the beginning of the section, the calculation just given is still  valid if we replace f by some general function h ≥ 0 and replace |DN|2 by some more general  determinant det[γ(xi − xj)]1≤i,j≤N, where γ(x − y) is the kernel of some rank N projection (for  instance the one-particle density matrix of a Slater determinant of N particles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The criterion  for absolute convergence reads  sup  x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',xn  �  1≤i<j≤n  h(xi − xj) ≤ Cn for all n ∈ N,  1  L3  �  k∈ 2π  L Z3  |ˆγ(k)|  ˆ  Λ  ��h2 − 1  �� dx  ˆ  Λ  |γ| dy < c  for some constants c, C > 0, where the first condition is the “stability condition” of the tree-  graph bound [PU09, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Uel18] and ˆγ(k) :=  ´  Λ γ(x)e−ikx dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We give the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' There exists a constant c > 0 such that if sa3ρ log(b/a)(log N)3 < c, then  CN  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' = exp  \uf8eb  \uf8ed  ∞  �  p=2  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈Lp  Γπ,G  \uf8f6  \uf8f8 ,  ρ(1)  Jas = ρ(1) +  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L1p  Γ1  π,G,  ρ(2)  Jas = f 2  12  \uf8ee  \uf8f0ρ(2) +  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2p  Γ2  π,G  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ρ(3)  Jas = f 2  12f 2  13f 2  23  �  ρ(3) + ρ  �  p≥1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2p  �  Γ2  π,G(x1, x2) + Γ2  π,G(x1, x3) + Γ2  π,G(x2, x3)  �  +  �  p≥1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L3p  Γ3  π,G  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3)  The first three formulas are the same as those of [GGR71, Equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='19), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9) and (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Our main contribution is to give a criterion for convergence, and hence for validity of the  formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The factor s(log N)3 results from the bound in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' If we had not  introduced the Fermi polyhedron, and instead used the Fermi ball, we would instead have a  factor N1/3 as mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is, the condition for absolute convergence would  be N1/3a3ρ log(b/a) < c for some constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  In either case, the N-dependence prevents us from taking a thermodynamic limit directly,  and we instead use a box method of gluing together multiple smaller boxes, where we may put  some finite number of particles in each box, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the case of using a Slater  22  determinant with momenta in the Fermi ball, there is no way to choose the number of particles  in each smaller box so that both the absolute convergence holds (N1/3a3ρ log(b/a) < c), and  the finite-size error made in the kinetic energy (∼ N2/3ρ2/3, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13) is smaller than  the claimed energy contribution from the interaction (∼ Na3ρ5/3, see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For this  reason we need the polyhedron of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The formulas for ρ(2)  Jas and ρ(3)  Jas only hold for periodic boundary conditions, since  in this case ρ(1)  Jas = ρ(1) = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For different boundary conditions, one has to take into account  that this equality is not valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In general one has for ρ(2)  Jas that  ρ(2)  Jas = f 2  12  \uf8ee  \uf8f0ρ(2) +  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2p  Γ2  π,G +  �  ρ(1)  Jas(x1)ρ(1)  Jas(x2) − ρ(1)(x1)ρ(1)(x2)  �  \uf8f9  \uf8fb  = f 2  12  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0ρ(2) +  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2p  Γ2  π,G +  �  p1,p2≥0  p1+p2≥1  1  p1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='p2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π1,G1)∈L1  p1  (π2,G2)∈L1  p2  Γ1  π1,G1(x1)Γ1  π2,G2(x2)  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  One of the reasons we work with periodic boundary conditions is that by doing so, we don’t  have the complication of dealing with the additional term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By following the same procedure as in the previous sections, one can equally  well get formulas for the higher order reduced particle densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Similarly one can extend the  absolute convergence, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2, to any q, only one may have to change the constant c > 0  to depend on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1  Absolute convergence  We now prove Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' that the appropriate sums are absolutely convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We consider the four sums �  p≥0  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ���  �  (π,G)∈Lq  p Γq  π,G  ���, q = 0, 1, 2, 3 one by  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1  Absolute convergence of the Γ-sum  Consider first  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈Lp Γπ,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Split the sum according to the number of connected compo-  nents of G, labelled as (G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , Gk) of sizes n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We call these clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (Note that  “connected” only refers to the graph G, and is independent of the permutation π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Name the  vertices in G1 as {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , n1}, in G2 as {n1 +1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , n1 +n2} and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then we have (for p ≥ 2)  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈Lp  Γπ,G =  ∞  �  k=1  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk≥2  1  n1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' · · ·nk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='χ(� nℓ=p)  �  G1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',Gk  Gℓ∈Cnℓ  �  π∈Sp  (−1)πχ((π,∪Gℓ)∈Lp)  ×  ˙  k  �  ℓ=1  �  e∈Gℓ  ge  p  �  j=1  γ(1)  N (xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xπ(j)) dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxp,  where Cn denotes the set of connected graphs on n (labelled) vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The factorial factors are  similar to those of Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Indeed, the factor 1/(k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') comes from counting the possible  23  1  2  3  4  5  6  7  8  9  10  11  G1  G2  G3  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2: A linked diagram (π, G) decomposed into clusters G1, G2, G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Dashed  lines denote g-edges, and arrows (i, j) denote that π(i) = j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  labelling of the clusters, and the factors 1/(n1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , 1/(nk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') come from counting the number of  ways to distribute the p = � nℓ vertices into the clusters and using the factor 1/(p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') already  present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For the analysis we will need the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Let A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , Ak denote disjoint non-empty sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The truncated correlation  function is  ρ(A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',Ak)  t  :=  �  π∈S∪ℓAℓ  (−1)πχ((π,∪Gℓ) linked)  �  j∈∪ℓAℓ  γ(1)  N (xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xπ(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4)  for some choice of connected graphs Gℓ ∈ CAℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The definition does not depend on the choice of  graphs Gℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  If the underlying sets A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , Ak are clear we will simply denote the truncated correlation  by their sizes,  ρ(|A1|,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',|Ak|)  t  = ρ(A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',Ak)  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The truncated correlation functions are also sometimes referred to as connected correlation  functions [GMR21, Appendix D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We write the characteristic function in Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4) as χ((π,∪Gℓ) linked) for ease  of generalizability to the cases in Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3 where we will need the notion of trun-  cated correlations also for some of the vertices being external.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the truncated correlations  it doesn’t matter which (if any) vertices are external, only which vertices are in which clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Since 0 ≤ f ≤ 1 we have −1 ≤ g ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, by the tree-graph bound [PU09, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Uel18] we have  �����  �  G∈Cn  �  e∈G  ge  ����� ≤  �  T∈Tn  �  e∈T  |ge|,  where Tn is the set of all trees on n (labelled) vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus we get  ∞  �  p=2  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ������  �  (π,G)∈Lp  Γπ,G  ������  ≤  ∞  �  k=1  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk≥2  1  n1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' · · ·nk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ˙  �  T1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',Tk  Tℓ∈Tnℓ  k  �  ℓ=1  �  e∈Tℓ  |ge|  ���ρ(n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  t  ��� dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dx� nℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5)  24  Here the vertices in Tℓ are the same as in Gℓ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' T1 has vertices {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , n1}, T2 has vertices  {n1 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , n1 + n2} and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  In [GMR21, Equation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='53)] the following formula, known as the Brydges-Battle-Federbush  (BBF) formula, is shown for the truncated correlation functions  ρ(A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',Ak)  t  =  �  τ∈A(A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',Ak)  �  (i,j)∈τ  γ(1)  N (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj)  ˆ  dµτ(r) det N (r),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6)  where A(A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',Ak) is the set of all anchored trees on k clusters with vertices A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (If the sets  A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , Ak are clear we will write A(|A1|,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',|Ak|) = A(A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',Ak) as for the truncated correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  An anchored tree is a directed graph on all the ∪ℓAℓ vertices, such that each vertex has at  most one incoming and at most one outgoing edge (note that these are all γ(1)  N -edges, and that  the g-edges don’t matter for this construction) and such that upon identifying all vertices in  each cluster, the resulting graph is a (directed) tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The measure µτ is a probability measure  on {(rℓℓ′)1≤ℓ≤ℓ′≤k : 0 ≤ rℓℓ′ ≤ 1} = [0, 1]k(k−1)/2 and depends only on τ but not on the factors  γ(1)  N (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Finally, N is an I × J (square) matrix with entries Nij = rc(i)c(j)γ(1)  N (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj), where  c(i) is the (label of the) cluster containing the vertex {i} and rℓℓ′ := rℓ′ℓ if ℓ > ℓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Here  I =  �  i ∈ �k  ℓ=1 Aℓ : ∄j : (i, j) ∈ τ  �  ,  J =  �  j ∈ �k  ℓ=1 Aℓ : ∄i : (i, j) ∈ τ  �  ,  are the set of i’s (respectively j’s) not appearing as i’s (respectively j’s) in the anchored tree  τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  T1  T2  T3  T4  T5  T6  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3: An anchored tree τ (arrows) and trees T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , T6 (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  From [GMR21, Equation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='57)] it follows that |det N | ≤ ρ  � nℓ−(k−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' To see this, one has  to adapt the argument in [GMR21, Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2] slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We sketch the argument here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10 ([GMR21, Lemmas D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The matrix N (r) satisfies |det N (r)| ≤  ρ  � nℓ−(k−1) for all r ∈ [0, 1]k(k−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' First we bound ρ(p) = det[γ(1)  N (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj)]1≤i,j≤p following the strategy of [GMR21, Lemma  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This is done by writing (as in [GMR21, Equations (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8), (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9)])  γ(1)  N (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj) = ⟨αi|βj⟩ℓ2((2π/L)Z3) ,  where for k ∈ 2π  L Z3  αi(k) = L−3/2e−ikxiχ(k∈PF )  βj(k) = L−3/2e−ikxjχ(k∈PF ) = αj(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  25  By the the Gram-Hadamard inequality [GMR21, Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1] we have  ��ρ(p)�� =  ���det[γ(1)  N (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj)]1≤i,j≤p  ��� ≤  p  �  i=1  ∥αi∥ℓ2((2π/L)Z3) ∥βi∥ℓ2((2π/L)Z3) = ρp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  By modifying this argument exactly as described in the proof of [GMR21, Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6] and  noting that rℓℓ′ ≤ 1 one concludes the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We denote the functions as αj and βj (even though they denote the same  function) for ease of modifying the argument later in order to prove Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  In particular one concludes the bound  ���ρ(n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  t  ��� ≤ ρ  � nℓ−(k−1)  �  τ∈A(n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  �  (i,j)∈τ  ���γ(1)  N (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj)  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7)  Plugging this into Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5) above we get  ∞  �  p=2  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ������  �  (π,G)∈Lp  Γπ,G  ������  ≤  ∞  �  k=1  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk≥2  1  n1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' · · · nk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='ρ  � nℓ−(k−1) �  T1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',Tk  Tℓ∈Tnℓ  �  τ∈A(n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  ×  ˙  dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dx� nℓ  k  �  ℓ=1  �  e∈Tℓ  |ge|  �  (i,j)∈τ  ���γ(1)  N (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj)  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  To compute these integrals we note that by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2  ˆ  |g(x)| dx =  ˆ �  1 − f(x)2�  dx  ≤  4π  (1 − a3/b3)2  ˆ b  a  ��  1 − a3  b3  �2  −  �  1 − a3  r3  �2�  r2 dr ≤ Ca3 log b  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  That is, each factor of ge gives a contribution Ca3 log(b/a) after integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The γ(1)  N -factors  we can bound by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12 as  ˆ ���γ(1)  N (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' y)  ��� dy ≤ Cs(log N)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This takes care of all but one integration, which gives the volume factor L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We shall compute  the integrations in the following order:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Pick any leaf {j0} of the anchored tree τ lying in some cluster ℓ, meaning that there is  exactly one edge of τ incident in ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Consider {j0} as the root of Tℓ and pick any leaf {j} of Tℓ and integrate over xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Since  {j} is a leaf of Tℓ and {j0} is a leaf of τ we have that the only place xj appears in the  integrand is in some factor gij for {i} the unique vertex connected to {j} by a g-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Hence the xj-integral contributes  ´  |g| by the translation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remove {j} and its incident edge from Tℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Repeat for all vertices in the cluster until only {j0} remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (At this point the entirety  of Tℓ has been removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  26  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Integrate over xj0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Since {j0} is a leaf of τ the only place xj0 appears (in the remaining  integrand) is in the γ(1)  N -factor from τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, the xj0-integral gives a contribution  ´  |γ(1)  N |  by the translation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remove {j0} and its incident edge from τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Repeat steps (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') until all integrals have been computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The final integral gives the  volume factor L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Steps (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') compute all integrations in one cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Repeating this process we integrate  over the clusters one by one and thus compute all the integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note that each integration is  always over a coordinate associated to a leaf of the relevant graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This is a key point, since  then by translation invariance each integration contributes exactly  ´  |g| or  ´  |γ(1)  N | whichever is  appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In total we thus have the bound  ˙  dx1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dx� nℓ  k  �  ℓ=1  �  e∈Tℓ  |ge|  �  (i,j)∈τ  ���γ(1)  N (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj)  ��� ≤  �  Ca3 log(b/a)  �� nℓ−k �  Cs(log N)3�k−1 L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This bound is for each summand τ, T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , Tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By Cayley’s formula #Tn = nn−2 ≤ Cnn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=', and  by [GMR21, Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5] #A(n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk) ≤ k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4  � nℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, we get  ∞  �  p=2  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ������  �  (π,G)∈Lp  Γπ,G  ������  ≤ CN  ∞  �  k=1  �  Cs(log N)3�k−1  � ∞  �  n=2  (Ca3ρ log(b/a))n−1  �k  ≤ CNa3ρ log(b/a)  ∞  �  k=1  �  Csa3ρ log(b/a)(log N)3�k−1  ≤ CNa3ρ log(b/a) < ∞,  for sa3ρ log(b/a)(log N)3 sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This shows that �  p  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈Lp Γπ,G is absolutely  convergent under this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2  Absolute convergence of the Γ1-sum  Consider now  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L1p Γ1  π,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The argument is almost identical to the argument above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We  again split the sum according to the connected components of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Call these G∗, G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , Gk,  where G∗ is the distinguished connected component (cluster) containing the distinguished vertex  {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Exactly as for  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈Lp Γπ,G we have that (for p = 0 one has to interpret the empty  product of integrals as 1, so �  (π,G)∈L1  0 Γ1  π,G = ρ)  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L1p  Γ1  π,G(x1)  =  ∞  �  k=0  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  n∗≥0  �  n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk≥2  1  n∗!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='n1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' · · · nk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='χ(  �  ℓ{∗,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k} nℓ=p)  �  G1∈Cn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',Gk∈Cnk  G∗∈Cn∗+1  �  π∈Sp+1  (−1)π  × χ((π,∪ℓ∈{∗,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k}Gℓ)∈L1p)  ˙  �  ℓ∈{∗,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k}  �  e∈Gℓ  ge  p+1  �  j=1  γ(1)  N (xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xπ(j)) dx2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxp+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  27  Here for the k = 0 term one should think of the n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , nk-sums as being an empty product  before it is an empty sum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' it should give a factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is, the k = 0 term reads  �  G∗∈Cp+1  �  π∈Sp+1  (−1)π  ˙  �  e∈G∗  ge  p+1  �  j=1  γ(1)  N (xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xπ(j)) dx2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxp+1,  since (π, G∗) is trivially linked, since G∗ is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' From here on, we won’t write out the  k = 0 term separately to make the formulas more concise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' As before we use the tree-graph  inequality and the truncated correlation function (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9) to get  �  p≥0  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ������  �  (π,G)∈L1p  Γ1  π,G  ������  ≤  ∞  �  k=0  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  n∗≥0  �  n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk≥2  1  n∗!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='n1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' · · ·nk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  T1∈Tn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',Tk∈Tnk  T∗∈Tn∗+1  ×  ˙  �  ℓ∈{∗,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k}  �  e∈Tℓ  |ge|  ���ρ(n∗+1,n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  t  ��� dx2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dx�  ℓ∈{∗,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k} nℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  To bound this we use the same bound, Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7), on the truncated correlations as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  It reads  ���ρ(n∗+1,n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  t  ��� ≤ ρ  �  ℓ∈{∗,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k} nℓ+1−(k+1−1)  �  τ∈A(n∗+1,n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  �  (i,j)∈τ  ���γ(1)  N (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj)  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Computing the integrals is as before, with a few differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' During each repeat (apart from  the last one) of step (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') we pick not just any leaf j0 but a leaf j0 not in the cluster containing  {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (Since any tree has at least 2 leaves, this is always possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') For each of these repeats,  the argument is the same as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the last repeat of step (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') where only the cluster  containing {1} remains we follow step (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') with the slight change, that the root is chosen to be  {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (There are no γ(1)  N -factors left, so we are free to choose any vertex as the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') There is  then no step (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') since we do not integrate over x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This has the following effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' First, the last variable x1 is not integrated over, so there is  no volume factor L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' And second, there are k integrals  ´  |γ(1)  N | instead of k − 1 (since there are  k + 1 many clusters including the distinguished one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the bounds of the sum of all terms  we use that #A(n∗+1,n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk) ≤ (k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4  �  ℓ∈{∗,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k} nℓ+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, uniformly in x1  �  p≥0  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ������  �  (π,G)∈L1p  Γ1  π,G  ������  ≤ Cρ  � ∞  �  n∗=0  �  Ca3ρ log(b/a)  �n∗  � \uf8eb  \uf8ed  ∞  �  k=0  (k + 1)  �  Cs(log N)3�k  � ∞  �  n=2  (Ca3ρ log(b/a))n−1  �k\uf8f6  \uf8f8  ≤ Cρ < ∞,  for sa3ρ log(b/a)(log N)3 sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This shows that �  p  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L1p Γ1  π,G is absolutely  convergent under this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3  Absolute convergence of the Γ2-sum  For the third sum the argument is mostly analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  There are a few changes needed for  the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' First, one has to distinguish between the two cases of whether or not the two  28  distinguished vertices {1, 2} are in the same connected component (cluster) of the graph or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  One computes  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2p  Γ2  π,G = Σdifferent + Σsame,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8)  where  Σdifferent =  ∞  �  k=0  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  n∗,n∗∗≥0  �  n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk≥2  1  �  ℓ nℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='χ(� nℓ=p)  �  G1∈Cn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',Gk∈Cnk  G∗∈Cn∗+{1}  G∗∗∈Cn∗∗+{2}  ×  ˙ �  ℓ  �  e∈Gℓ  geρ(n∗+{1},n∗∗+{2},n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  t  dx3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxp+2,  Σsame =  ∞  �  k=0  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  n∗≥1  �  n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk≥2  1  �  ℓ nℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='χ(� nℓ=p)  �  G1∈Cn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',Gk∈Cnk  G∗∈Cn∗+{1,2}  (1,2)/∈G∗  ×  ˙ �  ℓ  �  e∈Gℓ  geρ(n∗+{1,2},n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  t  dx3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxp+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9)  Here �  ℓ and �  ℓ are over ℓ ∈ {∗, ∗∗, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=', k} or ℓ ∈ {∗, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , k}, whichever is appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  With a slight abuse of notation we write n∗ +{1} for the set of vertices in the cluster containing  the external vertex {1} (similarly for n∗∗ + {2}, n∗ + {1, 2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This set has exactly n∗ internal  vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For p = 0 one has to interpret the empty product of integrals as a factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The first part is the contribution where {1} and {2} are in distinct clusters (labelled ∗ and  ∗∗), the second part is the contribution from where they are in the same (labelled ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note  that in the second contribution we have n∗ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Indeed, {1} and {2} are connected, but not  by an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Hence they must be connected by a path of length ≥ 2, which necessarily goes  through at least one vertex {j}, j ̸= 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We treat the two cases separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In the case where the two distinguished vertices are  in different clusters we may readily apply both the tree-graph bound and the bound on the  truncated correlation Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The latter reads  ���ρ(n∗+{1},n∗∗+{2},n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  t  ��� ≤ ρ(�  ℓ∈{∗,∗∗,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k} +2)−(k+2−1)  �  τ∈A(n∗+{1},n∗∗+{2},n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  �  (i,j)∈τ  ���γ(1)  N (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj)  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The integration procedure is slightly modified compared to that of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In the anchored  tree there is a path between (the cluster containing) {1} and (the cluster containing) {2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For  the edge incident to (the cluster containing) {1} on this path, we bound |γ(1)  N | ≤ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This cuts  the anchored tree into two anchored trees τ1, τ2 such that (with a slight abuse of notation)  1 ∈ τ1 and 2 ∈ τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We may follow the integration procedure exactly as for the Γ1-sum for each  of the anchored trees τ1 and τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Recall the bound  #A(n∗+{1},n∗∗+{2},n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk) ≤ (k + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4  �  ℓ∈{∗,∗∗,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k} nℓ+2 ≤ C(k2 + 1)k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4  �  ℓ∈{∗,∗∗,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k} nℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We thus get for the contribution of all terms where the two distinguished vertices are in different  29  clusters (assuming that sa3ρ log(b/a)(log N)3 is sufficiently small)  |Σdifferent|  ≤ Cρ2  � ∞  �  n∗=0  �  Ca3ρ log(b/a)  �n∗  �2 \uf8eb  \uf8ed  ∞  �  k=0  (k2 + 1)  �  Cs(log N)3�k  � ∞  �  n=2  (Ca3ρ log(b/a))n−1  �k\uf8f6  \uf8f8  ≤ Cρ2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10)  Now we consider the case where {1} and {2} are in the same distinguished cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Here we  may readily apply the bound in Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7) on the truncated correlation but we need to  be a bit careful in applying the tree-graph bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Indeed, then the sum over graphs in the  cluster containing the two vertices is not �  G∗∈Cn∗+2, but instead �  G∗∈Cn∗+2,(1,2)/∈G∗, since in the  construction, no g-edges are allowed between {1} and {2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' To still apply the tree-graph bound,  we define  ˜ge :=  �  ge  e ̸= (1, 2)  0  e = (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Then −1 ≤ ˜ge ≤ 0 so we can apply the tree-graph bound with these edge-weights to get  ������  �  G∗∈Cn∗+2,(1,2)/∈G∗  �  e∈G∗  ge  ������  =  ������  �  G∗∈Cn∗+2  �  e∈G∗  ˜ge  ������  ≤  �  T∗∈Tn∗+2  �  e∈T∗  |˜ge| =  �  T∗∈Tn∗+2,(1,2)/∈T∗  �  e∈T∗  |ge|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We again have to modify the integrations slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The integrations over all clusters apart  from the distinguished one may be computed as for the Γ- and Γ1-sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the distinguished  cluster with {1} and {2} there is some path of g-edges connecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Pick the unique edge  on this path incident with {1} and bound |g| ≤ 1 for this factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This splits the tree T∗ into  two trees T 1  ∗ and T 2  ∗ with 1 ∈ T 1  ∗ and 2 ∈ T 2  ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We may compute the integrations over all the  variables with index in the distinguished cluster exactly as for the Γ1-sum for each tree T 1  ∗ and  T 2  ∗ separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' One gets for the contribution (assuming that sa3ρ log(b/a)(log N)3 is sufficiently  small)  |Σsame|  ≤ Cρ2  � ∞  �  n∗=1  �  Ca3ρ log(b/a)  �n∗  � \uf8eb  \uf8ed  ∞  �  k=0  (k + 1)  �  Cs(log N)3�k  � ∞  �  n=2  (Ca3ρ log(b/a))n−1  �k\uf8f6  \uf8f8  ≤ Ca3ρ3 log(b/a) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11)  We conclude that  �  p≥0  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ������  �  (π,G)∈L2p  Γ2  π,G  ������  ≤ Cρ2 < ∞,  uniformly in x1, x2 for sufficiently small sa3ρ log(b/a)(log N)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4  Absolute convergence of the Γ3-sum  The argument for the last sum is completely analogous to the argument for the Γ2-sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We have  to distinguish between different cases of the clusters containing the external vertices {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Either there is one cluster containing all of them, one cluster containing two of them and one  30  cluster containing the last vertex, or they are all in distinct clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' One then deals with the  different cases exactly as we did for the Γ2-sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We skip the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This concludes the proof  of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  4  Energy of the trial state  In this section we bound the energy of the trial state ψN defined in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Recall  Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4 we have (for sa3ρ log(b/a)(log N)3 sufficiently small)  ρ(2)  Jas(x1, x2) = f(x1 − x2)2  \uf8eb  \uf8edρ(2)(x1, x2) +  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2p  Γ2  π,G  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We can expand ρ(2) in x1 − x2 using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The second term is an error term we have to  control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Additionally, also the three-body term is an error we have to control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We claim that  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' There exist constants c, C > 0 such that if sa3ρ log(b/a)(log N)3 < c and N =  #PF > C, then  ������  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2p  Γ2  π,G  ������  ≤ Ca6ρ4(log(b/a))2�  s3a6ρ2(log b/a)2(log N)9 + 1  �  + Ca3ρ3+2/3|x1 − x2|2�  s5a12ρ4(log(b/a))5(log N)16 + b4ρ4/3 + log(b/a)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' There exists a constant c > 0 such that if sa3ρ log(b/a)(log N)3 < c, then  ρ(3)  Jas = f 2  12f 2  13f 2  23  �  ρ(3) + O  �  a3ρ4 log(b/a)  �  s3a6ρ2(log(b/a))2(log N)9 + 1  ���  where the error is uniform in x1, x2, x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We give the proof of these lemmas in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the three-body term, we  additionally have the bound ρ(3) ≤ Cρ5|x1 − x2|2|x1 − x3|2|x2 − x3|2 by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Com-  bining now Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='14, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4 and Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2) we thus get (for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='sa3ρ log(b/a)(log N)3 sufficiently small and N sufficiently large)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='⟨ψN|HN|ψN⟩ = 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5(6π)2/3ρ2/3N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 + O(N−1/3) + O(s−2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='+ L3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='ˆ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='dx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='|∇f(x)|2 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2v(x)f(x)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='×  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='(6π2)2/3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='ρ8/3|x|2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 − 3(6π2)2/3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='35  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='ρ2/3|x|2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='+ O(N−1/3) + O(s−2) + O(N−1/3ρ2/3|x|2) + O(ρ4/3|x|4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='+ O  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='a6ρ4(log(b/a))2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='s3a6ρ2(log b/a)2(log N)9 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='+ O  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='a3ρ3+2/3|x|2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='s5a12ρ4(log(b/a))5(log N)16 + b4ρ4/3 + log(b/a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='�� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='˚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='dx1 dx2 dx3f12∇f12f23∇f23f 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='O(ρ5|x1 − x2|2|x1 − x3|2|x2 − x3|2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='+ O  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='a3ρ4 log(b/a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='s3a6ρ2(log(b/a))2(log N)9 + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1)  31  We will choose N (really L, see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8) some large negative power of a3ρ, so errors with  N−1/3 are subleading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We may compute  ˆ  dx  �  |∇f(x)|2 + 1  2v(x)f(x)2  �  |x|2  =  1  (1 − a3/b3)2  ˆ  |x|≤b  dx  �  |∇f0(x)|2 + 1  2v(x)f0(x)2  �  |x|2  ≤ 12πa3 �  1 + O(a3/b3)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2)  by Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 since f =  1  1−a3/b3f0 for |x| ≤ b and b > R0, the range of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the higher  moments we recall that |∇f0| ≤ |∇fhc| = 3a3  |x|4 for |x| ≥ a by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then we have (for  n = 4, 6)  ˆ  dx  �  |∇f(x)|2 + 1  2v(x)f(x)2  �  |x|n  =  1  (1 − a3/b3)2  ˆ  |x|≤b  dx  �  |∇f0(x)|2 + 1  2v(x)f0(x)2  �  |x|n  ≤ 1  2Rn−2  0  ˆ  v|f0|2|x|2 dx +  ˆ  |x|≥a  � 3a3  |x|4  �2  |x|n dx + an−2  ˆ  |x|≤a  |∇f0|2|x|2 dx  ≲ CRn−2  0  a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3)  For n = 4 we have more precisely  ˆ  dx  �  |∇f(x)|2 + 1  2v(x)f(x)2  �  |x|4 ≤  ˆ  dx  �  |∇f0(x)|2 + 1  2v(x)f0(x)2  �  |x|4 �  1 + O(a3b−3)  �  = 36πa3a2  0 + O(a6a2  0b−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For the lower moment, we have by Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4)  ˆ  dx  �  |∇f(x)|2 + 1  2v(x)f(x)2  �  = 4π  ˆ b  0  �  |∂rf|2r2 + r2f∂2  rf + 4rf∂rf  �  dr  = 12πa3/b2  1 − a3/b3 + 8π  ˆ b  0  rf∂rf dr  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4)  where ∂r denotes the radial derivative, and we integrated by parts using that f(r) = 1−a3/r3  1−a3/b3  outside the support of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 we have  2  ˆ b  0  rf∂rf dr = b −  ˆ b  0  f 2 dr ≤ b −  1  (1 − a3/b3)2  ˆ b  a  �  1 − a3  r3  �2  dr ≤ Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5)  Hence  ˆ  dx  �  |∇f(x)|2 + 1  2v(x)f(x)2  �  ≤ Ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This concludes the bounds on all the terms in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1) arising from the 2-body term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' To  bound those arising from the 3-body term we bound |x1 − x3| ≤ 2b in the support of ∇f12∇f23  and f13 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By the translation invariance one integration gives a volume factor L3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The  32  remaining two integrals then both give the same contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is,  ˚  dx1 dx2 dx3f12∇f12f23∇f23f 2  13  �  O(ρ5|x1 − x2|2|x1 − x3|2|x2 − x3|2)  + O  �  a3ρ4 log(b/a)  �  s3a6ρ2(log(b/a))2(log N)9 + 1  ���  ≤ CNρ4b2  �ˆ  |x|2f∂rf dx  �2  + CNa3ρ3 log(b/a)  �  s3a6ρ2(log(b/a))2(log N)9 + 1  � �ˆ  f∂rf dx  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Using integration by parts and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2, we have that  1  4π  ˆ  |x|nf∂rf dx =  ˆ b  0  rn+2f∂rf dr = bn+2  2  − n + 2  2  ˆ b  0  rn+1f 2 dr  ≤ bn+2  2  − n + 2  2  ˆ b  a  rn+1  �1 − a3/r3  1 − a3/b3  �2  dr ≤  �  Ca2  n = 0,  Ca3b  n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6)  Plugging all this into Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1) we thus get for the energy density  ⟨ψN|HN|ψN⟩  L3  = 3  5(6π)2/3ρ5/3 + 12π  5 (6π2)2/3a3ρ8/3 − 108π(6π2)4/3  175  a3a2  0ρ10/3  + O  �  s−2ρ5/3�  + O  �  N−1/3ρ5/3�  + O  �  a6b−3ρ8/3�  + O  �  a6a2  0b−3ρ10/3�  + O  �  R4  0a3ρ4�  + O  �  a7ρ4(log(b/a))2�  s3a6ρ2(log b/a)2(log N)9 + 1  ��  + O  �  a6ρ3+2/3�  s5a12ρ4(log(b/a))5(log N)16 + b4ρ4/3 + log(b/a)  ��  + O  �  a6b4ρ5�  + O  �  a7ρ4 log(b/a)  �  s3a6ρ2(log(b/a))3(log N)9 + 1  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7)  We choose L ∼ a(a3ρ)−10 still ensuring that  LkF  2π  is rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (More precisely one chooses  L ∼ a(kFa)−30, since ρ is defined in terms of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Then N ∼ (a3ρ)−29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Choose moreover  b = a(a3ρ)−β,  s ∼ (a3ρ)−α| log(a3ρ)|−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Note that we need  2  9 < β < 5  12,  5  6 < α < 13  15  for the error terms to be smaller than the desired accuracy of order a3a2  0ρ10/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We get  ⟨ψN|HN|ψN⟩  L3  = 3  5(6π)2/3ρ5/3 + 12π  5 (6π2)2/3a3ρ8/3 − 108π(6π2)4/3  175  a3a2  0ρ10/3  + O(ρ5/3(a3ρ)γ1| log(a3ρ)|γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  where  γ1 = min  �  2α, 1 + 3β, 13  3 − 3α, 6 − 5α, 10  3 − 4β  �  ,  and γ2 is given by the power of the logarithmic factors of the largest error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Optimising in  α, β, δ we see that for the choice  β = 1  3,  α = 6  7,  δ = 3  33  we have γ1 = 12  7 and γ2 = 6, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ⟨ψN|HN|ψN⟩  L3  = 3  5(6π)2/3ρ5/3  + 12π  5 (6π2)2/3a3ρ8/3  �  1 − 9  35(6π2)2/3a2  0ρ2/3 + O((a3ρ)2/3+1/21| log(a3ρ)|6)  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8)  for a3ρ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note that for this choice of s, N we have s ∼ N6/203(log N)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus any  Q with N4/3 ≪ Q ≤ CNC satisfies the condition Q−1/4 ≤ Cs−1 of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1  Thermodynamic limit via a box method  In this section we construct a trial state in the thermodynamic limit using a box method of  gluing trial states for finite n together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' First we show that we may choose periodic boundary  conditions in the small boxes instead of using Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The setting and  argument is due to Robinson [Rob71, Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We present a slightly modified  version in [MS20, Section C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3 ([MS20;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Rob71]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Let 0 < d < L/2 be a cut-off, let HD  N,L+2d = �N  j=1 −∆D  j,L+2d +  �  i<j v(xi −xj) denote the N-particle Hamiltonian with Dirichlet boundary conditions on a box  of sides L + 2d, and let Hper  N,L = �N  j=1 −∆per  j,L + �  i<j vper(xi − xj) denote the N-particle Hamil-  tonian with periodic boundary conditions on a box of sides L, with the interaction vper(x) =  �  n∈Z3 v(x + nL), the periodized interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Then, there exists an isometry V : L2  a(ΛN  L ) → L2  a(ΛN  L+2d) such that for all ψ in the form-  domain of Hper  N,L we have V ψ in the form-domain of HD  N,L+2d and  �  V ψ  ��HD  N,L+2d  ��V ψ  �  ≤  �  ψ  ��Hper  N,L  ��ψ  �  + 6N  d2 ∥ψ∥2  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This is a trivial modification of [MS20, Lemma 4], noting that the explicitly constructed  V respects the anti-symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We now glue together trial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For any (sufficiently small) density ρ we have above found  that we may construct a (normalized) trial state ψn on the torus Λℓ = [−ℓ/2, ℓ/2]3 satisfying  Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8) with ℓ ∼ a(a3ρ)−10, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  n ∼ (a3ρ)−29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We now use the isometry V from  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3 to find a trial state V ψn with Dirichlet boundary conditions on Λℓ+2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Our trial  state ΨN for N = M3n is then obtained by gluing together M3 copies of V ψn arranged in  boxes, with a distance b between them, so that there is no interaction between the boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We  choose the same b as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' More precisely, for configurations where the first n particles are  in box 1 and so on,  ΨN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xN) =  M3  �  j=1  V ψn(xn(j−1)+1 − τj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xnj − τj),  where τj ∈ R3 denotes the centre of box number j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The state ΨN is then the antisymmetrization  of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Its energy is  �  ΨN  ��HD  N,M(ℓ+2d+b)  ��ΨN  �  = M3 �  V ψn  ��HD  n,ℓ+2d  ��V ψN  �  ≤ M3  ��  ψn  ��Hper  n,ℓ  ��ψn  �  + 6n  d2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  34  The particle density of the state ΨN is ˜ρ =  n  (ℓ+2d+b)3 = ρ(1 + O(d/ℓ) + O(b/ℓ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The energy  density is  e(˜ρ) ≤ lim inf  M→∞  �  ΨN  ���HD  N,M(ℓ+2d+b)  ���ΨN  �  M3(ℓ + 2d + b)3  =  �  V ψn  ��HD  n,ℓ+2d  ��V ψN  �  (ℓ + 2d + b)3  ≤  �  ψn  ��Hper  n,ℓ  ��ψn  �  ℓ3  [1 + O(d/ℓ) + O(b/ℓ)] + O(ρd−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9)  Choosing d = a(a3ρ)−5 and using Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8) we conclude that for a3ρ sufficiently small  e(˜ρ) ≤ 3  5(6π)2/3ρ5/3  + 12π  5 (6π2)2/3a3ρ8/3  �  1 − 9  35(6π2)2/3a2  0ρ2/3 + O  �  (a3ρ)2/3+1/21| log(a3ρ)|6��  = 3  5(6π)2/3˜ρ5/3  + 12π  5 (6π2)2/3a3˜ρ8/3  �  1 − 9  35(6π2)2/3a2  0˜ρ2/3 + O  �  (a3˜ρ)2/3+1/21| log(a3˜ρ)|6��  since ˜ρ = ρ(1 + O((a3ρ)5)), so ρ = ˜ρ(1 + O((a3˜ρ)5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This concludes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  It remains to give the proofs of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2  Subleading 2-particle diagrams (proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1)  In this section we give the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Before doing this, we first discuss why we don’t  just use the bounds of these terms from the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4 (Why not use bounds of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Inspecting the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 (more  precisely Equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11) of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3) we can extract the following bound  ������  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2p  Γ2  π,G  ������  ≤ Csa3ρ3 log(b/a)(log N)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This is immediate by considering the bounds Equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11) and noting that since  we have p ≥ 1 in the sum �∞  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2p Γ2  π,G, the summands either have k ≥ 1 or n∗ ≥ 1  or n∗∗ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Using this bound we would thus get for the error in the ground state energy  density the bound ∼ sa4ρ3 (ignoring the log-factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' However, as we saw in Section 4, using  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13, the s-dependent error of the kinetic energy density is ∼ s−2ρ5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' There is no way  to choose s, such that both of these errors are smaller that a3ρ8/3, which is the precision we  need in order to prove the leading correction to the kinetic energy in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Similarly, by following the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 one could get the bound  ρ(3)  Jas ≤ Cρ3f 2  12f 2  13f 2  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This bound is not problematic in terms of getting all the error terms in the energy density  smaller than a3ρ8/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' However, we improve on this bound in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 in order to get a better  error bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note first that by translation invariance  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2p  Γ2  π,G  35  is a function of x1 − x2 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Recall Equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We split the diagrams in L2  p  into three groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' To define these three groups we first define for any diagram (π, G) ∈ L2  p the  number k = k(π, G) as the number of clusters entirely containing internal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (This k is  exactly the same k as in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Then we define  ng = ng(π, G) = �k  ℓ=1 nℓ − 2k,  n∗  g = n∗  g(π, G) = n∗ + n∗∗,  with the understanding that n∗∗ = 0 if {1} and {2} are in the same cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We think of  ng + n∗  g as the “number of added vertices”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Indeed, given a k the smallest number of vertices  in a diagram (π, G) ∈ L2  p with k clusters is 2k + 2 and in this case we have p = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For such  a diagram, there are p = 2k internal vertices and 2 external vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The graph G of such a  diagram looks like  k  1  2  G =  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Then ng + n∗  g is the number of (internal) vertices a diagram has more than this lowest number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  By following the bound in Equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11) we see that for p = ng0 + 2k0  �����������  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2  p  ng(π,G)+n∗  g(π,G)=ng0  k(π,G)=k0  Γ2  π,G  �����������  ≤ Cρ2(Cs(log N)3)k0(Ca3ρ log(b/a))k0+ng0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10)  We split diagrams into different groups depending on whether or not they are “large” and  whether or not we will do a Taylor expansion of their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We first give some motivation for  what “large” means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Here “large” should be thought of in terms of the bound in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Recall that the (s-dependent) error in the kinetic energy density is s−2ρ5/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For this error to be  smaller than the desired accuracy of order a3a2  0ρ10/3 we need s ≫ (a3ρ)−5/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' If we think of, say  s ∼ (a3ρ)−6/7, then (ignoring log-factors) Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10) reads ≲ ρ2(a3ρ)k0/7+ng0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The large  diagrams are those for which this bound (or the differentiated version, where one effectively  gains a power a2ρ2/3, see the details of the proof) gives a contribution to the energy density  ≪ a5ρ10/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We split the diagrams into three (exhaustive) groups:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Small diagrams with  (A) {1} and {2} in different clusters and 1 ≤ k ≤ 4, ng = 0, n∗  g = 0,  (B) {1} and {2} in different clusters and 0 ≤ k ≤ 2, ng = 0, n∗  g = 1,  (C) {1} and {2} in the same cluster and 0 ≤ k ≤ 2, ng = 0, n∗  g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Large diagrams with n∗  g = 0 (in particular {1} and {2} are in different clusters) and  (A) k ≥ 5, ng = 0 or  (B) k ≥ 1, ng ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Large diagrams with n∗  g ≥ 1 and  36  (A) k ≥ 3, ng = 0 or  (B) k ≥ 1, ng ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Note that we have p ≥ 1, so the diagrams with k = 0, ng = 0, n∗  g = 0 are not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Moreover,  if k = 0 then clearly also ng = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For drawings of the small diagrams see Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 in  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We then write  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2p  Γ2  π,G = ξsmall,0 + ξsmall,≥1 + ξ0 + ξ≥1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11)  where ξsmall,0 is the contribution of all small diagrams of types (A) and (B), ξsmall,≥1 is the  contribution of all small diagrams of type (C), ξ0 is the contribution of all large diagrams with  n∗  g = 0, and ξ≥1 is the contribution of all large diagrams with n∗  g ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The notation is motivated by that of the large diagrams, which were split into two groups  depending on whether n∗  g = 0 or n∗  g ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We will treat the small diagrams of types (A) and (B)  somewhat similar to the large diagrams in ξ0 (hence the notation ξsmall,0) and the small diagrams  of type (C) somewhat similar to the large diagrams in ξ≥1 (hence the notation ξsmall,≥1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Indeed,  we will do a Taylor expansion of ξsmall,0 and ξ0 but not of ξsmall,≥1 or ξ≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Using the bound in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10) and the absolute convergence (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2) we get  |ξ≥1(x1, x2)| ≤ Cρ2(s(log N)3)3(a3ρ log(b/a))4  �  ��  �  type (A) diagrams  + Cρ2s(log N)3(a3ρ log(b/a))3  �  ��  �  type (B) diagrams  ≤ Ca9ρ5(log(b/a))3s(log N)3 �  s2(log N)6a3ρ log(b/a) + 1  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12)  uniformly in x1, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For ξsmall,≥1 we have  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the small diagrams of type (C) we have the bound  |ξsmall,≥1| ≤ Ca3b4ρ5|x1 − x2|2 + Ca6ρ4(log(b/a))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The proof of this lemma is a (not very insightful) computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We give it in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Slightly more insightful however, is why we split off the small diagrams from the large diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7 (Why one gets better bounds by computing small diagrams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We could treat all  the small diagrams exactly as we treat ξ0 and ξ≥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We do however gain better error bounds  by treating them more directly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' computing more precisely what the values of these small  diagrams are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  In exact calculations we can make use the fact that  ´  γ(1)  N  = 1, instead of  bounding the absolute value as  ´  |γ(1)  N | ≤ Cs(log N)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We Taylor expand ξsmall,0 and ξ0 to second order around the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We first claim that  ξsmall,0(x2, x2) + ξ0(x2, x2) + ξsmall,≥1(x2, x2) + ξ≥1(x2, x2) = 0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13)  Indeed by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4 we have  ξsmall,0 + ξ0 + ξsmall,≥1 + ξ≥1 = ρ(2)  Jas  f 2  12  − ρ(2) = N(N − 1)  CN  ˙  �  i<j  (i,j)̸=(1,2)  f 2  ijDN dx3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxN − ρ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (Formally to do the division by f in the first equality in case f = 0 somewhere one uses  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4 with all instances of f replaced by ˜f (n) for some sequence ˜f (n) > 0 with ˜f (n) ց f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  37  Then one readily applies the Lebesgue dominated convergence theorem to exchange the limit  ˜f (n) → f with the relevant sums and integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Taking x1 = x2 in this we have DN = 0 and  ρ(2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This shows Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We may thus bound the zeroth order term of ξsmall,0 and  ξ0 by  |ξsmall,0(x2, x2) + ξ0(x2, x2)|  ≤ |ξsmall,≥1(x2, x2)| + |ξ≥1(x2, x2)|  ≤ Ca6ρ4(log(b/a))2 �  s3(log N)9a6ρ(log(b/a))2 + s(log N)a3ρ log(b/a) + 1  �  ≤ Ca6ρ4(log(b/a))2 �  s3(log N)9a6ρ(log(b/a))2 + 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='14)  Since both ξsmall,0 and ξ0 are symmetric in x1 and x2 all first order terms vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We are left  with bounding the second derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For ξsmall,0 we have  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For any µ, ν = 1, 2, 3 we have  ��∂µ  x1∂ν  x1ξsmall,0  �� ≤ Ca3ρ3+2/3 log(b/a)  uniformly in x1, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Here ∂µ  x1 denotes the derivative in the xµ  1-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The proof of this lemma is a (not very insightful) computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We give it in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Next we consider ∂µ  x1∂ν  x1ξ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We write ξ0 in terms of truncated densities as in Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ξ0 =  ∞  �  k=0  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk≥2  χ(k≥5,nℓ=2 or k≥1,� nℓ=2k+1)  1  �  ℓ nℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  G1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',Gk  Gℓ∈Cnℓ  ×  ˙ �  ℓ  �  e∈Gℓ  geρ({1},{2},n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  t  dx3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dxp+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Since we consider terms with n∗ = n∗∗ = 0, there are no g-factors that depend on x1 and  thus all derivatives are of ρ({1},{2},n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We thus need to calculate ∂µ  x1∂ν  x1ρ({1},{2},n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For  this we use the definition in Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4) rather than the formula in Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In  Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4) the variable x1 appears exactly twice: Once in an outgoing γ(1)  N -edge from {1}  and once in an incoming γ(1)  N -edge to {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Taking the derivatives then amounts to replacing  either one of these edges by its second derivative or both of them by their first derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Thus, using that γ(1)  N (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj) = γ(1)  N (xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xi) since γ(1)  N is real, and that for (π, ∪Gℓ) to be linked  necessarily π(1) ̸= 1, we have (for p = 2 + �  ℓ nℓ)  ∂µ  x1∂ν  x1ρ({1},{2},n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  t  = ∂µ  x1∂ν  x1  �  π∈Sp  (−1)πχ((π,∪Gℓ)∈Lp)  p  �  j=1  γ(1)  N (xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xπ(j))  =  �  π∈Sp  (−1)πχ((π,∪Gℓ)∈Lp)  �  j̸=1,j̸=π−1(1)  γ(1)  N (xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xπ(j))  �  2∂µ  x1∂ν  x1γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xπ(1))γ(1)  N (xπ−1(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' x1)  +2∂µ  x1γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xπ(1))∂ν  x1γ(1)  N (xπ−1(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' x1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='15)  With this formula we may then redo the computation of [GMR21, Equation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='53)] only now  some of the γ(1)  N -factors (precisely 1 or 2 of them) carry derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The γ(1)  N -factors with  derivatives may end up in the anchored tree, or they may end up in the matrix N (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' If they  end up in N (r) it is explained around [GMR21, Equation (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9)] how to modify Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  38  One simply includes factors ikµ in the definition of (some of) the functions αi (and not of βj)  in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Since we may bound |k| ≤ Cρ1/3 for k ∈ PF we get  ���det ˜  N(r)  ��� ≤  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  ρ  � nℓ+2−(k+2−1)  if no derivatives end up in N ,  Cρ  � nℓ+2−(k+2−1)+1/3  if one derivative ends up in N,  Cρ  � nℓ+2−(k+2−1)+2/3  if two derivatives end up in N ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='16)  where ˜  N(r) is the appropriate modification of N (r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' To get the formula for ρt we need also to  consider two cases for how the anchored tree looks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' There could be both an incoming and an  outgoing edge to/from the vertex {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' And if there is just one edge to/from {1} it could be  either an incoming or an outgoing edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Since γ(1)  N is real, incoming and outgoing edges gives  the same factor γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' A simple calculation (essentially just undoing the product rule)  then shows that  ∂µ  x1∂ν  x1ρ({1},{2},n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  t  =  �  ∂∈{1,∂µ  x1,∂νx1,∂µ  x1∂νx1}  \uf8ee  \uf8ef\uf8ef\uf8f0  �  τ∈A({1},{2},n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  two edges to/from {1}  ∂  �  γ(1)  N (xj2, x1)γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj1)  �  +  �  τ∈A({1},{2},n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  one edge to/from {1}  ∂γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj1)  \uf8f9  \uf8fa\uf8fa\uf8fb  �  (i,j)∈τ  i,j̸=1  γ(1)  N (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj)  ˆ  dµτ(r) det ˜  N∂(r),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='17)  where j1 and j2 denote the vertices connected to {1} by the relevant edges in τ and ˜  N∂ is  the appropriately modified version of N , where the derivatives not in ∂ end up in N , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='16) reads  ���det ˜  N∂(r)  ��� ≤ Cρ  � nℓ+2−(k+2−1)+(2−#∂)/3,  where #∂ denotes the number of derivatives in ∂, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' #1 = 0, #∂µ  x1 = 1 and #∂µ  x1∂ν  x1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We denote the contribution of the two terms in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='17) to ∂µ  x1∂ν  x1ξ0 by (∂µ  x1∂ν  x1ξ0)→•→  and (∂µ  x1∂ν  x1ξ0)•→ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We first deal with the second term of Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='17) where there is just one edge to/from  {1} in the anchored tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We may bound the contribution of this term almost exactly as in the  proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We give a sketch here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Using Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='16) we get the bound  ≤ C  �  ∂∈{1,∂µ  x1,∂νx1,∂µ  x1∂νx1}  ρ(2−#∂)/3  �  τ∈A({1},{2},n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk)  ���∂γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj1)  ���  ×  �  (i,j)∈τ  i,j̸=1  ���γ(1)  N (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj)  ��� ρ(�  ℓ nℓ+2)−(k+2���1),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='18)  where again #∂ denotes the number of derivatives in ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  To bound the integrations we again follow the strategy of the proof of the Γ2-sum of  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The only difference is that the γ(1)  N -edge on the path in the an-  chored tree between {1} and {2} incident to {1} is the edge with derivatives, ∂γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This we bound by |∂γ(1)  N | ≤ Cρ1+#∂/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The integrations can then be performed exactly as in  39  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We conclude the bound  ˙  k  �  ℓ=1  �  e∈Tℓ  |ge|  ���∂γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj1)  ���  �  (i,j)∈τ  i,j̸=1  ���γ(1)  N (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj)  ��� dx3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' dx� nℓ+2  ≤ ρ1+#∂/3 �  Ca3 log(b/a)  �� nℓ−k �  Cs(log N)3�k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Again, as in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 we have by Cayley’s formula that #Tn = nn−2 ≤ Cnn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  and by [GMR21, Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5] that #A({1},{2},n1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',nk) ≤ (k + 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4  � nℓ+2 ≤ C(k2 + 1)k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4  � nℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Following the same arguments as for Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10) and recalling that the diagrams in ξ0 have  either k ≥ 5 or ng ≥ 1 we get the contribution to ∂µ  x1∂ν  x1ξ0 of  ��(∂µ  x1∂ν  x1ξ0)•→�� ≤ Cρ2+2/3�  (s(log N)3)5(a3ρ log(b/a))5  �  ��  �  type (A) diagrams  + s(log N)3(a3ρ log(b/a))2  �  ��  �  type (B) diagrams  �  ≤ Ca6ρ4+2/3(log(b/a))2s(log N)3 �  s4(log N)12a9ρ3(log(b/a))3 + 1  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='19)  uniformly in x1, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Next consider the first term of Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The argument is almost the same, only  we have to distinguish between which γ(1)  N -factor(s) the derivatives in ∂ hits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We consider the  case ∂ = ∂µ  x1∂ν  x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The other cases are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Suppose that the γ(1)  N -edge on the path (in the  anchored tree) from {1} to {2} is γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj1) and the γ(1)  N -factor not on the path is γ(1)  N (xj2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We distinguish between three cases:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' If both derivatives are on γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj1) we may bound this exactly as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' If one derivative is on γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj1) (say ∂ν  x1) and one derivative (say ∂µ  x1) is on γ(1)  N (xj2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' x1)  we bound |∂ν  x1γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj1)| ≤ Cρ4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Then the argument is similar, only now one of  the γ(1)  N -integrations is with ∂µγ(1)  N  instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, in the computation leading to Equa-  tion (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='19) we should replace one factor Csρ1/3(log N)3 with  ´  |∂µγ(1)  N | dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' If both derivatives are on γ(1)  N (xj2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' x1) then analogously we bound |γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xj1)| ≤ Cρ and  in the computation leading to Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='19) we should replace one factor Csρ2/3(log N)3  with  ´  |∂µ∂νγ(1)  N | dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  In total we have the contribution to ∂µ  x1∂ν  x1ξ0 of  ≤ Cρ2  �  ρ2/3s(log N)3 + ρ1/3  ˆ  Λ  ���∂µγ(1)  N  ��� dx + ρ1/3  ˆ  Λ  ���∂νγ(1)  N  ��� dx +  ˆ  Λ  ���∂µ∂νγ(1)  N  ��� dx  �  ×  �  (s(log N)3)4(a3ρ log(b/a))5 + (a3ρ log(b/a))2�  uniformly in x1, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' One may do a similar computation for the other cases of ∂ and conclude  that  ��(∂µ  x1∂ν  x1ξ0)→•→��  ≤ Cρ2  �  ρ2/3s(log N)3 + ρ1/3  ˆ  Λ  ���∂µγ(1)  N  ��� dx + ρ1/3  ˆ  Λ  ���∂νγ(1)  N  ��� dx +  ˆ  Λ  ���∂µ∂νγ(1)  N  ��� dx  �  ×  �  (s(log N)3)4(a3ρ log(b/a))5 + (a3ρ log(b/a))2�  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='20)  40  uniformly in x1, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, we need to bound the integrals  ˆ  Λ  ���∂µγ(1)  N  ��� dx =  ˆ  Λ  1  L3  �����  �  k∈PF  kµeikx  ����� dx =  1  (2π)2L  ˆ  [0,2π]3  �������  �  q∈  � LkF  2π P  �  ∩Z3  qµeiqu  �������  du,  and  ˆ  Λ  ���∂µ∂νγ(1)  N  ��� dx =  ˆ  Λ  1  L3  �����  �  k∈PF  kµkνeikx  ����� dx =  1  2πL2  ˆ  [0,2π]3  �������  �  q∈  � LkF  2π P  �  ∩Z3  qµqνeiqu  �������  du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Here we have  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The polyhedron P from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7 satisfies for any µ, ν = 1, 2, 3 that  ˆ  [0,2π]3  �������  �  q∈  � LkF  2π P  �  ∩Z3  qµeiqu  �������  du ≤ CsN1/3(log N)3,  ˆ  [0,2π]3  �������  �  q∈  � LkF  2π P  �  ∩Z3  qµqνeiqu  �������  du ≤ CsN2/3(log N)4  for sufficiently large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9 is a long and technical computation, which we give in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Applying the lemma we conclude that  ˆ  Λ  ���∂µγ(1)  N  ��� dx ≤ Csρ1/3(log N)3,  ˆ  Λ  ���∂µ∂νγ(1)  N  ��� dx ≤ Csρ2/3(log N)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  By combining this with Equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='19) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='20) we get  ��∂µ  x1∂ν  x1ξ0  �� ≤ Ca6ρ4+2/3(log(b/a))2s(log N)4 �  s4(log N)12a9ρ3(log(b/a))3 + 1  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='21)  uniformly in x1, x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Combining Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8 and Equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='14) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='21)  and using that for any real number t > 0 and integer n ≥ 1 we may bound t ≲ tn + 1 this  shows the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10 (Treating more diagrams as small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' One can improve the error bound in The-  orem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3 slightly by treating more diagrams as small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This is similar to what is done in  [BCGOPS22] for the dilute Bose gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We sketch the overall idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  If we choose s ∼ (a3ρ)−1+ε/2 then the error from the s-dependent term in the kinetic energy  to the energy density is ρ5/3(a3ρ)2−ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then choose as “large” the diagrams for which the bound  in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10) gives contributions to the energy density much smaller than ρ5/3(a3ρ)2−ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This happens for k0 > K for some large K ∼ ε−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We can then evaluate all small diagrams  as in Appendix A and conclude that their contributions are as given in Appendix A only with  some K-dependent constants, since there is some K-dependent number of small diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In  total we would then get an error of size Oε(ρ5/3(a3ρ)2−ε) in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  41  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3  Subleading 3-particle diagrams (proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2)  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Recall the formula for ρ(3)  Jas of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In this formula there are terms  like ρ �  p≥1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2p Γ2  π,G(x2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We have ρ = ρ(1)  Jas(x1) = �  p′≥1  1  p′!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π′,G′)∈L1  p′ Γ1  π′,G′(x1) by  translation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Joining the two diagrams (π, G) ∈ L2  p and (π′, G′) ∈ L1  p′ we get a new  (no longer linked) diagram (π′′, G′′) ∈ D3  p+p′ with two linked components, one of which contains  the vertices {2} and {3} and one of which contains the vertex {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Doing this for all three  terms of this type, we are led to define the set  ˜L3  p := L3  p ∪  �  q+q′=p  (L2  q ⊕ L1  q′),  where ⊕ refers to the operation of joining two diagrams as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The set ˜L3  p is then the set of  diagrams on 3 external and p internal vertices such that there is at most two linked components,  and that each linked component contains at least one external vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' With this, the formula  for ρ(3)  Jas of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4 reads (assuming that sa3ρ log(b/a)(log N)3 is sufficiently small)  ρ(3)  Jas = f 2  12f 2  13f 2  23  �  ρ(3) +  �  p≥1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈ ˜L3p  Γ3  π,G  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We split the diagrams in ˜L3  p into two groups, large and small similarly to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  To do this, we similarly define for a diagram (π, G) ∈ ˜L3  p the number k = k(π, G) as the number  of clusters entirely containing internal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (This k is exactly the same k as in the proof of  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Then we define  ng = ng(π, G) = �k  ℓ=1 nℓ − 2k,  n∗  g = n∗  g(π, G) = n∗ + n∗∗ + n∗∗∗,  where we understand n∗∗ = 0 and/or n∗∗∗ = 0 if {1, 2, 3} are not all in different clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (One  defines n∗∗∗ as the number of internal vertices in the cluster containing {3} if all {1, 2, 3} are in  different clusters, exactly as for the n∗∗ and n∗ of Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') We may still think  of ng +n∗  g as the “number of added vertices”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' As for Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10) we have (for p = 2k0+ng0)  ������������  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈ ˜L3  p  ng(π,G)+n∗  g(π,G)=ng0  k(π,G)=k0  Γ3  π,G  ������������  ≤ Cρ3(Cs(log N)3)k0(Ca3ρ log(b/a))k0+ng0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='22)  The main difference compared to Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10) is that we here allow diagrams that are not  linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This doesn’t matter, since when we compute the integrals (as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3) we  anyway have to cut the diagram up into 3 parts (either by bounding g-edges or γ(1)  N -edges) as  described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We split the diagrams into two (exhaustive) groups:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Small diagrams with 1 ≤ k ≤ 2, ng = 0, n∗  g = 0, and  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Large diagrams as the rest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' with  (A) k ≥ 3, or  42  (B) ng + n∗  g ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  As in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2, the splitting is motivated by counting powers in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note that  for p ≥ 1 the diagrams with k = 0, ng = 0, n∗  g = 0 are not present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We write  �  p≥1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈ ˜L3p  Γ3  π,G = ξ3  small + ξ3  large,  where ξ3  small and ξ3  large are the contributions of small and large diagrams respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Exactly  as in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12) we may bound, using Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='22)  |ξ3  large| ≤ Cρ3(s(log N)3)3(a3ρ log(b/a))3  �  ��  �  type (A) diagrams  + Cρ3a3ρ log(b/a)  �  ��  �  type (B) diagrams  ≤ Ca3ρ4 log(b/a)  �  s3a6ρ2(log(b/a))2(log N)9 + 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For the small diagrams we have  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We have  |ξ3  small| ≤ Ca3ρ4 log(b/a)  uniformly in x1, x2, x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  As with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8, the proof is simply a computation, which we give in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We  conclude the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  5  One and two dimensions  In this section we sketch the necessary changes one needs to make for the argument to apply in  dimensions d = 1 and d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We will abuse notation slightly and denote by the same symbols  as in Sections 2, 3 and 4 the relevant 1- and 2-dimensional analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1  Two dimensions  Similarly to the 3-dimensional setting, the p-wave scattering function f0 in 2 dimensions is  radial and solves the equation  − ∂2  rf0 − 3  r∂rf0 + 1  2vf0 = 0,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1)  see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 and recall Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, it is the same as the s-wave scattering function  in 4 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In particular it satisfies the bound  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 ([LY01, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1], Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The scattering function satisfies  �  1 − a2  |x|2  �  + ≤  f0(x) ≤ 1 for all x and |∇f0(x)| ≤ 2a2  |x|3 for |x| > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  As for the 3-dimensional setting we consider the trial state  ψN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xN) =  1  √CN  �  i<j  f(xi − xj)DN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xN),  where f is a rescaled scattering function  f(x) =  �  1  1−a2/b2f0(|x|)  |x| ≤ b,  1  |x| ≥ b  43  and  DN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xN) = det[uk(xi)]1≤i≤N  k∈PF  ,  uk(x) = 1  Leikx,  N = #PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Here PF denotes the “Fermi polygon”, the 2-dimensional analogue of the “Fermi polyhedron”  PF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' It is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (Compare to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The polygon P is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  First pick Q, satisfying  Q−1/2 ≤ Cs−2,  N3/2 ≪ Q ≤ CNC  in the limit N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (The exponents arise as −1  2 =  −1  2(d−1), −2 =  −2  d−1 and 3  2 = d+1  d  for d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  Pick two distinct primes Q1, Q2 ∼ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Place s evenly distributed points κR  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κR  s on the circle of radius Q��1/2 such that the points  are invariant under the symmetries (k1, k2) �→ (±ka, ±kb) for {a, b} = {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (Here the  exponent arises as −1  2 =  1  2(d−1) − 1 for d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  Evenly distributed means that the distance between any pair of points is d ≳ s−1Q−1/2 and  that for any k on the sphere of radius Q−1/2 the distance from k to the nearest point is  ≲ s−1Q−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (On the circle we can naturally order the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then the condition for being  evenly distributed reads that consecutive points are separated by a distance ∼ s−1Q−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  Find now points κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κs of the form  κj =  � p1  j  Q1  , p2  j  Q2  �  ,  pµ  j ∈ Z,  µ = 1, 2,  j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , s  such that the points are invariant under the symmetries (k1, k2) �→ (±k1, ±k2) and such that  for any j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , s we have  ��κj − κR  j  �� ≲ Q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Define ˜P as the convex hull of the points κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , κs and P = σ ˜P where σ is such that  Vol(P) = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Define the centre z = σ(1/Q1, 1/Q2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The “Fermi polygon” is the rescaled version defined as PF = kFP ∩ 2π  L Z2, where L is chosen  large (depending on kF) such that kF L  2π is rational and large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Similarly as in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9 we have that σ is irrational and σ = Q1/2(1+O(s−2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In particular,  any point on the boundary ∂P has radial coordinate 1 + O(s−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (The power of s here comes  from the circle being locally quadratic and the distance between close points being ∼ s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Compare to Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Moreover, PF is almost symmetric under the map (k1, k2) �→ (k2, k1)  similarly to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Let F12 be the map (k1, k2) �→ (k2, k1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For any function t ≥ 0 we have  �  k∈ 2π  L Z2  ��χ(k∈PF ) − χ(k∈F12(PF ))  �� t(k) ≲ Q−1/2N sup  |k|∼kF  t(k) ≲ N1/4 sup  |k|∼kF  t(k),  where Q is as in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The analogue of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12 is then  44  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The Lebesgue constant of the Fermi polygon satisfies  ˆ  Λ  1  L2  �����  �  k∈PF  eikx  ����� dx =  1  (2π)2  ˆ  [0,2π]2  �������  �  q∈  � LkF  2π P  �  ∩Z2  eiqu  �������  du ≤ Cs(log N)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We can again compute the kinetic energy of the Slater determinant analogously to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13  and its 2-particle reduced density analogously to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The kinetic energy of the (Slater determinant with momenta in the) Fermi polygon  satisfies  �  k∈PF  |k|2 =  �  k∈BF  |k|2 �  1 + O(N−1/2) + O(s−4)  �  = π  8 ρN  �  1 + O(N−1/2) + O(s−4)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The 2-particle reduced density of the (normalized) Slater determinant satisfies  ρ(2)(x1, x2) = π  16ρ3|x1 − x2|2 �  1 + O(N−1/2) + O(s−4) + O(ρ|x1 − x2|2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The computations in Section 3 make no reference to the dimension and are thus also valid in  dimension d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the absolute convergence in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 one should simply  replace occurrences of g and γ(1)  N with their 2-dimensional analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Here we have the bounds  (using Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4)  ˆ  Λ  |g| ≲ a2 +  1  (1 − a2/b2)2  ˆ b  a  ��  1 − a2  b2  �2  −  �  1 − a2  x2  �2�  x dx ≲ a2 log(b/a),  ˆ  Λ  |γ(1)  N | ≲ s(log N)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2)  Thus the absolute convergence holds as long as sa2ρ log(b/a)(log N)2 is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That  is, the analogue of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4 reads  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' There exists a constant c > 0 such that if sa2ρ log(b/a)(log N)2 < c, then  the formulas in Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3) hold (with ρ(n)  Jas and Γn  π,G interpreted as appropriate in the two-  dimensional setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The analogues of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 read  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' There exist constants c, C > 0 such that if sa2ρ log(b/a)(log N)2 < c and N =  #PF > C, then  ������  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2p  Γ2  π,G  ������  ≤ Ca4ρ4(log(b/a))2�  s3a4ρ2(log b/a)2(log N)6 + 1  �  + Ca2ρ4|x1 − x2|2�  s5a8ρ4(log(b/a))5(log N)11 + b4ρ2 + log(b/a)  �  ,  and  ρ(3)  Jas ≤ Cf 2  12f 2  13f 2  23  �  ρ6|x1 − x2|2|x1 − x3|2|x2 − x3|2  + a2ρ4 log(b/a)  �  s3a4ρ2(log(b/a))2(log N)6 + 1  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  45  The proof is again similar to the 3-dimensional case replacing the bounds on  ´  |g| and  ´  |γ(1)  N |  as in Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Apart from this, there are two main changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The first is in the  proof of the analogue of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6, namely Equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1), where one bounds  ´  |x|2|1 − f 2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  In two dimensions this bound is, using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1,  ˆ  R2  �  1 − f(x)2�  |x|2 dx ≤ Ca4 +  C  (1 − a2/b2)2  ˆ b  a  ��  1 − a2  b2  �2  −  �  1 − a2  r2  �2�  r3 dr ≤ Ca2b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The other main difference is for the analogue of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Here the 2-dimensional analogue  reads  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The polygon P from Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 satisfies for any µ, ν = 1, 2 that  ˆ  [0,2π]2  �������  �  q∈  � LkF  2π P  �  ∩Z2  qµeiqu  �������  du ≤ CsN1/2(log N)2,  ˆ  [0,2π]2  �������  �  q∈  � LkF  2π P  �  ∩Z2  qµqνeiqu  �������  du ≤ CsN(log N)3  for sufficiently large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The proof is similar to that of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9 given in Appendix B only one skips Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2  and notes that R = LkF  2π ∼ N1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Putting together the formulas in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7 with the bounds in Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8  we easily find the analogue of Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We then need to bound a few terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Following  the type of arguments of Section 4, namely Equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6) and  using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 we get the bounds  ˆ �  |∇f(x)|2 + 1  2v(x)f(x)2  �  |x|n dx ≤  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  C,  n = 0,  4πa2 + O(a4b−2),  n = 2,  Ca4 log(b/a) + CR2  0a2,  n = 4,  ˆ  |x|nf∂rf dx ≤  �  Ca,  n = 0,  Ca2b,  n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Plugging this into the analogue of Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1) we get the analogue of Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7),  ⟨ψN|HN|ψN⟩  L2  = π  8 ρ2 + π2  4 a2ρ3  + O  �  s−4ρ2�  + O  �  N−1/2ρ2�  + O  �  a4b−2ρ3�  + O  �  a4ρ4 log(b/a)  �  + O  �  R2  0a2ρ4�  + O  �  a4ρ4(log(b/a))2�  s3a4ρ2(log b/a)2(log N)6 + 1  ��  + O  �  a4ρ4�  s5a8ρ4(log(b/a))5(log N)11 + b4ρ2 + log(b/a)  ��  + O  �  a4b4ρ6�  + O  �  a4ρ4 log(b/a)  �  s3a4ρ2(log(b/a))3(log N)6 + 1  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3)  As above, we can choose L ∼ a(a2ρ)−10 still ensuring that LkF  2π is rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (More precisely one  chooses L ∼ a(kFa)−20, since ρ is defined in terms of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Then N ∼ (a2ρ)−19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Choose moreover  b = a(a2ρ)−β,  s ∼ (a2ρ)−α| log(a2ρ)|−γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  46  Optimising in α, β, γ we see that for the choice  β = 1  2,  α = 4  7,  γ = 10  7  we have  ⟨ψN|HN|ψN⟩  L3  = π  8ρ2 + π2  4 a2ρ3  �  1 + O  �  a2ρ| log(a2ρ)|2��  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4)  for a2ρ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note that for this choice of s, N we have s ∼ N4/133(log N)10/7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus  any Q with N3/2 ≪ Q ≤ CNC satisfies the condition Q−1/2 ≤ Cs−2 of Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The extension to the thermodynamic limit of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 is readily generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We thus  conclude the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2  One dimension  Similarly to the 2- and 3-dimensional settings, the p-wave scattering function f0 in 1 dimension  is even and solves the equation (here ∂2 denotes the second derivative)  − ∂2f0 − 2  r∂f0 + 1  2vf0 = 0,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5)  see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 and recall Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, it is the same as the s-wave scattering function  in 3 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In particular it satisfies the bound  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10 ([LY01, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1], Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The scattering function satisfies  �  1 −  a  |x|  �  + ≤  f0(x) ≤ 1 for all x and |∂f0(x)| ≤  a  |x|2 for |x| > a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Before giving the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9 we first compare our definition of the scattering length  to that of [ARS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In [ARS22] the following definition is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11 ([ARS22, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The odd-wave scattering length aodd is given by  4  R − aodd  = inf  �ˆ R  −R  �  2|∂h|2 + v|h|2�  dx : h(R) = −h(−R) = 1  �  for any R > R0, the range of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The value of aodd is independent of R > R0 so aodd is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We claim that  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The p-wave scattering length a defined in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8 and the odd-wave  scattering length aodd defined in Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11 agree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' a = aodd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note first that h �→ E(h) =  ´ R  −R (2|∂h|2 + v|h|2) dx is convex, so by replacing h by  (h(x) − h(−x))/2 we can only lower its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, we have  4  R − aodd  = inf  �ˆ R  −R  �  2|∂h|2 + v|h|2�  dx : h(x) = −h(−x), h(R) = 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Any h we write as h(x) = xf(x)  R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Using this and integration by parts we get  4  R − aodd  = 1  R2 inf  �ˆ R  −R  �  2|f|2 + 4xf∂f + 2|x|2|∂f|2 + v|f|2|x|2�  dx : f(x) = f(−x), f(R) = 1  �  = 4  R + 2  R2 inf  �ˆ R  −R  �  |∂f|2 + 1  2v|f|2  �  |x|2 dx : f(x) = f(−x), f(R) = 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  47  That is,  2R  �  1  1 − aodd/R − 1  �  = inf  �ˆ R  −R  �  |∂f|2 + 1  2v|f|2  �  |x|2 dx : f(x) = f(−x), f(R) = 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Taking R → ∞ in this we recover the definition of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We conclude that a = aodd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Concerning the assumption on v that  ´ �1  2vf 2  0 + |∂f0|2�  dx < ∞ we have the following two  propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Suppose that v ≥ 0 is even and compactly supported and that for some  interval [x1, x2], 0 ≤ x1 < x2 we have v(x) = ∞ for x1 ≤ x ≤ x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then  ´ � 1  2vf 2  0 + |∂f0|2�  dx <  ∞, where f0 denotes the p-wave scattering function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Let [x1, x2] be an interval where v(x) = ∞ for x1 ≤ x ≤ x2 and note that f0(x) = 0 for  all |x| ≤ x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then we have  ˆ �1  2vf 2  0 + |∂f0|2  �  dx ≤ 1  x2  2  ˆ  |x|≥x2  �1  2vf 2  0 + |∂f0|2  �  |x|2 dx = 2ax−2  2  < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Suppose that v ≥ 0 is even, compactly supported and smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Then  ´ � 1  2vf 2  0 + |∂f0|2�  dx < ∞, where f0 denotes the p-wave scattering function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For smooth v also the scattering function f0 is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Recall the scattering equation  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then a simple calculation using integration by parts shows that  ˆ �1  2vf 2  0 + |∂f0|2  �  dx = 2  ˆ ∞  0  �  f0∂2f0 + 2f0∂f0  x  + (∂f0)2  �  dx = 2  ˆ ∞  0  f0(x)2 − f0(0)2  x2  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The function f0 is smooth and even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus for small x we have f(x) = f(0) + O(|x|2), hence  the integral converges around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By the decay of  1  x2 the integral converges at ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We conclude  the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We now give the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We consider the trial state given in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1)  where f is a rescaled scattering function  f(x) =  �  1  1−a/bf0(|x|)  |x| ≤ b,  1  |x| ≥ b  and  DN(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xN) = det[uk(xi)]1≤i≤N  k∈BF  ,  uk(x) =  1  L1/2eikx,  N = #BF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  In 1 dimension, there is no difference between a ball and a polyhedron, so we may use the Fermi  ball BF = {k ∈ 2π  L Z : |k| ≤ kF} for the momenta in the Slater determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In this case we  have (see [KL18, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2] or Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11)  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The Lebesgue constant of the Fermi ball satisfies  ˆ L/2  −L/2  1  L  �����  �  k∈BF  eikx  ����� dx = 1  2π  ˆ 2π  0  �������  �  q∈  �  B  � LkF  2π  ��  ∩Z2  eiqu  �������  du ≤ C log N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  48  As for the 2-dimensional setting one easily generalizes the computation of the kinetic energy in  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13 and the calculation of the 2-particle reduced density for a Slater determinant in  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is,  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The kinetic energy of the (Slater determinant with momenta in the) Fermi ball  satisfies  �  k∈BF  |k|2 = π2  3 ρ2N  �  1 + O(N−1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The 2-particle reduced density of the (normalized) Slater determinant satisfies  ρ(2)(x1, x2) = π2  3 ρ4|x1 − x2|2 �  1 + O(N−1) + O(ρ2|x1 − x2|2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For the Gaudin-Gillespie-Ripka-expansion we replace occurrences of g and γ(1)  N  with their 1-  dimensional analogues as for the 2-dimensional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Here we have the bounds (using Lem-  mas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='15)  ˆ  Λ  |g| ≲ a log(b/a),  ˆ  Λ  |γ(1)  N | ≲ log N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6)  Then, the 1-dimensional analogue of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4 reads  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' There exists a constant c > 0 such that if aρ log(b/a) log N < c, then the  formulas in Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3) hold (with ρ(n)  Jas and Γn  π,G interpreted as appropriate for the 1-  dimensional setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  For the analogues of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 we have to a bit more careful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In order to get errors  smaller than the desired accuracy of the leading interaction term (of order aρ4 for the energy  density) we need to also do a Taylor expansion of (some of) the 3-particle diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (Pointwise  we only have the bound  ��Γ3  π,G  �� ≤ Caρ4 log(b/a) log N (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3 and Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2) for  any subleading diagram (π, G), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' for (π, G) ∈ ˜L3  p with p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='19 (Why this was not a problem for dimensions d = 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In dimensions d = 1, 2, 3  the analoguous bound reads  ��Γ3  π,G  �� ≤ Csadρ4 log(b/a)(log N)d (if d = 1 then there is no s) for  any subleading diagram, see Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This bound should be compared to the energy  density of the leading interaction term of order adρ2+2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Considering just the power of ρ, we  see that such terms are subleading compared to the interaction term for d ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Similarly the argument for Γ2 is also slightly different compared to that of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We  have the bounds  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' There exists a constant c > 0 such that if aρ log(b/a) log N < c, then  ������  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2p  Γ2  π,G  ������  ≤ Ca2ρ4 �  aρ(log(b/a))3(log N)3 + b4ρ4 �  1 + b2ρ2��  + Caρ5|x1 − x2|2 �  b4ρ4 + Nab6ρ7 + log(b/a)  �  and  ρ(3)  Jas ≤ Cf 2  12f 2  13f 2  23  �  ρ9|x1 − x2|2|x1 − x3|2|x2 − x3|2 + a2ρ5(log(b/a))2(log N)2  + aρ6 �  (b4ρ4 + log(b/a))  � �  |x1 − x2|2 + |x1 − x3|2 + |x2 − x3|2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  49  The proof is similar to that of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We postpone it to the end of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Note here that the N-dependence is not just via logarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, we need to be more  careful in choosing the size of the smaller boxes when applying the box method arguments of  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' With this we get the analogue of Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1) in 1 dimension,  ⟨ψN|HN|ψN⟩  = π2  3 ρ2N  �  1 + O(N−1)  �  + L  ˆ  dx  �  |∂f(x)|2 + 1  2v(x)f(x)2  �  ×  �  π2  3 ρ4|x|2 �  1 + O(N−1) + O(ρ2|x|2)  �  + O  �  aρ5|x|2�  b4ρ4 + Nab6ρ7 + log(b/a)  ��  + O  �  a2ρ4�  aρ(log b/a)3(log N)3 + b4ρ4 �  1 + b2ρ2��� �  +  ˚  dx1 dx2 dx3 f12∂f12f23∂f23f 2  13  ×  �  O(ρ9|x1 − x2|2|x1 − x3|2|x2 − x3|2) + O  �  a2ρ5(log(b/a))2(log N)2�  + O  �  aρ6 �  b4ρ4 + log(b/a)  � �  |x1 − x2|2 + |x1 − x3|2 + |x2 − x3|2��  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7)  For the 2-body error terms we may follow the type of arguments of Section 4, namely Equa-  tions (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6) exactly as for the 2-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  By using  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10 we get the bounds  ˆ �  |∂f(x)|2 + 1  2v(x)f(x)2  �  |x|n dx ≤  �  2a,  n = 2,  Ca2b,  n = 4,  ˆ  |x|nf∂f dx ≤  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  C,  n = 0,  Ca log(b/a),  n = 1,  Cab,  n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Define a0 by  1  2a0  =  ˆ �  |∂f(x)|2 + 1  2v(x)f(x)2  �  dx  and recall by assumption on v that a0 > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' that 1/a0 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the 3-body terms we may  do as for the 3-dimensional case, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the first term we bound |x1 − x3| ≤ 2b in the  support of ∂f12∂f23 and f13 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the other terms we bound f13 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By the translation  50  invariance one integration gives a volume (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' length) factor L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is,  ˚  dx1 dx2 dx3  ��f12∂f12f23∂f23f 2  13  ��  ×  �  O(ρ9|x1 − x2|2|x1 − x3|2|x2 − x3|2) + O  �  a2ρ5(log(b/a))2(log N)2�  + O  �  aρ6 �  b4ρ4 + log(b/a)  � �  |x1 − x2|2 + |x1 − x3|2 + |x2 − x3|2��  �  ≤ CNb2ρ8  �ˆ b  0  |x|2f∂f dx  �2  + CNa2ρ4(log(b/a))2(log N)2  �ˆ b  0  f∂f  �2  + CNaρ5 �  b4ρ4 + log(b/a)  � �ˆ b  0  |x|2f∂f dx  � �ˆ b  0  f∂f dx  �  + CNaρ5 �  b4ρ4 + log(b/a)  �  ��ˆ b  0  |x|2f∂f dx  � �ˆ b  0  f∂f dx  �  +  �ˆ b  0  |x|f∂f dx  �2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ≤ CNa2ρ4 �  b4ρ4 + (log(b/a))2(log N)2 + bρ  �  b4ρ4 + log(b/a)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We conclude the analogue of Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7) in dimension 1  ⟨ψN|HN|ψN⟩  L  = π2  3 ρ3 + 2π2  3 aρ4 + O  �  N−1ρ3�  + O  �  a2b−1ρ4�  + O  �  a2bρ6�  + O  �  a2ρ4a−1  0  �  aρ(log(b/a))3(log N)3 + b4ρ4 �  1 + b2ρ2���  + O  �  a2ρ5 �  b4ρ4 + Nab6ρ7 + log(b/a)  ��  + O  �  a2ρ5 �  b4ρ4 + (log(b/a))2(log N)2 + bρ  �  b4ρ4 + log(b/a)  ���  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8)  We need to be careful how we choose N (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' how we choose L), since the error depends on N  not just via logarithmic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We choose  N = (aρ)−α,  α > 1  b = a(aρ)−β,  0 < β < 1  where the bounds on α, β are immediate for all the error-terms to be smaller than the desired  accuracy (there is similarly also an upper limit for α, which we do not write).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Keeping then  only the leading error terms we get  ⟨ψN|HN|ψN⟩  L  = π2  3 ρ3 + 2π2  3 aρ4 + O  �  N−1ρ3�  + O  �  a2b−1ρ4�  + O  �  a2a−1  0 b4ρ8�  + O  �  Na3b6ρ12�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9)  Using the box method similarly as in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 we also have to be careful with how we choose  the parameter d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' As in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9) we get  e(˜ρ) ≤ ⟨ψn|Hn|ψn⟩  ℓ  [1 + O(d/ℓ) + O(b/ℓ)] + O  �  ρd−2�  ≤ π2  3 ρ3 + 2π2  3 aρ4 + O  �  n−1ρ3�  + O  �  a2b−1ρ4�  + O  �  a2a−1  0 b4ρ8�  + O  �  na3b6ρ12�  + O  �  dℓ−1ρ3�  + O  �  bℓ−1ρ3�  + O  �  ρd−2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Here we change notation from N to n and choose d = a(aρ)−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' To get the error smaller than  desired, we see that we need to choose δ > 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In particular then the error is O(ρ3(aρ)γ),  where  γ = min{1 + β, 5 − 4β, 9 − α − 6β, α + 1 − δ, 2δ − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  51  Then, also ˜ρ = ρ (1 + O((aρ)γ)) so ρ = ˜ρ(1 + O((a˜ρ)γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Optimising in α, β, δ we see that for  α = 12  5 ,  β = 13  17,  δ = 32  17  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10)  we get γ = 30/17, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  e(˜ρ) ≤ π2  3 ˜ρ3 + 2π2  3 a˜ρ4 �  1 + O  �  (a˜ρ)13/17��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This concludes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  It remains to give the  Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note first that, completely analogously to Equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='22),  we have  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �����������  �  (π,G)∈L2  p  ng(π,G)+n∗  g(π,G)=ng0  k(π,G)=k0  Γ2  π,G  �����������  ≤ Cρ2(C log N)k0(Caρ log(b/a))k0+ng0,  p = 2k0 + ng0  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ������������  �  (π,G)∈ ˜L3  p  ng(π,G)+n∗  g(π,G)=ng0  k(π,G)=k0  Γ3  π,G  ������������  ≤ Cρ3(C log N)k0(Caρ log(b/a))k0+ng0,  p = 2k0 + ng0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11)  We will use this to split the diagrams of L2  p and ˜L3  p into groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We split diagrams in L2  p into  three (exhaustive) groups:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Small diagrams with 1 ≤ k + ng + n∗  g ≤ 2, {1} and {2} in different clusters  (A) and k ≥ 1,  (B) and k = 0, n∗  g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Small diagrams with 1 ≤ k + ng + n∗  g ≤ 2 and  (A) {1} and {2} in different clusters and k = 0, n∗  g = 2,  (B) {1} and {2} in the same cluster,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Large diagrams with k + ng + n∗  g ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We then split  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2p  Γ2  π,G = ξsmall,0 + ξsmall,≥1 + ξ≥1,  where ξsmall,0 is the contribution of all small diagram in the first group, ξsmall,≥1 is the contribu-  tion of all small diagrams in the second group and ξ≥1 is the contribution of all large diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We will then do a Taylor expansion of ξsmall,0 but not of the other terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We split diagrams in ˜L3  p into three (exhaustive) groups:  52  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Small diagrams with k + ng + n∗  g = 1 and {1}, {2} and {3} in 3 different clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (Then  ng = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Small diagrams with k + ng + n∗  g = 1 and {1}, {2} and {3} in < 3 different clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (Then k = ng = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Large diagrams with k + ng + n∗  g ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We then split  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈ ˜L3p  Γ3  π,G = ξ3  small,0 + ξ3  small,≥1 + ξ3  ≥1,  where ξ3  small,0 is the contribution of all small diagram in the first group, ξ3  small,≥1 is the contribu-  tion of all small diagrams in the second group and ξ3  ≥1 is the contribution of all large diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Again, we do a Taylor expansion of ξ3  small,0 but not of the other terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For simplicity we will  only compute the derivatives ∂2  x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' With this bound the error term for the energy density is  O(a2bρ6 log(b/a)) and so it is even smaller than the accuracy a2ρ5 with b chosen as in Equa-  tion (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (By the symmetry, we could bound ξsmall,0 by bounding its 6th derivative ∂2  x1∂2  x2∂2  x3  instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') To keep the result symmetric in x1, x2, x3 we will symmetrize the result afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We have immediately by Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11) that  |ξ≥1| ≤ Ca3ρ5(log(b/a))3(log N)3,  ��ξ3  ≥1  �� ≤ Ca2ρ5(log(b/a))2(log N)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12)  Similarly as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 we have for x1 = x2  ξsmall,0(x2, x2) + ξsmall,≥1(x2, x2) + ξ≥1(x2, x2) = 0,  ξ3  small,0(x2, x2, x3) + ξ3  small,≥1(x2, x2, x3) + ξ3  ≥1(x2, x2, x3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Hence we may bound the zeroth order by  |ξsmall,0(x2, x2)| ≤ |ξsmall,≥1(x2, x2)| + |ξ≥1(x2, x2)| ,  ��ξ3  small,0(x2, x2, x3)  �� ≤  ��ξ3  small,≥1(x2, x2, x3)  �� +  ��ξ3  ≥1(x2, x2, x3)  �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For the diagrams in ξsmall,0 and ξ3  small,0 we have similarly to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8 that  ��∂2  x1ξsmall,0  �� ≤ Caρ5 log(b/a),  ��∂2  x1ξ3  small,0  �� ≤ Caρ6 log(b/a)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13)  uniformly in x1, x2, x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For the diagrams in ξsmall,≥1 and ξ3  small,≥1 the analysis is somewhat  similar to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We have  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the small diagrams in ξsmall,≥1 and ξ3  small,≥1 we have the bounds  |ξsmall,≥1| ≤ Ca2b4ρ8(1 + b2ρ2) + Cab4ρ9|x1 − x2|2 �  1 + Nab2ρ3�  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='14)  ��ξ3  small,≥1  �� ≤ Cab4ρ10 �  |x1 − x2|2 + |x1 − x3|2 + |x2 − x3|2�  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='15)  uniformly in x1, x2, x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We give the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='21 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Combining Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='21 and Equa-  tions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13) concludes the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Acknowledgements  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' would like to thank Johannes Agerskov and Jan Philip Solovej for valuable discus-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We thank Alessandro Giuliani for helpful discussions and for pointing out the reference  [GMR21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Funding from the European Union’s Horizon 2020 research and innovation pro-  gramme under the ERC grant agreement No 694227 is acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  53  A  Small diagrams  In this appendix we compute the contributions of all the small diagrams of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8,  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We first consider those of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1  Small 2-particle diagrams (proof of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8)  Recall from the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 that  ξsmall,0 + ξsmall,≥1 =  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈L2  p  (π,G) small  Γ2  π,G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The criterion for being small is defined in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 around Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11),  and will be recalled below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The diagrams are split into types (A), (B) and (C) according their  underlying graphs G as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We further split the type (B) into two types  (B1) and (B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The diagrams of type (B1) are those diagrams for which the extra vertex {3} in  the distinguished clusters is in the cluster containing {1}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' connected to {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The diagrams  of type (B2) are those diagrams for which the extra vertex {3} is in the cluster containing {2},  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' connected to {2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is, the different types are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' See also Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (A) {1} and {2} in different clusters and 1 ≤ k ≤ 4, ng = 0, n∗  g = 0,  (B) {1} and {2} in different clusters and 0 ≤ k ≤ 2, ng = 0, n∗  g = 1,  (B1) and n∗ = 1, n∗∗ = 0,  (B2) and n∗ = 0, n∗∗ = 1,  (C) {1} and {2} in the same cluster and 0 ≤ k ≤ 2, ng = 0, n∗  g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  k  1  2  (a) Type (A), 1 ≤ k ≤ 4  k  1  2  3  (b) Type (B1), 0 ≤ k ≤ 2  k  1  2  3  (c) Type (B2), 0 ≤ k ≤ 2  k  1  2  3  (d) Type (C), 0 ≤ k ≤ 2  Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1: g-graphs of small diagrams of different types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For each diagram only  the graph G is drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The relevant diagrams come with permutations π such that  the diagrams are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We first give the  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Consider first all diagrams of type (C) of smallest size, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' with g-graph  G0 =  1  2  54  Since this graph is connected, all π ∈ S3 give rise to a linked diagram (π, G0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By Wick’s rule,  the π-sum then gives the factor ρ(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is,  �  π∈S3:(π,G0)∈L2  1  Γ2  π,G0 =  ˆ  g13g23  �  π∈S3  (−1)π  3  �  j=1  γ(1)  N (xj, xπ(j)) dx3 =  ˆ  g13g23ρ(3) dx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Recall the bound ρ(3) ≤ Cρ5|x1 − x2|2|x1 − x3|2|x2 − x3|2 from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Now we bound  |g23| ≤ 1 and |x2 − x3| ≤ b in the support of g23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus  ������  �  π∈S3:(π,G0)∈L2  1  Γ2  π,G0  ������  ≤ Cb2ρ5|x1 − x2|2  ˆ �  1 − f(x)2�  |x|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Recalling Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 we may bound  ˆ �  1 − f(x)2�  |x|2 dx ≤ Ca5 +  C  (1 − a3/b3)2  ˆ b  a  ��  1 − a3  b3  �2  −  �  1 − a3  r3  �2�  r4 dr ≤ Ca3b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1)  We conclude that all diagrams of smallest size contribute ≤ Ca3b4ρ5|x1 − x2|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For the larger diagrams, we consider an example diagram  (π, G) =  1  2  3  For this diagram we have  Γ2  π,G = (−1)π  ˚  γ(1)  N (x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' x4)γ(1)  N (x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' x3)γ(1)  N (x3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' x1)γ(1)  N (x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' x5)γ(1)  N (x5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' x2)g13g23g45 dx3 dx4 dx5  = −1  L15  �  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k5∈PF  ˚  eik1(x1−x4)eik2(x4−x3)eik3(x3−x1)eik4(x2−x5)eik5(x5−x2)g13g23g45 dx3 dx4 dx5  = −1  L15  �  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k5∈PF  ei(k1−k3)x1ei(k4−k5)x2  ˆ  dx3 ei(k3−k2)x3g(x1 − x3)g(x2 − x3)  ×  ˆ  dx4  �  ei(k2−k1+k5−k4)x4  ˆ  dx5 e−i(k5−k4)(x4−x5)g(x4 − x5)  �  = −1  L12  �  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k5∈PF  ei(k1−k3)x1ei(k4−k5)x2  ˆ  dx3 ei(k3−k2)x3g(x1 − x3)g(x2 − x3)  × χ(k2−k1=k4−k5)ˆg(k5 − k4),  where ˆg(k) :=  ´  Λ g(x)e−ikx dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Bounding |g23| ≤ 1 and |ˆg(k)| ≤  ´  |g| ≤ a3 log(b/a) we get that  ��Γ2  π,G  �� ≤ Ca6ρ4(log(b/a))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  One may do a similar computation for all the remaining diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By computing the integra-  tions of the vertices in the internal clusters first, these give some factor ˆg(ki − kj) and a factor  L3χ(ki−kj=ki′−kj′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By bounding as above we conclude that the contribution of small diagrams  of type (C) is bounded as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  55  1  2  (a) Example of a type (A) di-  agram of smallest size  1  2  3  (b) Example of a type (B1) di-  agram of smallest size  1  2  3  (c) Example of a type (B2) di-  agram of smallest size  Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2: Exemplary diagrams of types (A), (B1) and (B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The dashed lines  denote g-edges, and the arrows denote (directed) edges of the permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' As with the larger diagrams of type (C) we only give calculations for a  few example diagrams and explain how the calculation for the remaining diagrams are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We consider the examples in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The contribution of the diagram in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2a to ∂µ  x1∂ν  x1ξsmall is  1  2∂µ  x1∂ν  x1Γ2  π,G  = −1  2L12  �  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k4∈PF  (kµ  1 − kµ  2)(kν  1 − kν  2)  ¨  eik1(x1−x3)eik2(x3−x1)eik3(x2−x4)eik4(x4−x2)g34 dx3 dx4  = −1  2L12  �  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k4∈PF  (kµ  1 − kµ  2)(kν  1 − kν  2)ei(k1−k2)x1ei(k3−k4)x2  ×  ¨  e−i(k4−k3)(x3−x4)ei(k2−k1+k4−k3)x3g(x3 − x4) dx3 dx4  = −1  2L9  �  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k4∈PF  (kµ  1 − kµ  2)(kν  1 − kν  2)ei(k1−k2)x1ei(k3−k4)x2χ(k2−k1=k3−k4)ˆg(k4 − k3)  = O(ρ3+2/3a3 log(b/a))  using that ˆg(k) =  ´  Λ g(x)e−ikx dx satisfies |ˆg(k)| ≤  ´  |g| ≤ Ca3 log(b/a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The same type of  computation is valid for all other diagrams of type (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Consider now the diagram in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2c of type (B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This contributes  ∂µ  x1∂ν  x1  −1  L9  �  k1,k2,k3∈PF  ˆ  eik1(x1−x2)eik2(x2−x1)g(x2 − x3) dx3  = 1  L9  �  k1,k2,k3∈PF  (kµ  1 − kµ  2)(kν  1 − kν  2)ei(k1−k2)x1ei(k2−k1)x2  ˆ  g(x2 − x3) dx3  = O(ρ3+2/3a3 log(b/a))  exactly as for type (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Similarly, all other diagrams of type (B2) may be bounded using the  same method as for types (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  56  Finally, we consider the diagram in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2b of type (B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Here we have  ∂µ  x1∂ν  x1Γ2  π,G = ∂µ  x1∂ν  x1  1  L9  �  k1,k2,k3∈PF  ˆ  eik1(x1−x3)eik2(x3−x2)eik3(x2−x1)g(x1 − x3) dx3  = ∂µ  x1∂ν  x1  1  L9  �  k1,k2,k3∈PF  ei(k2−k3)x1ei(k3−k2)x2  ˆ  e−i(k2−k1)(x1−x3)g(x1 − x3) dx3  = −1  L9  �  k1,k2,k3∈PF  (kµ  2 − kµ  3)(kν  2 − kν  3)ei(k2−k3)x1ei(k3−k2)x2ˆg(k2 − k1)  = O(ρ3+2/3a3 log(b/a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  All larger diagrams of type (B1) may be bounded similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We conclude the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2  Small 3-particle diagrams (proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11)  We now give the  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Recall that  ξ3  small =  ∞  �  p=1  1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  �  (π,G)∈ ˜L3  p  (π,G) small  Γ3  π,G,  where “small” refers to diagrams with G-graph  G =  k = 1, 2  k  1  2  3  and permutation π such that (π, G) has at most two linked components, both of which contain  at least one external vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  As in the proof of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8 in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 we  compute the value of a few examples and explain how to compute the value of the remaining  diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We consider the examples of Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3  1  2  3  (a) Example of a diagram of smallest size  with one linked component  1  2  3  (b) Example of a diagram of smallest size  with two linked components  Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3: Exemplary small diagrams in ˜L3  2  The contribution of the diagram in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3a is  Γ3  π,G = −1  L15  �  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k5∈PF  ¨  eik1(x1−x4)eik2(x4−x2)eik3(x2−x1)eik4(x3−x5)eik5(x5−x3)g45 dx4 dx5  = −1  L12  �  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k5∈PF  ei(k1−k3)x1ei(k3−k2)x2ei(k4−k5)x3χ(k2−k1=k4−k5)ˆg(k5 − k4)  = O(a3ρ4 log(b/a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  57  Similarly, the contribution of the diagram in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3b is  Γ3  π,G = −1  L15  �  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k5∈PF  ¨  eik1(x1−x4)eik2(x4−x5)eik3(x5−x1)eik4(x2−x3)eik5(x3−x2)g45 dx4 dx5  = −1  L12  �  k1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',k5∈PF  ei(k1−k3)x1ei(k4−k5)x2ei(k5−k4)x3χ(k2−k1=k2−k3)ˆg(k3 − k2)  = O(a3ρ4 log(b/a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  One may follow this kind of computation for any diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The central property we used is  that the internal vertices are all in the same linked component as some external vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This  means that the integrals over internal vertices either gives a factor of ˆg(ki − kj) or a factor of  L3χ(ki−kj=ki′−kj′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We conclude the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3  Small diagrams in 1 dimension (proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='21)  We now give the  Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We first give the proof of Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We split the two cases (A)  and (B) of small diagrams further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' They are given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (A) {1} and {2} in different clusters and k = 0, n∗  g = 2,  (A1) n∗ = 2, n∗∗ = 0 (or n∗ = 0, n∗∗ = 2),  (A2) n∗ = 1, n∗∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (B) {1} and {2} in the same cluster and 1 ≤ k + ng + n∗  g ≤ 2,  (B1) k = 0,  (B2) k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  See also Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  1  2  (a) Type (A1)  1  2  (b) Type (A2)  1  2  (c) Type (B1)  1  2  (d) Type (B2)  Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4: g-graphs of small diagrams of different types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For each diagram only  the graph G is drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The relevant diagrams come with permutations π such the  the diagrams are linked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The diagrams of type (A1) and (B1) may have some of  the drawn g-edges not present, but the same connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Moreover,  the diagrams of type (B1) may have one of the internal vertices drawn not present  (indicated by a ◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' With the modification of the drawings described here these are  all small diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We will consider some examples of diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Namely those drawn in Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4 (but not  modified as described in the caption), except for the diagram of type (B1), where we will  consider diagrams of smallest size, with g-graph  1  2  G0 =  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2)  58  All other diagrams can be treated in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the argument we will need a different  formula for ρt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Recall the definition in Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We may write the characteristic function  as  χ((π, ∪Gℓ) linked) = 1 − χ((π, ∪Gℓ) not linked).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  That is,  ρ(A1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=',Ak)  t  =  �  π∈S∪Aℓ  (−1)π �  j∈∪Aℓ  γ(1)  N (xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xπ(j)) −  �  π∈S∪Aℓ  (−1)πχ((π,∪Gℓ) not linked)  �  j∈∪Aℓ  γ(1)  N (xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xπ(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For our case we only need to consider cases where there are at most two clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' If there is  just one cluster then ρ(A)  t  = ρ(|A|)((xj)j∈A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' So suppose we have two clusters A1, A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Here, all  the π’s for which (π, ∪Gℓ) is not linked are exactly those arising as products π = π1π2, where  π1 ∈ SA1 and π2 ∈ SA1 are permutations of the vertices in the 2 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus,  ρ(A1,A2)  t  ((xj)j∈A1∪A2) =  �  π∈SA1∪A2  (−1)π  �  j∈A1∪A2  γ(1)  N (xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xπ(j))  −  �  π1∈SA1  (−1)π1 �  j∈A1  γ(1)  N (xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xπ1(j))  �  π2∈SA2  (−1)π2 �  j∈A2  γ(1)  N (xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' xπ2(j))  = ρ(|A1|+|A2|)((xj)j∈A1∪A2) − ρ(|A1|)((xj)j∈A1)ρ(|A2|)((xj)j∈A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3)  We now consider the diagrams in Figures A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4a, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4b and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4d and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We get  Type (A1) :  �  π∈S4:(π,G0)∈L2  2  Γ2  π,G0 =  ¨  g13g14g34ρ({1,3,4},{2})  t  dx3 dx4,  Type (A2) :  �  π∈S4:(π,G0)∈L2  2  Γ2  π,G0 =  ¨  g13g24ρ({1,3},{2,4})  t  dx3 dx4,  Type (B1) :  �  π∈S3:(π,G0)∈L2  1  Γ2  π,G0 =  ˆ  g13g23ρ(3) dx3,  Type (B2) :  �  π∈S5:(π,G0)∈L2  3  Γ2  π,G0 =  ˚  g13g23g45ρ({1,2,3},{4,5})  t  dx3 dx4 dx5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4)  Using Equation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3) and (the 1-dimensional versions of) Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='14 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='15 and simi-  lar bounds for the 4- and 5-particle reduced densities we get the bounds on the truncated  correlations  (A1)  ���ρ({1,3,4},{2})  t  ��� ≤ ρ(4)(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , x4) + ρ(3)(x1, x3, x4)ρ(1)(x2)  ≤ Cρ10|x1 − x3|2|x1 − x4|2|x3 − x4|2,  (A2)  ���ρ({1,3},{2,4})  t  ��� ≤ ρ(4)(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , x4) + ρ(2)(x1, x3)ρ(2)(x2, x4)  ≤ Cρ8|x1 − x3|2|x2 − x4|2,  (B1)  ρ(3) ≤ Cρ9|x1 − x2|2|x1 − x3|2|x2 − x3|2,  (B2)  ���ρ({1,2,3},{4,5})  t  ��� ≤ ρ(5)(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , x5) + ρ(3)(x1, x2, x3)ρ(2)(x4, x5)  ≤ Cρ13|x1 − x2|2|x1 − x3|2|x2 − x3|2|x4 − x5|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  59  Bounding moreover, g34|x3−x4|2 ≤ b2 for the diagram of type (A1) we thus get by the translation  invariance  ��������  �  π∈S4:(π,G0)∈L2  2  type (A1)  Γ2  π,G0  ��������  ≤ Cb2ρ10  �ˆ  |g(x)||x|2 dx  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For the diagram of type (A2) we get  ��������  �  π∈S4:(π,G0)∈L2  2  type (A2)  Γ2  π,G0  ��������  ≤ Cρ8  �ˆ  |g(x)||x|2 dx  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For the diagram of type (B1) we get by bounding g23|x2−x3|2 ≤ b2 (as in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6)  ��������  �  π∈S3:(π,G0)∈L2  1  type (B1)  Γ2  π,G0  ��������  ≤ Cb2ρ9|x1 − x2|2  ˆ  |g(x)||x|2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Finally, for the diagram of type (B2) we get in the same way  ��������  �  π∈S5:(π,G0)∈L2  3  type (B2)  Γ2  π,G0  ��������  ≤ CNb2ρ12|x1 − x2|2  �ˆ  |g(x)||x|2 dx  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We may bound  ´  |x|2|g| dx similarly as in 3 and 2 dimensions,  ˆ  R  �  1 − f(x)2�  |x|2 dx ≤ Ca3 +  C  (1 − a/b)2  ˆ b  a  ��  1 − a  b  �2  −  �  1 − a  r  �2�  r2 dr ≤ Cab2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The other diagrams of types (A1) and (B1) (there are no other diagrams of type (A2) or (B2))  we may treat similarly by bounding some of the g-edges by |g| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Combining these bounds  we conclude the proof of Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  To prove Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='15) we recall that we consider all diagrams with g-graph  G0 =  1  2  3  or  G1 =  1  2  3  (and graphs that look like G0 where {1, 2, 3} are permuted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' One may treat this similarly as  the diagrams above, with the result that  ������  �  π∈S4:(π,G0)∈L3  1  Γ3  π,G0  ������  ≤  ˆ  |g14||g24|  ���ρ({1,2,4},{3})  t  ��� dx4 ≤ Cab4ρ10|x1 − x2|2  ������  �  π∈S4:(π,G1)∈L3  1  Γ3  π,G1  ������  ≤  ˆ  |g14||g24||g34|ρ(4) dx4 ≤ Cab4ρ10|x1 − x2|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Summing this over all the permutations of {1, 2, 3} we conclude the proof of Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  60  B  Derivative Lebesgue constants (proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9)  In this appendix we give the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We recall the statement in slightly different  notation for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The polyhedron P from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7 satisfies for any µ, ν = 1, 2, 3 that  ˆ  [0,2π]3  �����  �  k∈RP ∩Z3  kµeikx  ����� dx ≤ CsR(log R)3,  ˆ  [0,2π]3  �����  �  k∈RP ∩Z3  kµkνeikx  ����� dx ≤ CsR2(log R)4  for sufficiently large R = LkF  2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Recall that by construction R ∼ N1/3 is rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The proof follows quite closely the argument in [KL18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In particular the structure is that  of induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The 3-dimensional integral is bounded one dimension at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We start by  introducing some notation from [KL18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Notation B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For any real number x we will write [x] for either ⌊x⌋ or ⌈x⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Similarly we will  write ⟨x⟩ = x − [x], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' ⟨x⟩ is either the fractional part {x} = x − ⌊x⌋ or x − ⌈x⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For any  computation we do below, the definition of [x] is fixed, but the computations hold with either  choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Additionally for a d-dimensional vector x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , xd) we write x( ˜d) = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , x  ˜d) for the  first ˜d ≤ d components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We emphasize that expressions like k2, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' do not denote squares or cubes of numbers  k, x, but instead refer to coordinates of vectors k, x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The instances where we do want to denote  a square, cube or higher power should be clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  By potentially relabelling the coordinates it suffices to consider the cases µ = 1, µ = ν = 1  and µ = 1, ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (Alternatively, by appealing to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11 and choosing Q ≳ N4 in  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7 we have a symmetry of coordinates up to error-terms which are subleading  compared to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Hence define  t1(k) = k1,  t2(k) = k1k1 = (k1)2,  t3(k) = k1k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We want to show that  ˆ  [0,2π]3  �����  �  k∈RP ∩Z3  tj(k)eikx  ����� dx ≤  �  CsR(log R)3  j = 1,  CsR2(log R)4  j = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  As in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12 we write RP as a union of O(s) closed tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We also recall  that Rz /∈ Z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' As in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12 we get by the inclusion exclusion principle O(s)  terms with tetrahedra of lower dimension (triangles or line segments).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' All the 3-dimensional  (closed) tetrahedra are convex and hence of the form  T =  �  k ∈ Z3 : λ1 ≤ k1 ≤ Λ1, λ2(k1) ≤ k2 ≤ Λ2(k1), λ3(k1, k2) ≤ k3 ≤ Λ3(k1, k2)  �  ,  for some piecewise affine functions λi, Λi, i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  They are the equations of the planes  bounding the tetrahedron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Since any k ∈ T has integer coordinates we can replace Λj by  ⌊Λj⌋ and λj by ⌈λj⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' It will be convenient to not distinguish between ⌊·⌋ and ⌈·⌉ and use  instead the notation [·] introduced in Notation B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then the tetrahedra are of the form  T =  �  k ∈ Z3 : [λ1] ≤ k1 ≤ [Λ1], [λ2(k1)] ≤ k2 ≤ [Λ2(k1)], [λ3(k1, k2)] ≤ k3 ≤ [Λ3(k1, k2)]  �  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1)  61  where we allow [·] to be different in any of the 6 instances it appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Sums over lower-dimensional tetrahedra can be written as differences of sums over 3-  dimensional tetrahedra (with potentially different meanings of [·]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We will thus only consider  3-dimensional tetrahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is, for a tetrahedron T of the form Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1), we need to  bound  ˆ  [0,2π]3  �����  �  k∈T∩Z3  tj(k)eikx  ����� dx ≤  �  CR(log R)3  j = 1,  CR2(log R)4  j = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2)  Gluing together tetrahedra as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12 we conclude the desired bound, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The  remainder of this section gives the proof of Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1  Reduction to simpler tetrahedron  We first reduce to the case of a simpler tetrahedron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Consider what happens by shifting all  k’s by some fixed lattice vector κ ∈ Z3 with |κ| ≤ CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For t2 we have  �  k∈T∩Z3  (k1)2eikx =  �  k∈(T−κ)∩Z3  (k1 + κ1)2eikx  =  �  k∈(T−κ)∩Z3  (k1)2eikx + 2κ1  �  k∈(T−κ)∩Z3  k1eikx + (κ1)2  �  k∈(T−κ)∩Z3  eikx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  A similar computation holds for t1, t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We may bound |κ| ≤ CR and thus we may assume that  T ⊂ [0, CR]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (Recall that  ´  [0,2π]3  ���  k∈T∩Z3 eikx�� dx ≤ C(log R)3 by [KL18, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1], see  the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  For any tetrahedron of the form (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1) we may write the k-sum as three 1-dimensional sums  �  k∈T∩Z3  =  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  [Λ3(k1,k2)]  �  k3=[λ3(k1,k2)]  =  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  \uf8eb  \uf8ed  [Λ3(k1,k2)]  �  k3=0  −  [λ3(k1,k2)−1]  �  k3=0  \uf8f6  \uf8f8 ,  where the λj’s and Λj’s are the equations of the planes bounding the tetrahedron T, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' piece-  wise affine functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' As in Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1) each instance of [·] may be either of the definitions  of Notation B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By splitting the k1, k2 sums into at most 4 parts, we may ensure that both Λ3  and λ3 − 1 are only from one bounding plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' they are affine functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' When we do this  splitting, we have to choose (in each new tetrahedron) which definition of [·] to use for the new  bounding plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This may give rise to some “boundary term”, if we choose definitions of [·] in  the new tetrahedra such that the k’s on the splitting face are either in both or in neither of  the two tetrahedra sharing this face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' These boundary terms are sums over lower-dimensional  tetrahedra, and may thus be bounded by sums over 3-dimensional ones as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' One may similarly let the k1- and k2-sums go from 0 by writing e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  [Λ2(k1)]  �  k2=[λ2(k1)]  =  [Λ2(k1)]  �  k2=0  −  [λ2(k1)−1]  �  k2=0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  However, the upper limits Λ3(k1, k2) and λ3(k1, k2)−1 for the k3-sum may become much larger  than R for k2 ≤ λ2(k1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This is why we don’t do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The terms with Λ3 and λ3 − 1 may be treated the same way, so we just look at the one with  Λ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We thus want to bound  ˆ  [0,2π]3  ������  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  tj(k1, k2)  [Λ3(k1,k2)]  �  k3=0  eikx  ������  dx ≲  �  R(log R)3  j = 1,  R2(log R)4  j = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  62  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2  Reduction from d = 3 to d = 2  We show that we may bound the three-dimensional integrals by analogous two-dimensional  integrals up to a factor of (log R + log Q) ∼ log N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  First, before shifting by a constant κ ∈ Z3, Λ3 is given by either the plane through 3 close  corners of RP (points Rσ(p1/Q1, p2/Q2, p3/Q3)) or of two close corners and the centre Rz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This  follows from the construction of P in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7, since forming the edges between pairs of  close points constructs a triangulation of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The equation for a plane through the three points Rσ(p1  j/Q1, p2  j/Q2, p3  j/Q3), j = 1, 2, 3 is  given by  α1  Q2Q3  k1 +  α2  Q1Q3  k2 +  α3  Q1Q2  k3 = Rσγ  where by construction of P, see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7, we have  σ /∈ Q,  γ ∈ Q,  αj ∈ Z,  |αj| ≤ C  �  Q,  j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We might have that αj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' If α3 = 0 then this plane is parallel to the k3-axis and so does not  give rise to a bound on the k3-sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Hence α3 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By choice of L, we have that R is rational,  and so Rσγ /∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (The choice of L such that R is rational, is exactly so that Rσγ /∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') The  equation for Λ3 is an integer shift of this plane, hence it is of the form  Λ3(k1, k2) = n3 − m1k1 − m2k2 = n3 − Q1α1  Q3α3  k1 − Q2α2  Q3α3  k2,  n3 /∈ Q,  |αj| ≤ C  �  Q, j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3)  Define for j = 1, 2, 3 the quantities  Dj  3(x) :=  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  tj(k1, k2)  [Λ3(k1,k2)]  �  k3=0  eik(3)x(3),  ˜Dj  2(x) :=  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  tj(k1, k2)eik(2)x(2),  Gj  3(x) :=  1  eix3 − 1  �  ei(n3+1)x3 ˜Dj  2(x(2) − m(2)x3) − ˜Dj  2(x(2))  �  ,  F j  3 (x) := ei(n3+1)x3  eix3 − 1  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  tj(k1, k2)eik(2)(x(2)−m(2)x3) �  e−i⟨Λ3(k(2))⟩x3 − 1  �  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4)  where m(2) = (m1, m2) is defined in Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We shall prove the following bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We have for some k(2)  0  ∈ Z2, some (non-zero) κ = κ(2) ∈ Z2 and a h ∈ Z, h ≥ 0  with |k(2)  0 | ≤ CR and h|κ(2)| ≤ CR that for any j = 1, 2, 3  ˆ  [0,2π]3  ��Dj  3(x(3))  �� dx(3)  ≲ (log R + log Q)  ˆ  [0,2π]2  ��� ˜Dj  2(x(2))  ��� dx(2) + 1 +  ˆ 2π  0  �����  h  �  τ=0  tj  �  k(2)  0  + τκ(2)�  eiτ|κ(2)|x1  ����� dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  63  As a first step, consider the case where both α1 = α2 = 0 in Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then the k3-sum  and x3-integral in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3 factors out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Using [KL18, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2] to evaluate the k3-sum  and x3-integral we conclude the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Hence we can assume that at most one of α1, α2 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (This will be relevant for Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8, but only then.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  A simple calculation shows that [KL18, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1]  Dj  3(x) = Gj  3(x) + F j  3(x),  j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5)  By a straightforward modification of the argument in [KL18, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3] (including the factor  tj) we have  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4 ([KL18, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For any j = 1, 2, 3 we have  ˆ  [0,2π]3  ��Gj  3(x)  �� dx ≲ log R  ˆ  [0,2π]2  ��� ˜Dj  2(x(2))  ��� dx(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We thus want to bound the integral of F j  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Again, by a straightforward modification of the  argument in [KL18, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7] (including the factor tj) we have  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5 ([KL18, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For any j = 1, 2, 3 we have  ˆ  [0,2π]3  ��F j  3 (x)  �� dx ≲  ∞  �  r=1  (2π)r  r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ˆ  [0,2π]2  ������  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  tj(k1, k2)eik(2)x(2) �  Λ3(k(2))  �r  ������  dx(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  To bound the right hand side of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5 we bound either definition of ⟨·⟩ by the fractional  part {·}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This follows the strategy in [KL18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In analogy with [KL18, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6] we have  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6 ([KL18, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For either definition of ⟨·⟩ we have the bound  ˆ  [0,2π]2  ������  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  tj(k1, k2)eik(2)x(2) �  Λ3(k(2))  �r  ������  dx(2)  ≤  ˆ  [0,2π]2  ��� ˜Dj  2(x(2))  ��� dx(2) +  r  �  ν=1  �r  ν  � ˆ  [0,2π]2  ������  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  tj(k1, k2)eik(2)x(2){Λ3(k(2))}ν  ������  dx(2)  uniformly in (integer) r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' If ⟨·⟩ = {·} this is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Hence suppose that ⟨x⟩ = x − ⌈x⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then  ⟨x⟩ = {x} − 1 +  �  1  if x ∈ Z  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  By construction Λ3(k1, k2) /∈ Z for k1, k2 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, ⟨Λ3(k1, k2)⟩ = {Λ3(k1, k2)} − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  tj(k1, k2)eik(2)x(2) �  Λ3(k(2))  �r  =  r  �  ν=0  (−1)r−ν  �r  ν  �  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  tj(k1, k2)eik(2)x(2){Λ3(k(2))}ν  = (−1)r ˜Dj  2(x(2)) +  r  �  ν=1  (−1)r−ν  �r  ν  �  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  tj(k1, k2)eik(2)x(2){Λ3(k(2))}ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  64  We now bound the second summand of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6 similarly to [KL18, Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We first define ˜Λ3, a rational approximation of Λ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Recall the definition of Λ3 in Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  By Dirichlet’s approximation theorem we may for any Q∞ find integers p, q with 1 ≤ q ≤ Q∞  such that  γ3 := n3 − p  q  satisfies  |γ3| <  1  qQ∞  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We will choose Q∞ = Q3α3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Define then  ˜Λ3(k(2)) = Λ3(k(2)) − γ3 = p  q − Q1α1  Q3α3  k1 − Q2α2  Q3α3  k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6)  Note that this takes values in  1  qQ∞Z for integers k1, k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  In particular (for integers k1, k2)  {˜Λ3(k(2))} ∈ {0,  1  qQ∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , qQ∞−1  qQ∞ }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, since |γ3| <  1  qQ∞ we have  {Λ3(k(2))} = γ3 + {˜Λ3(k(2))} +  �  1  if γ3 < 0 and ˜Λ3(k(2)) ∈ Z,  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7)  We claim that  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For N sufficiently large, we have uniformly in (integer) r ≥ 1 that  ˆ  [0,2π]2  ������  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  tj(k1, k2)eik(2)x(2){Λ3(k(2))}r  ������  dx(2)  ≲ log(rQ)  ˆ  [0,2π]2  ��� ˜Dj  2(x(2))  ��� dx(2) +  ˆ  [0,2π]2  ���������  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  ˜Λ3(k1,k2)∈Z  tj(k1, k2)eik(2)x(2)  ���������  dx(2) + 2r  The proof differs from that of [KL18, Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9] in a few key location, so we give it  here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Using Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7) we have  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  tj(k1, k2)eik(2)x(2){Λ3(k(2))}r  =  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  tj(k1, k2)eik(2)x(2) �  γ3 + {˜Λ3(k(2))}  �r  + χ(γ3<0)  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  ˜Λ3(k1,k2)∈Z  tj(k1, k2)eik(2)x(2) + mixed terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  All the mixed terms have at least one power of γ3 + {˜Λ3(k(2))} = γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (Indeed, in the mixed  terms we have ˜Λ3(k(2)) ∈ Z so {˜Λ3(k(2))} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Since |γ3| < 1/(qQ∞) ≤ 1/Q the sum of all  mixed terms may be bounded by 2rR4Q−1 ≲ 2r for N sufficiently large (independent of r) by  65  our choice of Q, see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Similarly expanding the first summand, all the terms with  at least one power of γ3 may be bounded the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We thus have  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  tj(k1, k2)eik(2)x(2){Λ3(k(2))}r  =  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  tj(k1, k2)eik(2)x(2){˜Λ3(k(2))}r  + χ(γ3<0)  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  ˜Λ3(k1,k2)∈Z  tj(k1, k2)eik(2)x(2) + O(2r),  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8)  where the error is O(2r) uniform in x(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the first summand we have by a simple modification  of [KL18, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8] (including the factor tj) that  ˆ  [0,2π]2  ������  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  tj(k1, k2)eik(2)x(2){˜Λ3(k(2))}r  ������  dx(2) ≲ log(rqQ∞)  ˆ  [0,2π]2  ��Dj  2(x(2))  �� dx(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This importantly uses that {˜Λ3(k(2))} ∈ {0,  1  qQ∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , qQ∞−1  qQ∞ } for integers k1, k2, so that on  can find some smartly chosen function h(u) ≈ ur on [0, 1] but with a smooth cut-off at 1 and  h({˜Λ3(k(2))}) = {˜Λ3(k(2))}r for which one can bound Fourier coefficients, see [KL18, Lemma  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We have q ≤ Q∞ = Q3α3 ≤ CQ3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We conclude the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Next we bound the second term in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7, where ˜Λ3 is integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' If there are no valid choices  of k1, k2 for which ˜Λ3(k1, k2) is an integer, then this term is clearly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Otherwise we have  the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Let N be sufficiently large and suppose that the set  I0 =  �  (k1, k2) ∈ Z2 : [λ1] ≤ k1 ≤ [Λ1], [λ2(k1)] ≤ k2 ≤ [Λ2(k1)], ˜Λ3(k1, k2) ∈ Z  �  is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then we may find a point k(2)  0  ∈ I0, a (non-zero) lattice vector κ = κ(2) ∈ Z2  and an integer h ≥ 0 with k(2)  0  + hκ ∈ I0 (in particular h|κ| ≲ R) such that I0 = {k(2)  0  + τκ(2) :  τ ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , h}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In particular  ˆ  [0,2π]2  ���������  [Λ1]  �  k1=[λ1]  [Λ2(k1)]  �  k2=[λ2(k1)]  ˜Λ3(k1,k2)∈Z  tj(k1, k2)eik(2)x(2)  ���������  dx(2) ≲  ˆ 2π  0  �����  h  �  τ=0  tj  �  k(2)  0  + τκ(2)�  eiτ|κ(2)|x  ����� dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9)  The proof is an exercise in elementary number theory analysing the set I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Define k(2)  0  to be any point in the (non-empty) set I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Recall Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6), and that  ���˜Λ3(k1, k2)  ��� ≤ CR for any k(2) ∈ I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (This follows since the relevant tetrahedron is contained  in [0, CR]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') By redefining αj as αj/ gcd(α1, α2, α3) we may assume that α1, α2, α3 have no  shared prime factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (This only decreases their values, so that still |αj| ≤ C√Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') In case one  of the αj’s is zero we will use the convention that gcd(α, β, 0) = gcd(α, β) and gcd(α, 0) = α  for α, β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  66  Solving the general problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We first consider the general problem of finding all k1, k2 ∈ Z  for which ˜Λ3(k1, k2) is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This set has the form k(2)  0 +Γ for some two-dimensional lattice  Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We now find spanning lattice vectors of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Define αij = gcd(αi, αj) for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (Note that the αj’s are not necessarily pairwise coprime,  only all 3 αj’s have no shared factor by the reduction above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Also, since α1 and α2 are not  both 0, we have α12 ̸= 0 is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Shifting k(2)  0  by κ0 := (Q2  α2  α12 , −Q1  α1  α12) we have  ˜Λ3(k(2)  0  + bκ0) = ˜Λ3(k(2)  0 ) ∈ Z,  b ∈ Z  and κ0 is the shortest lattice vector with this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' One should note here that κ0 is not  “short”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Indeed |κ0| ≳ Q since both Q1, Q2 ≳ Q, see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7, and α1, α2 are not both  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We now look for the lattice vector in Γ giving the smallest possible (integer) increase of ˜Λ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This lattice vector together with κ0 spans Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note that  δ˜Λ3(κ) := ˜Λ3(k(2)  0  + κ) − ˜Λ3(k(2)  0 ) = −Q1α1κ1 − Q2α2κ2  Q3α3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10)  Suppose first that either α1 = 0 or α2 = 0, say α2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Now, Q1 ̸= Q3 and |αj| ≤ C√Q so Qj  is not a factor of αi for any i = 1, 3, j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, gcd(Q3α3, Q1α1) = gcd(α1, α3) = 1 since  α2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the ratio δ˜Λ3(κ) to be an integer we need that the numerator is some multiple of  Q3α3, and thus that |κ| ≳ Q3 ≫ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus there is at most one k(2)  0  ∈ I0 and the lemma is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Suppose then that α1 ̸= 0, α2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Varying κ ∈ Z2 we have by B´ezout’s lemma that the  numerator in Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10) assumes as values all multiples of gcd(Q1α1, Q2α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We have  gcd(Q1α1, Q2α2) = gcd(α1, α2) = α12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the ratio δ˜Λ3(κ) to be an integer we need that the  numerator is some multiple of Q3α3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Since by assumption there are no prime factors shared by  all αj’s and Q3 is not a factor of α12 we have gcd(α12, Q3α3) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, the smallest integer  increase of ˜Λ3 is α12 ≥ 1 and this happens along some lattice vector κ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Immediately then  Γ ⊃ {aκ1 + bκ0 : a, b ∈ Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' To see that Γ ⊂ {aκ1 + bκ0 : a, b ∈ Z} note that by B´ezout’s lemma  the (integer) solutions to the equation  −Q1α1κ1 − Q2α2κ2 = Q3α3A,  for some integer A ∈ Z, is exactly (κ1, κ2) ∈  �  A  α12 κ1 + bκ0 : b ∈ Z  �  if α12 divides A and there  are no solutions otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In summary then  Γ = {aκ1 + bκ0 : a, b ∈ Z},  ˜Λ3(k(2)  0  + aκ1 + bκ0) = ˜Λ3(k(2)  0 ) + aα12,  a, b ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11)  Moreover  I0 =  �  k(2)  0  + Γ  �  ∩  �  (k1, k2) ∈ Z2 : [λ1] ≤ k1 ≤ [Λ1], [λ2(k1)] ≤ k2 ≤ [Λ2(k1)]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Finding the candidate for κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We now find the candidate for the κ in the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Either  I0 = {k(2)  0 }, in which case the lemma is clear (take h = 0), or there exists some (non-zero)  κ = aκ1 + bκ0 ∈ Γ such that k(2)  0  + κ ∈ I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For such κ we have (for sufficiently large N) that  a ̸= 0 as |κ0| ≳ Q ≫ R and any such κ has |κ| ≤ CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Let κ2 = a2κ1 + b2κ0 be the κ such  that k(2)  0  + κ ∈ I0 with minimal value of |a2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (κ2 is unique up to potentially a sign if both  k(2)  0  − κ2 ∈ I0 and k(2)  0  + κ2 ∈ I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') It follows from Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11) that |a2| ≤ CR/α12 ≤ CR  since |δ˜Λ3(κ2)| ≤ CR as the tetrahedron is contained in [0, CR]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  If b2 = 0 then a2 = ±1, else if b2 ̸= 0 then gcd(a2, b2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Indeed, if a2 and b2 shared some  common factor, we could factor this out to find a κ with smaller value |a| contradicting the  minimality of |a2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  67  Characterizing all allowed κ’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We claim that by potentially redefining k(2)  0  to k(2)  0  − aκ2  with a ∈ Z largest such that still k(2)  0  − aκ2 ∈ I0 we have that  I0 = {k(2)  0  + τκ2 : τ ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' , h}},  for some h ∈ Z, h ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12)  (The intuition for the remainder of the argument is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Essentially, if some κ had  k(2)  0  + κ ∈ I0 but was not a multiple of κ2, it would have to differ from some multiple of κ2 by  at least κ0 or κ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Since |κ0| ≫ R and either κ1 = κ2 or |κ1| ≫ R, this is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=')  To prove Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12) we first introduce the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We view a lattice vector  κ ∈ Z2 as a vector κ ∈ R2 and write κ∥ for its component parallel to κ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Note that κ∥ need not  have integer coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Define the constant A such that κ∥  1 = Aκ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (Note that A need not  be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Let 0 ̸= κ = aκ1 + bκ0 ∈ Γ with k(2)  0  + κ ∈ I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We have  κ∥ = aκ∥  1 + bκ0 = (aA + b)κ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Thus, since |κ0| ≳ Q, |κ| ≲ CR and |a| ≥ 1 (since κ ̸= 0) we have  �� b  a + A  �� ≤ CR  Q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Using this also for κ2 = a2κ1 + b2κ0 we get  |ba2 − b2a| =  ����  b  a − b2  a2  ���� |aa2| ≤ |aa2|  �����  b  a + A  ���� +  ����−b2  a2  − A  ����  �  ≤ CR2R  Q ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  But ba2 − b2a is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Hence (for N sufficiently large) we have ba2 = b2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Now, if b2 = 0  then b = 0 and so a2 = ±1 is a divisor of a so κ = ±aκ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' If b2 ̸= 0 then gcd(a2, b2) = 1 and  thus a2 is again a divisor of a and a/a2 = b/b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then κ =  a  a2κ2 is a multiple of κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This shows  the desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Integral form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  To prove Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9) we do the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Define e2 = κ2/|κ2| as the unit  vector parallel to κ2 and e⊥  2 as the unit vector perpendicular to κ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then define the domain  S0 :=  �  x(2) ∈ R2 :  ��x(2) · e2  �� ≤ 4π,  ��x(2) · e⊥  2  �� ≤ 4π  �  and note that [0, 2π]2 ⊂ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, using Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12)  ˆ  [0,2π]2  �����  �  k∈I0  tj(k1, k2)eik(2)x(2)  ����� dx(2) ≤  ˆ  S0  �����  h  �  τ=0  tj  �  k(2)  0  + τκ(2)�  eiτκ(2)x(2)  ����� dx(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The integrand is constant in the e⊥  2 -direction, and 2π-periodic in the e2-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, com-  puting the integral in these coordinates we have  ˆ  S0  �����  h  �  τ=0  tj  �  k(2)  0  + τκ(2)�  eiτκ(2)x(2)  ����� dx(2) = 32π  ˆ 2π  0  �����  h  �  τ=0  tj  �  k(2)  0  + τκ(2)�  eiτ|κ(2)|x  ����� dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Combining Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8 the r- and ν-sums in Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6 are readily  bounded because of the factor 1/r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We conclude that  ˆ  [0,2π]3  ��F j  3(x)  �� dx ≲ log Q  ˆ  [0,2π]2  ��� ˜Dj  2(x)  ��� dx + 1 +  ˆ 2π  0  �����  h  �  τ=0  tj  �  k(2)  0  + τκ(2)�  eiτ|κ(2)|x1  ����� dx1,  where k(2)  0  and κ(2) are as in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' If the set I0 from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8 is empty, then the  bound is valid without the last term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In particular it is valid with any k(2)  0  ∈ [0, CR]2, (non-  zero) κ = κ(2) ∈ Z2 and h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus, by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4 and Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5) we prove the desired  bound, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  68  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3  Reduction from d = 2 to d = 1  For j = 1, 2 we will do one more step reducing the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The argument is basically the  same as for going from dimension d = 3 to d = 2 in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We sketch the main  differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  As we did in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 for d = 3 by adding and subtracting the lower tail of the sum,  we may assume that the k2-sum is �[Λ2(k1)]  k2=0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' It is valid here to make the k2-sum go from 0, since now the k2-sum is the inner-  most sum and we do not risk values of k3 much larger that R by doing so (as in Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Indeed, we already computed the sum over the relevant k3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We could at this point also do  the same splitting of the k1-sum, but we would have the same problems that Λ2(k1) or λ2(k1)  might be much larger than R for k1 ≤ λ1 as in Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Additionally, by splitting the k1-sum into at most 2 parts, we may assume that Λ2 is just the  equation for a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Here again one needs to be careful with what to do with the boundary  terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This gives some sums over 1-dimensional tetrahedra (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' line segments), which we can  write as differences of sums over 2-dimensional tetrahedra exactly as for the 3-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We are led to define the quantities  Dj  2(x) :=  [Λ1]  �  k1=[λ1]  tj(k1)  [Λ2(k1)]  �  k2=0  eik(2)x(2),  ˜Dj  1(x) :=  [Λ1]  �  k1=[λ1]  tj(k1)eik1x1,  Gj  2(x) :=  1  eix2 − 1  �  ei(n2+1)x2 ˜Dj  1(x1 − m1x2) − ˜Dj  1(x1)  �  ,  F j  2(x) := ei(n2+1)x2  eix2 − 1  [Λ1]  �  k1=[λ1]  tj(k1)eik1(x1−m1x2) �  e−i⟨Λ2(k1)⟩x2 − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We claim the following inductive bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For j = 1, 2 we have for N sufficiently large that  ˆ  [0,2π]2  ��Dj  2(x(2))  �� dx(2) ≲ (log R + log Q)  ˆ  [0,2π]  ��� ˜Dj  1(x1)  ��� dx1 +  �  R  j = 1,  R2  j = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' As for Λ3, we have that the equation of a line between any two points (p1  i /Q1, p2  i /Q2),  i = 1, 2 is given by  p1  1 − p1  2  Q2  k1 + p2  2 − p2  1  Q1  k2 = const .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  If we choose the points to be either corners of RP or the central point Rz we get the equation  α1  Q2  k1 + α2  Q1  k2 = Rσγ /∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Here we might have that α1 = 0 or α2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  If α2 = 0 this line is parallel to the k2-axis and so does not give rise to a bound for the  k2-sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus α2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' If α1 = 0 the sum in Dj  2(x) and integral thereof factorizes, and hence by  69  [KL18, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2] we have that  ˆ  [0,2π]2  ��Dj  2(x(2))  �� dx(2) ≤ C log R  ˆ 2π  0  ����� ˜Dj  1(x1)  ��� dx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Hence, this case yields the desired inductive bound, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Suppose then α1, α2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Then  Λ2(k1) = n2 − m1k1 = n2 − Q1α1  Q2α2  k1,  n2 /∈ Q,  |αj| ≤ CQ1/4,  j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6 are readily adapted and proven as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The adaptation of  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7 is then mostly analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' One chooses Q∞ = Q2α2 and finds the rational ap-  proximation of Λ2 as  ˜Λ2(k1) = Λ2(k1) − γ2 = p  q − Q1α1  Q2α2  k1,  |γ2| <  1  qQ∞  ≤ Q−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The rest of the argument follows exactly as for d = 3 only that the extra term of the sum where  ˜Λ2(k1) ∈ Z may be bounded as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ���������  [Λ1]  �  k1=[λ1]  ˜Λ2(k1)∈Z  tj(k1)eik1x1  ���������  ≤  �  R  j = 1  R2  j = 2  since there is at most one k1 such that ˜Λ2(k1) is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' To see this note that gcd(α1, Q2) = 1  since |α1| ≤ CQ1/4 ≪ Q2, hence the change in k1 to change ˜Λ2(k1) by an integer is at least  Q3 ≫ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We thus conclude the desired bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4  Bounding the one-dimensional integrals  Now we bound  ´  | ˜Dj  1| and  ´ ���  �h  τ=0 tj  �  k(2)  0  + τκ  �  eiτ|κ|x��� dx from the right-hand-sides of Lem-  mas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For ˜Dj  1 we may assume that the lower bound of the summations are at 0  by the same procedure as in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Expanding tj  �  k(2)  0  + τκ  �  we see that j = 1 gives  an affine expression in τ and j = 2, 3 give quadratic expressions in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For instance,  t2  �  k(2)  0  + τκ  �  = (k1  0)2 + 2k1  0κ1τ + (κ1)2τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Thus, bounding both the integrals amounts to bounding the following:  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Let M ≥ 2 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then  (1)  ˆ 2π  0  �����  M  �  k=0  eikx  ����� dx ≤ C log M,  (2)  ˆ 2π  0  �����  M  �  k=0  keikx  ����� dx ≤ CM log M,  (3)  ˆ 2π  0  �����  M  �  k=0  k2eikx  ����� dx ≤ CM2 log M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  70  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The bound (1) is elementary, see also [KL18, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For any M ∈ N and q ∈ C\\{1}  we have  M  �  k=0  qk = qM+1 − 1  q − 1  M  �  k=0  kqk =  q  (q − 1)2  �  qM(Mq − M − 1) + 1  �  M  �  k=0  k2qk =  q  (q − 1)3  �  qM �  M2(q − 1)2 − 2M(q − 1) + q + 1  �  − q − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13)  Consider now the integrals (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By symmetry of complex conjugation  ´ 2π  0  = 2  ´ π  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We  split the integrals according according to whether x ≤ 1/M or x ≥ 1/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For x ≤ 1/M we have  ˆ 1/M  0  �����  M  �  k=0  keikx  ����� dx ≲  ˆ 1/M  0  M2 dx ≲ M,  ˆ 1/M  0  �����  M  �  k=0  k2eikx  ����� dx ≲  ˆ 1/M  0  M3 dx ≲ M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For x ≥ 1/M we use Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13) and note that |eix − 1| ≥ cx for x ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Expanding the  exponentials eix = 1 + O(x) we thus have  ˆ π  1/N  �����  M  �  k=0  keikx  ����� dx ≲  ˆ π  1/M  1  x2  �  MeiMx(eix − 1) + 1 − eiMx�  dx  ≲  ˆ π  1/M  �M  x + 1  x2  �  dx ≲ M log M  and  ˆ π  1/N  �����  M  �  k=0  k2eikx  ����� dx ≲  ˆ π  1/M  1  x3  �  eiMx �  M2(eix − 1)2 − 2M(eix − 1) + eix + 1  �  − eix − 1  �  dx  ≲  ˆ π  1/M  �M2  x + M  x2 + 1  x3  �  dx ≲ M2 log M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  With this we may thus bound for (j = 2, say)  ˆ 2π  0  �����  h  �  τ=0  t2  �  k(2)  0  + τκ  �  eiτ|κ|x  ����� dx  =  ˆ 2π  0  �����  h  �  τ=0  �  (k1  0)2 + 2k1  0κ1τ + (κ1)2τ 2�  eiτ|κ|x  ����� dx  ≤ CR2  ˆ 2π  0  �����  h  �  τ=0  eiτ|κ|x  ����� dx + CR|κ|  ˆ 2π  0  �����  h  �  τ=0  τeiτ|κ|x  ����� dx + C|κ|2  ˆ 2π  0  �����  h  �  τ=0  τ 2eiτ|κ|x  ����� dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Substituting y = |κ|x, using Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11 and recalling that h|κ| ≲ R and |κ| ≥ 1 by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='8  we may bound this by R2 log R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' An analoguous bound holds for j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' This takes care of all  the one-dimensional integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' In combination with Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10 we get the bounds  for j = 1, 2 of Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' It remains to consider the two-dimensional integral for j = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  71  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='5  Bounding the j = 3 two-dimensional integral  We are left with bounding the integral  ´  | ˜D3  2| on the right-hand-side of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We  first reduce to the case of a simpler tetrahedron (triangle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By shifting the sums by a fixed  κ = (κ1, 0) ∈ Z2 and using the bounds in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11 to evaluate the extra contributions of the  shift, we may assume that the k1-sum starts at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' By splitting the k2-sum as in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1  we may assume that that k2-sum also starts at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' That is, we need to evaluate the integral  ¨  [0,2π]2  ������  [Λ1]  �  k=0  [Λ2(k)]  �  ℓ=0  kℓeikxeiℓy  ������  dx dy,  where Λ2(k) = n2 − Q1α1  Q2α2k for an irrational n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Recall that |Λ1| ≤ CR and for any 0 ≤ k ≤ [Λ1]  we have |Λ2(k)| ≤ CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The analysis given here is in spirit the same as given in Appendices B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' It is  sufficiently different that we find it easier to do the arguments separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We shall show the  following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We have the following bound  ¨  [0,2π]2  ������  [Λ1]  �  k=0  [Λ2(k)]  �  ℓ=0  kℓeikxeiℓy  ������  dx dy ≤ CR2(log R)2 log Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Combining then Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='10, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12 and choosing Q some sufficiently large  power of N as required in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7 we conclude the proof of Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2) and thus of  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' It remains to give the proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Denote M = [Λ1] and recall Λ2(k) = n2−m1k = n2− Q1α1  Q2α2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' First note that by mapping  ℓ �→ [Λ2(k)] − ℓ we may assume that m1 ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' If m1 = 0 the sum factors, and so does the  integral into two one-dimensional sums/integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' These may be bounded using Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  In this case we get the bound ≤ CR2(log R)2 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Hence assume that m1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Moreover,  if n2 > m1M we may split the (k, ℓ)-sum into two parts,  M  �  k=0  [Λ2(k)]  �  ℓ=0  =  M  �  k=0  [n2−m1M]  �  ℓ=0  +  M  �  k=0  [Λ2(k)]  �  ℓ=[n2−m1M]+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The first sum factors into one-dimensional integrals which we may bound using Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11  again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The second we may shift by a constant ℓ (again then using Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='11 to evaluate the  contribution of the shift) and assume that the lower limit of the ℓ-sum is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' The upper limit  then becomes [Λ(k)], where  Λ(k) = n2 − ([n2 − m1M] + 1) − m1k := n − mk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Geometrically, this means that the domain of the (k, ℓ)-sum is a triangle with two sides along  the axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We thus need to bound  ¨  [0,2π]2  ������  M  �  k=0  [Λ(k)]  �  ℓ=0  kℓeikxeiℓy  ������  dx dy,  where  Λ(k) = n − mk,  M ≤ R,  mM = n + O(1),  n ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  72  By the symmetries of translation invariance and complex conjugation we may integrate over  the domain [−π, π] × [0, π] instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We evaluate the ℓ-sum using Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Recall that  [Λ(k)] = Λ(k) − ⟨Λ(k)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We thus have  [Λ(k)]  �  ℓ=0  ℓeiℓy =  eiy  (eiy − 1)2  �  ei[Λ(k)]y([Λ(k)]eiy − [Λ(k)] − 1) + 1  �  =  eiy  (eiy − 1)2  ��  eiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1  �  +  �  e−i⟨Λ(k)⟩y − 1  � �  eiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1  �  −  �  ⟨Λ(k)⟩ ei(Λ(k)−⟨Λ(k)⟩)y(eiy − 1) + (e−i⟨Λ(k)⟩y − 1)  ��  =: (I) + (II) + (III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The third summand (III) may be calculated as  −eiy  (eiy − 1)2  �  ⟨Λ(k)⟩  �  ei[Λ(k)]y − 1  �  iy + O(y2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The factor  −eiy  (eiy−1)2 may be bounded by 1/y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For this term we split the y-integral according to  whether y ≤ 1/n or y ≥ 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For y ≤ 1/n we expand additionally ei[Λ(k)]y − 1 = O(ny).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We  get the contribution  ˆ π  −π  dx  ˆ 1/n  0  dy 1  y2  �����  M  �  k=0  keikx �  ⟨Λ(k)⟩  �  ei[Λ(k)]y − 1  �  y + O(y2)  �  ����� ≲ 1  nM2n + 1  nM2 ≲ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For y ≥ 1/n we bound ei[Λ(k)]y − 1 = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We get  ˆ π  −π  dx  ˆ π  1/n  dy 1  y2  �����  M  �  k=0  keikx �  ⟨Λ(k)⟩  �  ei[Λ(k)]y − 1  �  y + O(y2)  �  ����� ≲ (log n)M2 + M2 ≲ R2 log R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For the second summand (II) we again split the integral according to whether y ≤ 1/n or  y ≥ 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' If y ≤ 1/n we have  eiy  (eiy − 1)2  �  e−i⟨Λ(k)⟩y − 1  � �  eiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1  �  = O(Λ(k)2y) = O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Hence this contributes the term  ˆ π  −π  dx  ˆ 1/n  0  dy  �����  M  �  k=0  keikxO(Λ(k)2y)  ����� ≲ 1  nM2n ≲ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  For y ≥ 1/n we write  eiy  (eiy − 1)2  �  e−i⟨Λ(k)⟩y − 1  � �  eiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1  �  =  eiy  (eiy − 1)2  ∞  �  ν=1  (−iy)ν  ν!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ⟨Λ(k)⟩ν �  eiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  73  Again we bound the factor  eiy  (eiy−1)2 as 1/y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We treat each summand similarly as in Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6  and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7 (or rather, the 2-dimensional version of these as used in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=') Completely  analogously to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='6 we see that for any integer r ≥ 1 we have  ¨ �����  M  �  k=0  keikxeiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1  y2  ⟨Λ(k)⟩r  ����� dx dy  ≤  ¨ �����  M  �  k=0  keikxeiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1  y2  ����� dx dy  +  r  �  ν=1  �r  ν  � ¨ �����  M  �  k=0  keikxeiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1  y2  {Λ(k)}ν  ����� dx dy,  for either definition of ⟨·⟩ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  either ⟨·⟩ = {·} or ⟨·⟩ = · − ⌈·⌉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Also the application of  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7 is analogous to its use in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' There is at most one k such that ˜Λ(k) ∈ Z  for the appropriate rational approximation ˜Λ of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Using that eiy = 1 + O(y) we obtain the  bound  ����keikxeiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1  y2  ���� ≲ M ny + 1  y2  ≲ R2  y + R  y2,  valid for any k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Hence this error term contributes at most  ˆ π  −π  dx  ˆ π  1/n  dy  �R2  y + R  y2  �  ≲ R2 log n + Rn ≲ R2 log R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The rest of the argument in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='7 is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We conclude that we may bound the  contribution of the term (II) by that of (I) up to a factor of log Q and an error R2 log R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  ¨ �����  �  k  keikx(II)  ����� dx dy ≲ log Q  ¨ �����  �  k  keikx(I)  ����� dx dy + R2 log R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  In particular  ¨  [0,2π]2  ������  M  �  k=0  [Λ(k)]  �  ℓ=0  kℓeikxeiℓy  ������  dx dy  ≲ log Q  ˆ π  −π  dx  ˆ π  0  dy  �����  M  �  k=0  keikxeiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1  y2  ����� + R2 log R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='14)  In order to evaluate the integral on the right-hand side, we split the integration domain into 5  regions, see Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  I1 = {|x| ≤ 2/M, y ≤ 2/n},  I2 = {|x| ≤ 1/M, y ≥ 2/n},  I3 = {y ≤ 1/n, |x| ≥ 2/M},  I4 = {y ≥ 1/n, |x| ≥ 1/M, |x − my| ≥ 1/M}  I5 = {|x − my| ≤ 1/M, (x, y) /∈ I1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We will be a bit sloppy with notation and refer to both the domain of integration and the value  of the integration over that domain by Ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (I1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We expand  (∗) :=  M  �  k=0  keikxeiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1  y2  74  x  y  2/M  −2/M  2/n  x = my  I1  I2  I3  I3  I4  I4  I4  I5  Figure B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1: Decomposition of the domain [−π, π] × [0, π] into different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (or rather the numerator) to second order in y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Using that Λ(k) = O(n) we get that (∗) ≲ M2n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Thus the integral gives  I1 ≲  ˆ 2/M  −2/M  dx  ˆ 2/n  0  dyM2n2 ≲ Mn ≲ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (I2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We expand eiy = 1 + O(y) in (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Then (∗) ≲  M2n  y  + M2  y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  The integral is then  I2 ≲ R2 log R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (I3, I4, I5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' For the remaining integrals we use the explicit formula for Λ(k) = n−mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then  (∗) = 1  y2  M  �  k=0  keikx(eiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1)  = 1  y2  M  �  k=0  �  keikx + keik(x−my)einy(neiy − n − 1) + k2eik(x−my)einy(m − meiy)  �  = 1  y2  �  −i∂D(x) − ieiny(neiy − n − 1)∂D(x − my) + meiny(eiy − 1)∂2D(x − my)  �  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='15)  where we introduced D(z) = �M  k=0 eikz = ei(M+1)z−1  eiz−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' From Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='13) we conclude that  we may bound derivatives of D as  |∂D(z)| ≲ M  z + 1  z2,  |∂2D(z)| ≲ M2  z  + M  z2 + 1  z3,  |∂3D(z)| ≲ M3  z  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' + 1  z4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='16)  (I3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We have y ≤ 1/n and |x| ≥ 2/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We expand Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='15) to second order in y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Expanding first the exponentials and then derivatives of D where needed we get  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='15) = 1  y2  �  − i∂D(x) + i∂D(x − my) + imy∂2D(x − my)  + O(n2y2∂D(x − my)) + O(nmy2∂2D(x − my))  �  ≲ n2 sup  y |∂D(x − my)| + nm sup  y |∂2D(x − my)| + m2 sup  y |∂3D(x − my)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  75  Now we use the bounds Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='16) and use that z := x − my has |z| ≥ |x| − m/n =  |x| − 1/M + O(1/(Mn)) (recall that mM = n + O(1)) and |x| ≥ 2/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Thus  I3 ≲ 1  n  ˆ π  1/M  dz  �  n2|∂D(z)| + nm|∂2D(z)| + m2|∂3D(z)|  �  ≲ R2 log R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (I4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' We expand the exponentials eiy = 1 + O(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Then  |(B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='15)| ≤ |∂D(x)|  y2  + n|∂D(x − my)|  y  + |∂D(x − my)|  y2  + m|∂2D(x − my)|  y  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Using the bounds Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='16) as before and noting that |x| ≥ 2/M and z = x − my has  |z| ≥ 1/M one easily sees that I4 ≲ R2(log R)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (I5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Again, expanding the exponentials eiy = 1 + O(y) we have as for I4 that  |(B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='15)| ≤ |∂D(x)|  y2  + n|∂D(x − my)|  y  + |∂D(x − my)|  y2  + m|∂2D(x − my)|  y  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We use the bounds  |∂D(z)| =  �����  M  �  k=0  keikz  ����� ≤ M2,  |∂2D(z)| =  �����  M  �  k=0  k2eikz  ����� ≤ M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Thus  I5 ≲ 1  M  ˆ π  1/n  M2  y2 + nM2 + mM3  y  dy ≲ R2 log R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  We conclude that  ˆ π  −π  dx  ˆ π  0  dy  �����  M  �  k=0  keikxeiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1  y2  ����� ≲ R2(log R)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Together with Equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='14) this concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  References  [ARS22]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Agerskov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Reuvers, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Solovej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' “Ground state energy of dilute Bose gases in 1D”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='17183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  [BCGOPS22]  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Basti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Cenatiempo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Giuliani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Olgiati, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Pasqualetti, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Schlein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' “A simple  upper bound for the ground state energy of a dilute Bose gas of hard spheres”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='04431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  [BCS21]  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Basti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Cenatiempo, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Schlein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' “A new second-order upper bound for the ground  state energy of dilute Bose gases”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Forum of Mathematics, Sigma 9 (2021), e74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1017  /fms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  [CW68]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Clark and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Westhaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' “Cluster Expansions in Many-Fermion Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' “Factor-Cluster”  Formalisms”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 (1968), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' 131–148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1664466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  [Dys57]  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Dyson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' “Ground-State Energy of a Hard-Sphere Gas”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 (1957), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' 20–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1103/PhysRev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  [Efi66]  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Efimov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' “A Rarefied Fermi Gas and the Two-body Scattering Problem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  (JETP) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 (1966), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' url: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='jetp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='ras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='ru/cgi-bin/e/index/e/22/1/p135  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='a=list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  76  [EA65]  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Efimov and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Amus’ya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' “Ground State of a Rarefied Fermi Gas of Rigid Spheres”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' (JETP) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='2 (1965), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' 388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' url: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='jetp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='ras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='ru/cgi-bin/e/index/e  /20/2/p388?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='a=list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='  [FGHP21]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Falconi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content=' The Thermodynamic Pressure in Quantum Statistical Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content=' Springer, Berlin, Heidelberg, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content=' “An improved tree-graph bound”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Oberwolfach Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content='1 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' arXiv:1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content='  [WDS20]  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Wellenhofer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Drischler, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Schwenk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' “Dilute Fermi gas at fourth order in effective  field theory”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Phys Lett B 802 (2020), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content='  [WC68]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Clark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content=' Rearrange-  ments of Primitive Decomposition Equations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content=' Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
+page_content=' “The Second Order Upper Bound for the Ground Energy of a Bose Gas”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE4T4oBgHgl3EQfGgwU/content/2301.04894v1.pdf'}
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