diff --git "a/ENE4T4oBgHgl3EQfGgwU/content/tmp_files/2301.04894v1.pdf.txt" "b/ENE4T4oBgHgl3EQfGgwU/content/tmp_files/2301.04894v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/ENE4T4oBgHgl3EQfGgwU/content/tmp_files/2301.04894v1.pdf.txt" @@ -0,0 +1,8244 @@ +arXiv:2301.04894v1 [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. One of the main ingredients in the proof is a rigorous implementation of the +fermionic cluster expansion of Gaudin, Gillespie and Ripka (Nucl. Phys. A, 176.2 (1971), +pp. 237–260). +Contents +1 +Introduction and main results +2 +1.1 +Precise statement of results +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2 +Preliminary computations +6 +2.1 +The scattering function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +2.2 +The “Fermi polyhedron” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +2.3 +Reduced densities of the Slater determinant +. . . . . . . . . . . . . . . . . . . . +15 +3 +Gaudin-Gillespie-Ripka-expansion +17 +3.0.1 +Calculation of the normalization constant . . . . . . . . . . . . . . . . . . +17 +3.0.2 +Calculation of the 1-particle reduced density . . . . . . . . . . . . . . . . +19 +3.0.3 +Calculation of the 2-particle reduced density . . . . . . . . . . . . . . . . +20 +3.0.4 +Calculation of the 3-particle reduced density . . . . . . . . . . . . . . . . +21 +3.0.5 +Summarising the results +. . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +3.1 +Absolute convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +3.1.1 +Absolute convergence of the Γ-sum . . . . . . . . . . . . . . . . . . . . . +23 +3.1.2 +Absolute convergence of the Γ1-sum . . . . . . . . . . . . . . . . . . . . . +27 +3.1.3 +Absolute convergence of the Γ2-sum . . . . . . . . . . . . . . . . . . . . . +28 +3.1.4 +Absolute convergence of the Γ3-sum . . . . . . . . . . . . . . . . . . . . . +30 +∗alaurits@ist.ac.at +†robert.seiringer@ist.ac.at +1 + +4 +Energy of the trial state +31 +4.1 +Thermodynamic limit via a box method +. . . . . . . . . . . . . . . . . . . . . . +34 +4.2 +Subleading 2-particle diagrams (proof of Lemma 4.1) +. . . . . . . . . . . . . . . +35 +4.3 +Subleading 3-particle diagrams (proof of Lemma 4.2) +. . . . . . . . . . . . . . . +42 +5 +One and two dimensions +43 +5.1 +Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +43 +5.2 +One dimension +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +47 +A Small diagrams +54 +A.1 Small 2-particle diagrams (proof of Lemmas 4.6 and 4.8) . . . . . . . . . . . . . +54 +A.2 Small 3-particle diagrams (proof of Lemma 4.11) . . . . . . . . . . . . . . . . . . +57 +A.3 Small diagrams in 1 dimension (proof of Lemma 5.21) . . . . . . . . . . . . . . . +58 +B Derivative Lebesgue constants (proof of Lemma 4.9) +61 +B.1 Reduction to simpler tetrahedron . . . . . . . . . . . . . . . . . . . . . . . . . . +62 +B.2 Reduction from d = 3 to d = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . +63 +B.3 Reduction from d = 2 to d = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . +69 +B.4 Bounding the one-dimensional integrals . . . . . . . . . . . . . . . . . . . . . . . +70 +B.5 Bounding the j = 3 two-dimensional integral . . . . . . . . . . . . . . . . . . . . +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. We will mostly focus on the case d = 3. The particles interact via a two-body +interaction v, which we assume to be positive, radial and of compact support. In particular we +allow for v to have a hard core, i.e. v(x) = ∞ for |x| ≤ r for some r > 0. 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 0 (i.e., the error-term in the +theorem, O((a3ρ)2/3+1/21| log(a3ρ)|6) could be replaced by Oε((a3ρ)1−ε)). This is similar to the +recent work on the Bose gas [BCGOPS22]. We shall discuss this further in Remark 4.10. +We consider the lower-dimensional problems next. +We start with 2 dimensions, where the +scattering length is defined as follows. +Definition 1.6. 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. +With this, we may state the 2-dimensional analogue of Theorem 1.3. +Theorem 1.7 (Two dimensions). Suppose that v ≥ 0 is radial and compactly supported. 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�� +. +We sketch in Section 5.1 how to adapt the proof in the 3-dimensional setting to 2 dimensions. +Finally, we consider the 1-dimensional problem. The scattering length is defined as follows. +Definition 1.8. 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. +5 + +We show in Proposition 5.12 that Definition 1.8 agrees with the (seemingly different) definition +of the scattering length in [ARS22]. With this, we may state the 1-dimensional analogue of +Theorem 1.3. +Theorem 1.9 (One dimension). Suppose that v ≥ 0 is even and compactly supported. Suppose +moreover that +´ � 1 +2vf 2 +0 + |∂f0|2� +dx < ∞, where f0 denotes the (p-wave) scattering function. +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�� +. +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)). Compared to their +result we treat a slightly different class of potentials and obtain an improved error bound. The +conjectured next contribution is of order a2ρ5, see [ARS22]. +Remark 1.10 (On the assumptions on v). 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 5.13 and 5.14. +We sketch in Section 5.2 how to adapt the proof in the 3-dimensional setting to 1 dimension. +This turns out to be more involved than adapting the argument to 2 dimensions. +The paper is structured as follows. In Section 2 we give some preliminary computations +and in particular we introduce the “Fermi polyhedron”, a polyhedral approximation to the +Fermi ball. 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. +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. +Finally, in Section 5 we sketch how to adapt the argument to the lower-dimensional settings. In +Appendix A we give computations of “small diagrams” needed for some bounds in Sections 4 +and 5.2 and in Appendix B we give the proof of Lemma 4.9, an important lemma used in +Section 4. +2 +Preliminary computations +We will construct a trial state using a box method, and bound the energy of such trial state. +To use such a box method we need to use Dirichlet boundary conditions in each smaller box. In +Lemma 4.3 we show that we may construct trial states with Dirichlet boundary condition out of +trial states with periodic boundary conditions. We will thus use periodic boundary conditions +in the box Λ = [−L/2, L/2]3. For periodic boundary conditions, the Hamiltonian is given by +HN = Hper +N,L = +N +� +j=1 +−∆j + +� +i 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 2.2, and the number of +particles is N = #PF, the number of points in PF. We choose b to be larger than the range of +v; in particular, then f is continuous. (Note that the metric on the torus is d(x, y) = |x − y|. +We will abuse notation slightly and denote by |·| also the absolute value of some number or the +norm on R3.) +Before going further with the proof we first fix some notation. +Notation 2.1. We introduce the following. +• For any function h and edge (of some graph) e = (i, j) we will write he = hij = h(xi − xj). +• We denote by C a generic positive constant whose value may change line by line. +• For expressions A, B we write A ≲ B if there exists some constant C > 0 such that A ≤ CB. +If both A ≲ B and B ≲ A we write A ∼ B. +• For a vector x = (x1, . . . , xd) ∈ Rd we write x1, . . . , xd for its components. +We will fix the Fermi momentum kF and then choose L, N large but finite depending on kF. +The density of particles in the trial state ψN is ρ := N/L3. The limit of small density a3ρ → 0 +will be realized as kFa → 0. +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. +Using this on F = � +i a. +We give a short proof here for completeness. +Proof. From the radial Euler-Lagrange equation (2.4) we have ∂r(r4∂rf0) = vr4f0/2 ≥ 0. +Denote by fhc = +� +1 − a3 +|x|3 +� ++ the solution for a hard core potential of range a. Then +r4∂rfhc = +� +3a3 +r > a +0 +r < a +In particular ∂r(r4∂rfhc) = 0 for r > a. We thus see that ∂rf0 ≤ ∂rfhc = 3a3r−4 and f0 ≥ fhc +for r > a by integrating. Trivially f0 ≥ 0 = fhc for r ≤ a. +Remark 2.3. 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. +2.2 +The “Fermi polyhedron” +We now introduce a polyhedral approximation to the Fermi ball BF = {k ∈ 2π +L Z3 : |k| ≤ kF}. +We discuss why we need this in Remark 3.5. 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; Lif06] and references therein) is too big for our purposes. +Note +that this behaviour is a consequence of taking the absolute value. +In fact we have that +1 +L3 +´ � +k∈B(kF )∩ 2π +L Z3 eikx dx = 1. +8 + +This type of quantity is referred to as the Lebesgue constant [GL19; 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 3.1 and 4. For an overview of such +Lebesgue constants, see [GL19; Lif06]. +Of particular relevance for us is the fact that the +Lebesgue constants are much smaller for polyhedral domains than for balls. Hence we introduce +the polyhedron P = P(N) as an approximation of the unit ball. +Then the scaled version +PF = kFP ∩ 2π +L Z3 approximates the Fermi ball. We will refer to PF as the Fermi polyhedron. +In [KL18, Theorem 4.1] 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 +(2.5) +for any R > 2, in particular for R ∼ N1/3, where C(s) is some unknown function of s. 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.e. of C(s). We first give an almost correct definition of the +polyhedron P. +“Definition” 2.4 (Simple definition). The polyhedron P is chosen to be the convex hull of +s = s(N) points κ1, . . . , κs on a sphere of radius 1+δ, where δ is chosen such that Vol(P) = 4π/3. +We moreover choose the set of points to have the following properties. +• 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. That is, for some constants c, C > 0 we have d ≥ cs−1/2 and +infj |k − κj| ≤ Cs−1/2. +• P is invariant under any map (k1, k2, k3) �→ (±ka, ±kb, ±kc) for {a, b, c} = {1, 2, 3}, i.e. +reflection in or permutation of any of the axes. +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. +Remark 2.5. Note that the symmetry constraint adds a restriction on s. For instance, a +generic point away from any plane of symmetry (i.e. +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}. +For s points on a sphere of radius 1 + δ, the natural lengthscale is (1 + δ)s−1/2 ∼ s−1/2. 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. +Remark 2.6. For all purposes apart from the technical argument in Appendix B one may +take this as the definition. In particular, the convergence criterion of the cluster expansion +formulas of Gaudin, Gillespie and Ripka [GGR71], given in Theorem 3.4, holds also for this +simpler definition of P. We provide this simpler definition to better give an intuition of the +construction. +We now give the actual definition of P. We first give the construction. Then in Remark 2.9 we +give a few comments and in Remark 2.10 we give a short motivation. +9 + +Definition 2.7 (Actual definition). The polyhedron P with s corners and the “centre” z is +constructed as follows. +• 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 → ∞. +• Pick three large distinct primes Q1, Q2, Q3 with Qj ∼ Q. +• Place s evenly distributed points κR +1 , . . . , κR +s on the sphere of radius Q−3/4 and such that the +set of points {κR +1 , . . . , κR +s } is invariant under the symmetries (k1, k2, k3) �→ (±ka, ±kb, ±kc) +for {a, b, c} = {1, 2, 3}. +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. That is, d ≥ cs−1/2Q−3/4 and infj +��k − κR +j +�� ≤ Cs−1/2Q−3/4 for some constants +c, C > 0. +• Find points κ1, . . . , κs of the form +κj = +� p1 +j +Q1 +, p2 +j +Q2 +, p3 +j +Q3 +� +, +pµ +j ∈ Z, +µ = 1, 2, 3, +j = 1, . . . , s, +(2.6) +such that +��κj − κR +j +�� ≲ Q−1 for all j = 1, . . . , s and such that the set of points {κ1, . . . , κs} is +invariant under the symmetries (k1, k2, k3) �→ (±k1, ±k2, ±k3). +• Define ˜P as the convex hull of all the points κ1, . . . , κs. That is, ˜P = conv{κ1, . . . , κs}. +• Define P as σ ˜P, where σ is chosen such that Vol(P) = 4π/3. We will refer to the scaled +points σκj = σ(p1 +j/Q1, p2 +j/Q2, p3 +j/Q3) for j = 1, . . . , s as corners of P. +• Define P R = σ conv{κR +1 , . . . , κR +s } as the scaled convex hull of all the initial points κR +1 , . . . , κR +s . +• Define the centre as z = σ(1/Q1, 1/Q2, 1/Q3). +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. +We additionally define P R +F := kFP R ∩ 2π +L Z3. +Remark 2.8. We choose N := #PF, so that the Fermi polyhedron is filled. The dependence +in N of, for instance, Q should therefore more precisely be given in terms of a dependence on +kFL. Note that N = ρL3 ∼ (kFL)3 and kF = (6π2ρ)1/3(1 + O(N−1/3)). +We will choose also s depending on N (i.e. on kFL) satisfying s → ∞ as N → ∞. +Remark 2.9 (Comments on and properties of the construction). We collect here some prop- +erties of the Fermi polyhedron, some of which will only be needed in Appendix B. +• The points κ1, . . . , κs are evenly distributed on a thickened sphere of radius Q−3/4 – their +radial coordinates are |κj| = Q−3/4 + O(Q−1). +Indeed, the points κR +1 , . . . , κR +s are evenly +distributed and Q−1 ≪ s−1/2Q−3/4. For s points on a thickened sphere of radius Q−3/4, the +natural lengthscale between points is s−1/2Q−3/4. +10 + +• There is some constraint on the number of points s. A generic point κ ∈ S (with κ1, κ2, κ3 +all different and non-zero) has 48 images, including itself. The constraint on s is more or less +the same as for the simpler ‘‘Definition” 2.4. +• By choosing the points κ1, . . . , κs as in Equation (2.6) we break the symmetries of permuting +the coordinates, i.e. (k1, k2, k3) �→ (ka, kb, kc) if (a, b, c) ̸= (1, 2, 3). These symmetries are +however still almost satisfied, see Lemma 2.11. +• We choose s, Q such that Q−1/4 ≪ s−1/2 in the limit of large N. Hence, for N sufficiently +large, all the chosen points {κ1, . . . , κs} are extreme points of ˜P, i.e. all corners are extreme +points of the polyhedron P. That is, the name “corner” is well-chosen, and we do not have +any superfluous points in the construction. +• For any three points (xi, yi, zi) ∈ R3, i = 1, 2, 3 the plane through them is given by the +equation + + +(y2 − y1)(z3 − z1) − (y3 − y1)(z2 − z1) +(z2 − z1)(x3 − x1) − (z3 − z1)(x2 − x1) +(x2 − x1)(y3 − y1) − (x3 − x1)(y2 − y1) + + · + + +x +y +z + + = const. +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, +(2.7) +where +α1 = (p2 +2 − p2 +1)(p3 +3 − p3 +1) − (p2 +3 − p2 +1)(p3 +2 − p3 +1) ∈ Z +and similarly for α2, α3. From these formulas it is immediate that |αj| ≤ C√Q for j = 1, 2, 3. +For some planes we may have αj = 0 for some j. +• We claim that σ = Q3/4(1 + O(s−1)). In particular, that any point on the boundary ∂P has +radial coordinate 1 + O(s−1). 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). 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 . It +follows that σ ≥ Q3/4(1 − CQ−1/4). On the other hand, scaling Q3/4 ˜P by 1 + Cs−1 we have +that (1 + Cs−1)Q3/4 ˜P ⊃ B1(0). 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. It follows that +σ ≤ (1 + Cs−1)Q3/4. Since Q−1/4 ≤ Cs−1 this shows the desired. +• Note moreover that σ is irrational. Indeed, the volume of a polyhedron with rational corners +is rational. (This is easily seen for tetrahedra, of which any polyhedron is an essentially +disjoint union.) Thus σ3 = πr for a rational r. Hence the equations of the planes defined by +corners of P (i.e. scaled points) are of the form Equation (2.7) with an irrational constant σγ +on the right-hand side. 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). The equation of the plane through +three scaled points only differ by scaling the constant term. Since σ is irrational, and the +constant term was rational for the unscaled points, this shows the desired. +• We now construct a triangulation of ∂P. For all (2-dimensional) triangular faces of P simply +consider these as part of the triangulation. That is, we construct edges between any pair of +11 + +the three corners of such a triangle. Some of the (2-dimensional) faces of P may be polygons +of more than 3 sides (1-dimensional faces). Construct edges between all pairs of corners +sharing a side (i.e. a 1-dimensional face) and choose one corner and construct edges from +this corner to all other corners of the polygon. +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. Since the points {κ1, . . . , κs} are evenly dis- +tributed, that the distance between any pair of close corners is d ∼ s−1/2. +• Additionally, one may note that the corners of P have ≤ C many neighbours since the points +are evenly distributed. +• The reason we need +LkF +2π +rational will only become apparent in Appendix B and will be +explained there. +Remark 2.10 (Motivation of construction). The purpose of the construction is twofold. Firstly +we avoid a casework argument as in the proof of [KL18, Lemma 3.5] of whether the coefficients +of the planes are rational or not. The argument in Lemmas B.6 and B.7 is heavily inspired by +[KL18, Lemmas 3.6, 3.9], where such casework is required. Secondly we have good control over +how many (and which) lattice points (i.e. points in 2π +L Z3) can lie on each plane (or, rather, a +closely related plane, see Appendix B for the details). +These are technical details only needed in Appendix B. We reiterate, that apart from the +arguments in Appendix B, the reader may have the simpler ‘‘Definition” 2.4 in mind instead. +As mentioned in Remark 2.9 the Fermi polyhedron is almost symmetric under permutation of +the axes. This is formalized as follows. +Lemma 2.11. For µ ̸= ν let Fµν be the map that permutes kµ and kν (i.e. F12(k1, k2, k3) = +(k2, k1, k3), etc.). 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 2.7 and χ denotes the indicator function. +Proof. 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) +(2.8) +since P R +F is invariant under permutation of the axes, i.e. Fµν(P R +F ) = P R +F . The points {κj}j=1,...,s +only differ from {κR +j }j=1,...,s by at most ∼ Q−1 thus the points {σκj}j=1,...,s (the corners of P) +only differ from the points {σκR +j }j=1,...,s by ∼ Q−1/4. 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). That is, +supp +� +χ(k∈PF ) − χ(k∈P R +F ) +� +⊂ +� +k ∈ 2π +L Z3 : dist(k, ∂(kFP)) ≲ kFQ−1/4 +� +. +The surface ∂(kF P) has area ∼ k2 +F so +Vol +�� +k ∈ R3 : dist(k, ∂(kFP)) ≲ kFQ−1/4�� +≲ k3 +FQ−1/4. +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. 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). +The same argument applies to the second summand in Equation (2.8). We conclude the desired. +We now improve on Equation (2.5) for our polyhedron. +Lemma 2.12. 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. +The proof is (almost) the same as given in [KL18, Theorem 4.1]. We need to be a bit more +careful in the decomposition into tetrahedra. +Proof. Define R = LkF +2π . We decompose RP into tetrahedra using the “central” point z from +the construction of P. We triangulate the surface of RP as in Remark 2.9. For each triangle +in the triangulation add the point Rz to form a tetrahedron. Note that Rz /∈ Z3 since |Rz| ≤ +CRQ−1/4 ≪ 1 and z ̸= 0. This gives m = O(s) many (closed) tetrahedra {Tj} such that +RP = � Tj and that Tj ∩ Tj′ is a tetrahedron of lower dimension (i.e. the central point Rz, +a line segment or a triangle). Then, as in [KL18, Theorem 4.1] by the inclusion–exclusion +principle we have +L(RP) = +1 +(2π)3 +ˆ ������ +� +j +� +q∈Tj∩Z3 +eiqu − +� +j N we have ∆p = 0 for p > N. +17 + +Thus we may extend the summation to ∞. We now expand out the determinant and the Wp. +That is +CN +N! = 1 + +∞ +� +p=2 +1 +p! +� +G∈Gp +π∈Sp +(−1)π +˙ � +e∈G +ge +p +� +j=1 +γ(1) +N (xj, xπ(j)) dx1 . . . dxp, +where Sp denotes the symmetric group on p elements. We will consider π and G together as +a diagram (π, G). We give a slightly more general definition for what a diagram is, as we will +need such for the calculation of the reduced densities. +Definition 3.1. Define the set Gq +p as the set of all graphs on q “external” vertices {1, . . . , q} +and p “internal” vertices {q + 1, . . . , q + p} such that all internal vertices have degree at least +1, i.e. each internal vertex has at least one incident edge. The external edges are allowed to +have degree zero, i.e. have no incident edges. For q = 0 we recover G0 +p = Gp. +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, . . . , q + p}) and a graph G ∈ Gq +p. We denote the +set of all diagrams on q “external” vertices and p “internal” vertices by Dq +p. +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. The value of a diagram (π, G) ∈ Dq +p is the function +Γq +π,G(x1, . . . , xq) := (−1)π +˙ � +e∈G +ge +q+p +� +j=1 +γ(1) +N (xj, xπ(j)) dxq+1 . . . dxq+p. +For q = 0 we write Γπ,G = Γ0 +π,G and Dp = D0 +p. +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. The set of all linked diagrams on q “external” and p +“internal” vertices is denoted Lq +p. For q = 0 we write Lp = L0 +p. +By the translation invariance we have that Γ1 +π,G is a constant for any diagram (π, G). +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +(π1, G1) +(π2, G2) +(π3, G3) +Figure 3.1: A diagram (π, G) decomposed into linked components. The dashed lines +denote g-edges and the arrows (i, j) denote that π(i) = j. +In terms of diagrams we thus have +CN +N! = 1 + +∞ +� +p=2 +1 +p! +� +(π,G)∈Dp +Γπ,G. +If (π, G) is not linked we may decompose it into its linked components. Here the integration +factorizes. +We split the sum according to the number of linked components. +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. We get +CN +N! = 1 + +∞ +� +p=2 +∞ +� +k=1 +���� +# lnk. cps. +1 +k! +� +p1≥2 +· · · +� +pk≥2 +� +�� +� +sizes linked cps. +χ(� pℓ=p) +� +(π1,G1)∈Lp1 +· · · +� +(πk,Gk)∈Lpk +� +�� +� +linked components +Γπ1,G1 +p1! +· · · Γπk,Gk +pk! +(3.1) +Here χ is the indicator function. The factor 1/(k!) comes from counting the possible labellings of +the k linked components. The factors 1/(p1!), . . . , 1/(pk!) come from counting how to distribute +the p vertices {1, . . . , p} between the linked components of prescribed sizes p1, . . . , pk. This +gives the factor +� +p +p1,...,pk +� += +p! +p1!...pk!, which together with the factor 1/p! already present gives the +claimed formula. +We want to pull the p-summation inside the p1, . . . , pk-summation. This is allowed once we +check that � +p +1 +p! +� +(π,G)∈Lp Γπ,G is absolutely convergent. More precisely we need that the p-sum +is absolutely convergent, i.e. � +p +1 +p! +��� +� +(π,G)∈Lp Γπ,G +��� < ∞. This is the content of Lemma 3.2 +below. We conclude that if the assumptions of Lemma 3.2 are satisfied then +CN +N! = 1 + +∞ +� +k=1 +1 +k! +� +p1≥2 +· · · +� +pk≥2 +� +(π1,G1)∈Lp1 +· · · +� +(πk,Gk)∈Lpk +Γπ1,G1 +p1! +· · · Γπk,Gk +pk! += 1 + +∞ +� +k=1 +1 +k! + + +∞ +� +p=2 +1 +p! +� +(π,G)∈Lp +Γπ,G + + +k += exp + + +∞ +� +p=2 +1 +p! +� +(π,G)∈Lp +Γπ,G + + . +(3.2) +3.0.2 +Calculation of the 1-particle reduced density +We consider now the 1-particle reduced density of the Jastrow trial state. We have by the +translation invariance that ρ(1) +Jas = ρ(1) = ρ. We nonetheless compute it here, as we need the +formula in terms of (linked) diagrams. We have similarly as before +ρ(1) +Jas(x1) = N +CN +˙ +� +2≤i≤N +f 2 +1i +� +2≤i N. +We again expand out the determinant and the X1 +p’s. For each summand π ∈ Sp+1 and +G ∈ G1 +p we again think of them together as a diagram (π, G) ∈ D1 +p. The formula for ρ(1) +Jas in +terms of diagrams is +ρ(1) +Jas = N! +CN +∞ +� +p=0 +1 +p! +� +(π,G)∈D1p +Γ1 +π,G = N! +CN + +ρ(1) + +∞ +� +p=1 +1 +p! +� +(π,G)∈D1p +Γ1 +π,G + + . +19 + +As for the normalization we write out the diagrams in terms of their linked components. There +is a distinguished linked component, namely the one containing the vertex {1}. We will write +its size as p∗. It is convenient to take “size” to mean number of internal vertices, i.e. p∗ = 0 +if {1} is not connected to any other vertex by either an edge in G or an edge in π. Similarly +“number of linked components” means disregarding the distinguished one. +Analogously to the computation in Equation (3.1) we thus get for any p ≥ 0 +1 +p! +� +(π,G)∈D1p +Γ1 +π,G = +� +(π∗,G∗)∈L1p +Γ1 +π∗,G∗ +p! ++ + + +∞ +� +k=1 +1 +k! +� +p∗≥0 +� +p1≥2 +· · · +� +pk≥2 +χ(� +ℓ∈{∗,1,...,k} pℓ=p) +� +(π∗,G∗)∈L1p∗ +Γ1 +π∗,G∗ +p∗! +× +� +(π1,G1)∈Lp1 +· · · +� +(πk,Gk)∈Lpk +Γπ1,G1 +p1! +· · · Γπk,Gk +pk! + + , +where the superscript 1 refers to the slightly modified structure as described in Definition 3.1, +where there may be no g-edges connecting to {1}, and there is no integration over x1. Note +that (π1, G1) ∈ Lp1, . . . , (πk, Gk) ∈ Lpk only deal with internal vertices. +Again here we take the sum over p’s. We are allowed to permute the p-sum inside of the +p∗- and p1, . . . , pk-sums if the sums over linked diagrams are absolutely summable. That is, if +� +p≥0 +1 +p! +������ +� +(π,G)∈L1p +Γ1 +π,G +������ +< ∞, +� +p≥2 +1 +p! +������ +� +(π,G)∈Lp +Γπ,G +������ +< ∞, +then we have, as for the normalization in Equation (3.2), that +ρ(1) +Jas = N! +CN +∞ +� +p=0 +1 +p! +� +(π,G)∈D1p +Γ1 +π,G += N! +CN + +� +p∗≥0 +� +(π∗,G∗)∈Lp∗ +Γ1 +π∗,G∗ +p∗! + + × + + +∞ +� +k=0 +1 +k! + +� +p≥2 +� +(π,G)∈Lp +Γπ,G +p! + + +k + += +� +p≥0 +� +(π,G)∈L1p +Γ1 +π,G +p! += ρ(1) + +� +p≥1 +1 +p! +� +(π,G)∈L1p +Γ1 +π,G, +where we used Equation (3.2) and that the p = 0 term just gives the 1-particle density of the +Slater determinant. Thus, by translation invariance, we have +ρ = ρ(1) = ρ(1) +Jas = +� +p≥0 +1 +p! +� +(π,G)∈L1p +Γ1 +π,G. +3.0.3 +Calculation of the 2-particle reduced density +Let us now compute the 2-particle reduced density. As before by expanding all the f 2 = 1 + g +factors apart from the factor f12 we get +ρ(2) +Jas = N! +CN +f 2 +12 +∞ +� +p=0 +ˆ +X2 +p∆p+2 dx3 . . . dxp+2, +X2 +p = +� +G∈G2p +� +e∈G +ge, +20 + +where G2 +p is as in Definition 3.1. +We again decompose the diagrams into linked components. However, we need to distinguish +between the cases where {1} and {2} are both in the same component or in two different +components. The computation is analogous to the computation above. We get +ρ(2) +Jas(x1, x2) = f 2 +12 + + + + +� +p1,p2≥0 +� +(π1,G1)∈L1 +p1 +(π2,G2)∈L1 +p2 +Γ1 +π1,G1(x1)Γ1 +π2,G2(x2) +p1!p2! +� +�� +� +{1} and {2} in different linked components ++ +� +p12≥0 +� +(π12,G12)∈L2p12 +Γ2 +π12,G12(x1, x2) +p12! +� +�� +� +{1} and {2} in same linked component + + + + += f 2 +12 + +ρ(1) +Jas(x1)ρ(1) +Jas(x2) + +� +p12≥0 +1 +p12! +� +(π12,G12)∈L2p12 +Γ2 +π12,G12(x1, x2) + + +after pulling in the sum over p ≥ 0. 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. The condition of absolute convergence is +� +p≥2 +1 +p! +������ +� +(π,G)∈Lp +Γπ,G +������ +< ∞, +� +p≥0 +1 +p! +������ +� +(π,G)∈L1p +Γ1 +π,G +������ +< ∞, +� +p≥0 +1 +p! +������ +� +(π,G)∈L2p +Γ2 +π,G +������ +< ∞. +3.0.4 +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. We introduce the relevant diagrams and decompose these according +to their linked components. 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. They are either in 1, 2 or 3 different components. Thus, schematically +��(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 +� +. +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). Thus, +ρ(3) +Jas(x1, x2, x3) = f 2 +12f 2 +13f 2 +23 +� +ρ3 + ρ +� +p≥0 +1 +p! +� +(π,G)∈L2p +� +Γ2 +π,G(x1, x2) + Γ2 +π,G(x1, x3) + Γ2 +π,G(x2, x3) +� ++ +� +p≥0 +1 +p! +� +(π,G)∈L3p +Γ3 +π,G(x1, x2, x3) +� +. +All the p = 0-terms together give ρ(3) by Wick’s rule. The condition for absolute convergence +is that for any q ≤ 3 we have � +p≥0 +1 +p! +��� +� +(π,G)∈Lq +p Γq +π,G +��� < ∞. +3.0.5 +Summarising the results +For the absolute convergence we have +21 + +Lemma 3.2. There exists a constant c > 0 such that if sa3ρ log(b/a)(log N)3 < c, then +� +p≥0 +1 +p! +������ +� +(π,G)∈Lq +p +Γq +π,G +������ +< ∞ +for any 0 ≤ q ≤ 3. +Remark 3.3. 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). The criterion +for absolute convergence reads +sup +x1,...,xn +� +1≤i 0, where the first condition is the “stability condition” of the tree- +graph bound [PU09, Proposition 6.1; Uel18] and ˆγ(k) := +´ +Λ γ(x)e−ikx dx. +We give the proof of Lemma 3.2 in Section 3.1. Thus, we have the following. +Theorem 3.4. There exists a constant c > 0 such that if sa3ρ log(b/a)(log N)3 < c, then +CN +N! = exp + + +∞ +� +p=2 +1 +p! +� +(π,G)∈Lp +Γπ,G + + , +ρ(1) +Jas = ρ(1) + +∞ +� +p=1 +1 +p! +� +(π,G)∈L1p +Γ1 +π,G, +ρ(2) +Jas = f 2 +12 + +ρ(2) + +∞ +� +p=1 +1 +p! +� +(π,G)∈L2p +Γ2 +π,G + + . +ρ(3) +Jas = f 2 +12f 2 +13f 2 +23 +� +ρ(3) + ρ +� +p≥1 +1 +p! +� +(π,G)∈L2p +� +Γ2 +π,G(x1, x2) + Γ2 +π,G(x1, x3) + Γ2 +π,G(x2, x3) +� ++ +� +p≥1 +1 +p! +� +(π,G)∈L3p +Γ3 +π,G +� +. +(3.3) +The first three formulas are the same as those of [GGR71, Equations (3.19), (4.9) and (8.4)]. +Our main contribution is to give a criterion for convergence, and hence for validity of the +formulas. +Remark 3.5. The factor s(log N)3 results from the bound in Lemma 2.12. 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 2.2. That is, the condition for absolute convergence would +be N1/3a3ρ log(b/a) < c for some constant c > 0. +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 4.1. 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 2.13) is smaller than +the claimed energy contribution from the interaction (∼ Na3ρ5/3, see Theorem 1.3). For this +reason we need the polyhedron of Section 2.2. +Remark 3.6. 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. For different boundary conditions, one has to take into account +that this equality is not valid. In general one has for ρ(2) +Jas that +ρ(2) +Jas = f 2 +12 + +ρ(2) + +∞ +� +p=1 +1 +p! +� +(π,G)∈L2p +Γ2 +π,G + +� +ρ(1) +Jas(x1)ρ(1) +Jas(x2) − ρ(1)(x1)ρ(1)(x2) +� + + += f 2 +12 + +ρ(2) + +∞ +� +p=1 +1 +p! +� +(π,G)∈L2p +Γ2 +π,G + +� +p1,p2≥0 +p1+p2≥1 +1 +p1!p2! +� +(π1,G1)∈L1 +p1 +(π2,G2)∈L1 +p2 +Γ1 +π1,G1(x1)Γ1 +π2,G2(x2) + + . +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. +Remark 3.7. By following the same procedure as in the previous sections, one can equally +well get formulas for the higher order reduced particle densities. Similarly one can extend the +absolute convergence, Lemma 3.2, to any q, only one may have to change the constant c > 0 +to depend on q. +3.1 +Absolute convergence +We now prove Lemma 3.2, i.e. that the appropriate sums are absolutely convergent. +Proof of Lemma 3.2. We consider the four sums � +p≥0 +1 +p! +��� +� +(π,G)∈Lq +p Γq +π,G +���, q = 0, 1, 2, 3 one by +one. +3.1.1 +Absolute convergence of the Γ-sum +Consider first +1 +p! +� +(π,G)∈Lp Γπ,G. Split the sum according to the number of connected compo- +nents of G, labelled as (G1, . . . , Gk) of sizes n1, . . . , nk. We call these clusters. (Note that +“connected” only refers to the graph G, and is independent of the permutation π.) Name the +vertices in G1 as {1, . . . , n1}, in G2 as {n1 +1, . . . , n1 +n2} and so on. Then we have (for p ≥ 2) +1 +p! +� +(π,G)∈Lp +Γπ,G = +∞ +� +k=1 +1 +k! +� +n1,...,nk≥2 +1 +n1! · · ·nk!χ(� nℓ=p) +� +G1,...,Gk +Gℓ∈Cnℓ +� +π∈Sp +(−1)πχ((π,∪Gℓ)∈Lp) +× +˙ +k +� +ℓ=1 +� +e∈Gℓ +ge +p +� +j=1 +γ(1) +N (xj; xπ(j)) dx1 . . . dxp, +where Cn denotes the set of connected graphs on n (labelled) vertices. The factorial factors are +similar to those of Equation (3.1). Indeed, the factor 1/(k!) comes from counting the possible +23 + +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +G1 +G2 +G3 +Figure 3.2: A linked diagram (π, G) decomposed into clusters G1, G2, G3. Dashed +lines denote g-edges, and arrows (i, j) denote that π(i) = j. +labelling of the clusters, and the factors 1/(n1!), . . . , 1/(nk!) come from counting the number of +ways to distribute the p = � nℓ vertices into the clusters and using the factor 1/(p!) already +present. +For the analysis we will need the following. +Definition 3.8. Let A1, . . . , Ak denote disjoint non-empty sets. +The truncated correlation +function is +ρ(A1,...,Ak) +t +:= +� +π∈S∪ℓAℓ +(−1)πχ((π,∪Gℓ) linked) +� +j∈∪ℓAℓ +γ(1) +N (xj; xπ(j)). +(3.4) +for some choice of connected graphs Gℓ ∈ CAℓ. The definition does not depend on the choice of +graphs Gℓ. +If the underlying sets A1, . . . , Ak are clear we will simply denote the truncated correlation +by their sizes, +ρ(|A1|,...,|Ak|) +t += ρ(A1,...,Ak) +t +. +The truncated correlation functions are also sometimes referred to as connected correlation +functions [GMR21, Appendix D]. +Remark 3.9. We write the characteristic function in Equation (3.4) as χ((π,∪Gℓ) linked) for ease +of generalizability to the cases in Sections 3.1.2 and 3.1.3 where we will need the notion of trun- +cated correlations also for some of the vertices being external. For the truncated correlations +it doesn’t matter which (if any) vertices are external, only which vertices are in which clusters. +Since 0 ≤ f ≤ 1 we have −1 ≤ g ≤ 0. Thus, by the tree-graph bound [PU09, Proposition 6.1; +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. Thus we get +∞ +� +p=2 +1 +p! +������ +� +(π,G)∈Lp +Γπ,G +������ +≤ +∞ +� +k=1 +1 +k! +� +n1,...,nk≥2 +1 +n1! · · ·nk! +˙ +� +T1,...,Tk +Tℓ∈Tnℓ +k +� +ℓ=1 +� +e∈Tℓ +|ge| +���ρ(n1,...,nk) +t +��� dx1 . . . dx� nℓ. +(3.5) +24 + +Here the vertices in Tℓ are the same as in Gℓ, i.e. T1 has vertices {1, . . . , n1}, T2 has vertices +{n1 + 1, . . . , n1 + n2} and so on. +In [GMR21, Equation (D.53)] the following formula, known as the Brydges-Battle-Federbush +(BBF) formula, is shown for the truncated correlation functions +ρ(A1,...,Ak) +t += +� +τ∈A(A1,...,Ak) +� +(i,j)∈τ +γ(1) +N (xi; xj) +ˆ +dµτ(r) det N (r), +(3.6) +where A(A1,...,Ak) is the set of all anchored trees on k clusters with vertices A1, . . . , Ak. (If the sets +A1, . . . , Ak are clear we will write A(|A1|,...,|Ak|) = A(A1,...,Ak) as for the truncated correlations.) +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. 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; xj). Finally, N is an I × J (square) matrix with entries Nij = rc(i)c(j)γ(1) +N (xi; xj), where +c(i) is the (label of the) cluster containing the vertex {i} and rℓℓ′ := rℓ′ℓ if ℓ > ℓ′. 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. +T1 +T2 +T3 +T4 +T5 +T6 +Figure 3.3: An anchored tree τ (arrows) and trees T1, . . . , T6 (dashed lines). +From [GMR21, Equation (D.57)] it follows that |det N | ≤ ρ +� nℓ−(k−1). To see this, one has +to adapt the argument in [GMR21, Lemma D.2] slightly. We sketch the argument here. +Lemma 3.10 ([GMR21, Lemmas D.2 and D.6]). The matrix N (r) satisfies |det N (r)| ≤ +ρ +� nℓ−(k−1) for all r ∈ [0, 1]k(k−1)/2. +Proof. First we bound ρ(p) = det[γ(1) +N (xi; xj)]1≤i,j≤p following the strategy of [GMR21, Lemma +D.2]. This is done by writing (as in [GMR21, Equations (D.8), (D.9)]) +γ(1) +N (xi; 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). +25 + +By the the Gram-Hadamard inequality [GMR21, Lemma D.1] we have +��ρ(p)�� = +���det[γ(1) +N (xi; xj)]1≤i,j≤p +��� ≤ +p +� +i=1 +∥αi∥ℓ2((2π/L)Z3) ∥βi∥ℓ2((2π/L)Z3) = ρp. +By modifying this argument exactly as described in the proof of [GMR21, Lemma D.6] and +noting that rℓℓ′ ≤ 1 one concludes the desired. +Remark 3.11. 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 (4.16). +In particular one concludes the bound +���ρ(n1,...,nk) +t +��� ≤ ρ +� nℓ−(k−1) +� +τ∈A(n1,...,nk) +� +(i,j)∈τ +���γ(1) +N (xi; xj) +��� . +(3.7) +Plugging this into Equation (3.5) above we get +∞ +� +p=2 +1 +p! +������ +� +(π,G)∈Lp +Γπ,G +������ +≤ +∞ +� +k=1 +1 +k! +� +n1,...,nk≥2 +1 +n1! · · · nk!ρ +� nℓ−(k−1) � +T1,...,Tk +Tℓ∈Tnℓ +� +τ∈A(n1,...,nk) +× +˙ +dx1 . . . dx� nℓ +k +� +ℓ=1 +� +e∈Tℓ +|ge| +� +(i,j)∈τ +���γ(1) +N (xi; xj) +��� . +To compute these integrals we note that by Lemma 2.2 +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. +That is, each factor of ge gives a contribution Ca3 log(b/a) after integration. The γ(1) +N -factors +we can bound by Lemma 2.12 as +ˆ ���γ(1) +N (x; y) +��� dy ≤ Cs(log N)3. +This takes care of all but one integration, which gives the volume factor L3. We shall compute +the integrations in the following order: +(1.) 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. +(2.) Consider {j0} as the root of Tℓ and pick any leaf {j} of Tℓ and integrate over xj. 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. +Hence the xj-integral contributes +´ +|g| by the translation invariance. +Remove {j} and its incident edge from Tℓ. +Repeat for all vertices in the cluster until only {j0} remain. (At this point the entirety +of Tℓ has been removed.) +26 + +(3.) Integrate over xj0. 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. Thus, the xj0-integral gives a contribution +´ +|γ(1) +N | +by the translation invariance. +Remove {j0} and its incident edge from τ. +(4.) Repeat steps (1.)-(3.) until all integrals have been computed. The final integral gives the +volume factor L3. +Steps (1.)-(3.) compute all integrations in one cluster. Repeating this process we integrate +over the clusters one by one and thus compute all the integrals. Note that each integration is +always over a coordinate associated to a leaf of the relevant graphs. This is a key point, since +then by translation invariance each integration contributes exactly +´ +|g| or +´ +|γ(1) +N | whichever is +appropriate. In total we thus have the bound +˙ +dx1 . . . dx� nℓ +k +� +ℓ=1 +� +e∈Tℓ +|ge| +� +(i,j)∈τ +���γ(1) +N (xi; xj) +��� ≤ +� +Ca3 log(b/a) +�� nℓ−k � +Cs(log N)3�k−1 L3. +This bound is for each summand τ, T1, . . . , Tk. By Cayley’s formula #Tn = nn−2 ≤ Cnn!, and +by [GMR21, Appendix D.5] #A(n1,...,nk) ≤ k!4 +� nℓ. Thus, we get +∞ +� +p=2 +1 +p! +������ +� +(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. This shows that � +p +1 +p! +� +(π,G)∈Lp Γπ,G is absolutely +convergent under this condition. +3.1.2 +Absolute convergence of the Γ1-sum +Consider now +1 +p! +� +(π,G)∈L1p Γ1 +π,G. The argument is almost identical to the argument above. We +again split the sum according to the connected components of G. Call these G∗, G1, . . . , Gk, +where G∗ is the distinguished connected component (cluster) containing the distinguished vertex +{1}. Exactly as for +1 +p! +� +(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! +� +(π,G)∈L1p +Γ1 +π,G(x1) += +∞ +� +k=0 +1 +k! +� +n∗≥0 +� +n1,...,nk≥2 +1 +n∗!n1! · · · nk!χ( +� +ℓ{∗,1,...,k} nℓ=p) +� +G1∈Cn1,...,Gk∈Cnk +G∗∈Cn∗+1 +� +π∈Sp+1 +(−1)π +× χ((π,∪ℓ∈{∗,1,...,k}Gℓ)∈L1p) +�� +� +ℓ∈{∗,1,...,k} +� +e∈Gℓ +ge +p+1 +� +j=1 +γ(1) +N (xj; xπ(j)) dx2 . . . dxp+1. +27 + +Here for the k = 0 term one should think of the n1, . . . , nk-sums as being an empty product +before it is an empty sum, i.e. it should give a factor 1. 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; xπ(j)) dx2 . . . dxp+1, +since (π, G∗) is trivially linked, since G∗ is connected. From here on, we won’t write out the +k = 0 term separately to make the formulas more concise. As before we use the tree-graph +inequality and the truncated correlation function (see Remark 3.9) to get +� +p≥0 +1 +p! +������ +� +(π,G)∈L1p +Γ1 +π,G +������ +≤ +∞ +� +k=0 +1 +k! +� +n∗≥0 +� +n1,...,nk≥2 +1 +n∗!n1! · · ·nk! +� +T1∈Tn1,...,Tk∈Tnk +T∗∈Tn∗+1 +× +˙ +� +ℓ∈{∗,1,...,k} +� +e∈Tℓ +|ge| +���ρ(n∗+1,n1,...,nk) +t +��� dx2 . . . dx� +ℓ∈{∗,1,...,k} nℓ+1. +To bound this we use the same bound, Equation (3.7), on the truncated correlations as before. +It reads +���ρ(n∗+1,n1,...,nk) +t +��� ≤ ρ +� +ℓ∈{∗,1,...,k} nℓ+1−(k+1−1) +� +τ∈A(n∗+1,n1,...,nk) +� +(i,j)∈τ +���γ(1) +N (xi; xj) +��� . +Computing the integrals is as before, with a few differences. During each repeat (apart from +the last one) of step (1.) we pick not just any leaf j0 but a leaf j0 not in the cluster containing +{1}. (Since any tree has at least 2 leaves, this is always possible.) For each of these repeats, +the argument is the same as before. For the last repeat of step (1.) where only the cluster +containing {1} remains we follow step (2.) with the slight change, that the root is chosen to be +{1}. (There are no γ(1) +N -factors left, so we are free to choose any vertex as the root.) There is +then no step (3.) since we do not integrate over x1. +This has the following effect. First, the last variable x1 is not integrated over, so there is +no volume factor L3. And second, there are k integrals +´ +|γ(1) +N | instead of k − 1 (since there are +k + 1 many clusters including the distinguished one). For the bounds of the sum of all terms +we use that #A(n∗+1,n1,...,nk) ≤ (k + 1)!4 +� +ℓ∈{∗,1,...,k} nℓ+1. Thus, uniformly in x1 +� +p≥0 +1 +p! +������ +� +(π,G)∈L1p +Γ1 +π,G +������ +≤ Cρ +� ∞ +� +n∗=0 +� +Ca3ρ log(b/a) +�n∗ +�  + +∞ +� +k=0 +(k + 1) +� +Cs(log N)3�k +� ∞ +� +n=2 +(Ca3ρ log(b/a))n−1 +�k + +≤ Cρ < ∞, +for sa3ρ log(b/a)(log N)3 sufficiently small. This shows that � +p +1 +p! +� +(π,G)∈L1p Γ1 +π,G is absolutely +convergent under this condition. +3.1.3 +Absolute convergence of the Γ2-sum +For the third sum the argument is mostly analogous. +There are a few changes needed for +the argument. 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. +One computes +1 +p! +� +(π,G)∈L2p +Γ2 +π,G = Σdifferent + Σsame, +(3.8) +where +Σdifferent = +∞ +� +k=0 +1 +k! +� +n∗,n∗∗≥0 +� +n1,...,nk≥2 +1 +� +ℓ nℓ!χ(� nℓ=p) +� +G1∈Cn1,...,Gk∈Cnk +G∗∈Cn∗+{1} +G∗∗∈Cn∗∗+{2} +× +˙ � +ℓ +� +e∈Gℓ +geρ(n∗+{1},n∗∗+{2},n1,...,nk) +t +dx3 . . . dxp+2, +Σsame = +∞ +� +k=0 +1 +k! +� +n∗≥1 +� +n1,...,nk≥2 +1 +� +ℓ nℓ!χ(� nℓ=p) +� +G1∈Cn1,...,Gk∈Cnk +G∗∈Cn∗+{1,2} +(1,2)/∈G∗ +× +˙ � +ℓ +� +e∈Gℓ +geρ(n∗+{1,2},n1,...,nk) +t +dx3 . . . dxp+2. +(3.9) +Here � +ℓ and � +ℓ are over ℓ ∈ {∗, ∗∗, 1, . . ., k} or ℓ ∈ {∗, 1, . . . , k}, whichever is appropriate. +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}). This set has exactly n∗ internal +vertices. For p = 0 one has to interpret the empty product of integrals as a factor 1. +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 ∗). Note +that in the second contribution we have n∗ ≥ 1. Indeed, {1} and {2} are connected, but not +by an edge. Hence they must be connected by a path of length ≥ 2, which necessarily goes +through at least one vertex {j}, j ̸= 1, 2. +We treat the two cases separately. 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 (3.7). The latter reads +���ρ(n∗+{1},n∗∗+{2},n1,...,nk) +t +��� ≤ ρ(� +ℓ∈{∗,∗∗,1,...,k} +2)−(k+2−1) +� +τ∈A(n∗+{1},n∗∗+{2},n1,...,nk) +� +(i,j)∈τ +���γ(1) +N (xi; xj) +��� . +The integration procedure is slightly modified compared to that of Section 3.1.2. In the anchored +tree there is a path between (the cluster containing) {1} and (the cluster containing) {2}. For +the edge incident to (the cluster containing) {1} on this path, we bound |γ(1) +N | ≤ ρ. This cuts +the anchored tree into two anchored trees τ1, τ2 such that (with a slight abuse of notation) +1 ∈ τ1 and 2 ∈ τ2. We may follow the integration procedure exactly as for the Γ1-sum for each +of the anchored trees τ1 and τ2. Recall the bound +#A(n∗+{1},n∗∗+{2},n1,...,nk) ≤ (k + 2)!4 +� +ℓ∈{∗,∗∗,1,...,k} nℓ+2 ≤ C(k2 + 1)k!4 +� +ℓ∈{∗,∗∗,1,...,k} nℓ. +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  + +∞ +� +k=0 +(k2 + 1) +� +Cs(log N)3�k +� ∞ +� +n=2 +(Ca3ρ log(b/a))n−1 +�k + +≤ Cρ2 < ∞. +(3.10) +Now we consider the case where {1} and {2} are in the same distinguished cluster. Here we +may readily apply the bound in Equation (3.7) on the truncated correlation but we need to +be a bit careful in applying the tree-graph bound. 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}. To still apply the tree-graph bound, +we define +˜ge := +� +ge +e ̸= (1, 2) +0 +e = (1, 2). +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|. +We again have to modify the integrations slightly. +The integrations over all clusters apart +from the distinguished one may be computed as for the Γ- and Γ1-sums. For the distinguished +cluster with {1} and {2} there is some path of g-edges connecting them. Pick the unique edge +on this path incident with {1} and bound |g| ≤ 1 for this factor. This splits the tree T∗ into +two trees T 1 +∗ and T 2 +∗ with 1 ∈ T 1 +∗ and 2 ∈ T 2 +∗ . 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. 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∗ +�  + +∞ +� +k=0 +(k + 1) +� +Cs(log N)3�k +� ∞ +� +n=2 +(Ca3ρ log(b/a))n−1 +�k + +≤ Ca3ρ3 log(b/a) < ∞. +(3.11) +We conclude that +� +p≥0 +1 +p! +������ +� +(π,G)∈L2p +Γ2 +π,G +������ +≤ Cρ2 < ∞, +uniformly in x1, x2 for sufficiently small sa3ρ log(b/a)(log N)3. +3.1.4 +Absolute convergence of the Γ3-sum +The argument for the last sum is completely analogous to the argument for the Γ2-sum. We have +to distinguish between different cases of the clusters containing the external vertices {1, 2, 3}. +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. One then deals with the +different cases exactly as we did for the Γ2-sum. We skip the details. This concludes the proof +of Lemma 3.2. +4 +Energy of the trial state +In this section we bound the energy of the trial state ψN defined in Equation (2.1). Recall +Equation (2.2). By Theorem 3.4 we have (for sa3ρ log(b/a)(log N)3 sufficiently small) +ρ(2) +Jas(x1, x2) = f(x1 − x2)2 + +ρ(2)(x1, x2) + +∞ +� +p=1 +1 +p! +� +(π,G)∈L2p +Γ2 +π,G + + . +We can expand ρ(2) in x1 − x2 using Lemma 2.14. The second term is an error term we have to +control. Additionally, also the three-body term is an error we have to control. We claim that +Lemma 4.1. 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! +� +(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) +� +. +Lemma 4.2. 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. +We give the proof of these lemmas in Sections 4.2 and 4.3 below. For the three-body term, we +additionally have the bound ρ(3) ≤ Cρ5|x1 − x2|2|x1 − x3|2|x2 − x3|2 by Lemma 2.15. Com- +bining now Lemmas 2.13, 2.14, 4.1 and 4.2, Theorem 3.4 and Equation (2.2) we thus get (for +sa3ρ log(b/a)(log N)3 sufficiently small and N sufficiently large) +⟨ψN|HN|ψN⟩ = 3 +5(6π)2/3ρ2/3N +� +1 + O(N−1/3) + O(s−2) +� ++ L3 +ˆ +dx +� +|∇f(x)|2 + 1 +2v(x)f(x)2 +� +× +� +(6π2)2/3 +5 +ρ8/3|x|2 +� +1 − 3(6π2)2/3 +35 +ρ2/3|x|2 ++ O(N−1/3) + O(s−2) + O(N−1/3ρ2/3|x|2) + O(ρ4/3|x|4) +� ++ O +� +a6ρ4(log(b/a))2� +s3a6ρ2(log b/a)2(log N)9 + 1 +�� ++ O +� +a3ρ3+2/3|x|2� +s5a12ρ4(log(b/a))5(log N)16 + b4ρ4/3 + log(b/a) +�� � ++ +˚ +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 +��� +. +(4.1) +31 + +We will choose N (really L, see Remark 2.8) some large negative power of a3ρ, so errors with +N−1/3 are subleading. 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) +� +, +(4.2) +by Definition 1.1 since f = +1 +1−a3/b3f0 for |x| ≤ b and b > R0, the range of v. For the higher +moments we recall that |∇f0| ≤ |∇fhc| = 3a3 +|x|4 for |x| ≥ a by Lemma 2.2. 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. +(4.3) +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). +For the lower moment, we have by Equation (2.4) +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 +(4.4) +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. By Lemma 2.2 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. +(4.5) +Hence +ˆ +dx +� +|∇f(x)|2 + 1 +2v(x)f(x)2 +� +≤ Ca. +This concludes the bounds on all the terms in Equation (4.1) arising from the 2-body term. To +bound those arising from the 3-body term we bound |x1 − x3| ≤ 2b in the support of ∇f12∇f23 +and f13 ≤ 1. By the translation invariance one integration gives a volume factor L3. The +32 + +remaining two integrals then both give the same contribution. 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 +. +Using integration by parts and Lemma 2.2, 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. +(4.6) +Plugging all this into Equation (4.1) 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 +�� +. +(4.7) +We choose L ∼ a(a3ρ)−10 still ensuring that +LkF +2π +is rational. +(More precisely one chooses +L ∼ a(kFa)−30, since ρ is defined in terms of L.) Then N ∼ (a3ρ)−29. Choose moreover +b = a(a3ρ)−β, +s ∼ (a3ρ)−α| log(a3ρ)|−δ. +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. 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). +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. 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.e. +⟨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) +� +(4.8) +for a3ρ small enough. Note that for this choice of s, N we have s ∼ N6/203(log N)3. Thus any +Q with N4/3 ≪ Q ≤ CNC satisfies the condition Q−1/4 ≤ Cs−1 of Definition 2.7. +4.1 +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. First we show that we may choose periodic boundary +conditions in the small boxes instead of using Dirichlet boundary conditions. The setting and +argument is due to Robinson [Rob71, Lemmas 2.1.12 and 2.1.13]. We present a slightly modified +version in [MS20, Section C]. +Lemma 4.3 ([MS20; Rob71]). Let 0 < d < L/2 be a cut-off, let HD +N,L+2d = �N +j=1 −∆D +j,L+2d + +� +i 0 with ˜f (n) ց f. +37 + +Then one readily applies the Lebesgue dominated convergence theorem to exchange the limit +˜f (n) → f with the relevant sums and integrals.) Taking x1 = x2 in this we have DN = 0 and +ρ(2) = 0. This shows Equation (4.13). 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 +� +. +(4.14) +Since both ξsmall,0 and ξ0 are symmetric in x1 and x2 all first order terms vanish. We are left +with bounding the second derivatives. For ξsmall,0 we have +Lemma 4.8. For any µ, ν = 1, 2, 3 we have +��∂µ +x1∂ν +x1ξsmall,0 +�� ≤ Ca3ρ3+2/3 log(b/a) +uniformly in x1, x2. Here ∂µ +x1 denotes the derivative in the xµ +1-direction. +The proof of this lemma is a (not very insightful) computation. We give it in Appendix A.1. +Next we consider ∂µ +x1∂ν +x1ξ0. We write ξ0 in terms of truncated densities as in Equation (3.9), +i.e. +ξ0 = +∞ +� +k=0 +1 +k! +� +n1,...,nk≥2 +χ(k≥5,nℓ=2 or k≥1,� nℓ=2k+1) +1 +� +ℓ nℓ! +� +G1,...,Gk +Gℓ∈Cnℓ +× +˙ � +ℓ +� +e∈Gℓ +geρ({1},{2},n1,...,nk) +t +dx3 . . . dxp+2. +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,...,nk) +t +. We thus need to calculate ∂µ +x1∂ν +x1ρ({1},{2},n1,...,nk) +t +. For +this we use the definition in Equation (3.4) rather than the formula in Equation (3.6). In +Equation (3.4) 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}. Taking the derivatives then amounts to replacing +either one of these edges by its second derivative or both of them by their first derivatives. +Thus, using that γ(1) +N (xi; xj) = γ(1) +N (xj; 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,...,nk) +t += ∂µ +x1∂ν +x1 +� +π∈Sp +(−1)πχ((π,∪Gℓ)∈Lp) +p +� +j=1 +γ(1) +N (xj; xπ(j)) += +� +π∈Sp +(−1)πχ((π,∪Gℓ)∈Lp) +� +j̸=1,j̸=π−1(1) +γ(1) +N (xj; xπ(j)) +� +2∂µ +x1∂ν +x1γ(1) +N (x1; xπ(1))γ(1) +N (xπ−1(1); x1) ++2∂µ +x1γ(1) +N (x1; xπ(1))∂ν +x1γ(1) +N (xπ−1(1); x1) +� +. +(4.15) +With this formula we may then redo the computation of [GMR21, Equation (D.53)] only now +some of the γ(1) +N -factors (precisely 1 or 2 of them) carry derivatives. +The γ(1) +N -factors with +derivatives may end up in the anchored tree, or they may end up in the matrix N (r). If they +end up in N (r) it is explained around [GMR21, Equation (D.9)] how to modify Lemma 3.10. +38 + +One simply includes factors ikµ in the definition of (some of) the functions αi (and not of βj) +in the proof of Lemma 3.10. Since we may bound |k| ≤ Cρ1/3 for k ∈ PF we get +���det ˜ +N(r) +��� ≤ + + + + + +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 , +(4.16) +where ˜ +N(r) is the appropriate modification of N (r). To get the formula for ρt we need also to +consider two cases for how the anchored tree looks. There could be both an incoming and an +outgoing edge to/from the vertex {1}. And if there is just one edge to/from {1} it could be +either an incoming or an outgoing edge. Since γ(1) +N is real, incoming and outgoing edges gives +the same factor γ(1) +N (x1; xj). A simple calculation (essentially just undoing the product rule) +then shows that +∂µ +x1∂ν +x1ρ({1},{2},n1,...,nk) +t += +� +∂∈{1,∂µ +x1,∂νx1,∂µ +x1∂νx1} + + +� +τ∈A({1},{2},n1,...,nk) +two edges to/from {1} +∂ +� +γ(1) +N (xj2, x1)γ(1) +N (x1; xj1) +� ++ +� +τ∈A({1},{2},n1,...,nk) +one edge to/from {1} +∂γ(1) +N (x1; xj1) + + +� +(i,j)∈τ +i,j̸=1 +γ(1) +N (xi; xj) +ˆ +dµτ(r) det ˜ +N∂(r), +(4.17) +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.e. +Equation (4.16) reads +���det ˜ +N∂(r) +��� ≤ Cρ +� nℓ+2−(k+2−1)+(2−#∂)/3, +where #∂ denotes the number of derivatives in ∂, i.e. #1 = 0, #∂µ +x1 = 1 and #∂µ +x1∂ν +x1 = 2. +We denote the contribution of the two terms in Equation (4.17) to ∂µ +x1∂ν +x1ξ0 by (∂µ +x1∂ν +x1ξ0)→•→ +and (∂µ +x1∂ν +x1ξ0)•→ respectively. +We first deal with the second term of Equation (4.17) where there is just one edge to/from +{1} in the anchored tree. We may bound the contribution of this term almost exactly as in the +proof of Lemma 3.2. We give a sketch here. Using Equation (4.16) we get the bound +≤ C +� +∂∈{1,∂µ +x1,∂νx1,∂µ +x1∂νx1} +ρ(2−#∂)/3 +� +τ∈A({1},{2},n1,...,nk) +���∂γ(1) +N (x1; xj1) +��� +× +� +(i,j)∈τ +i,j̸=1 +���γ(1) +N (xi; xj) +��� ρ(� +ℓ nℓ+2)−(k+2−1), +(4.18) +where again #∂ denotes the number of derivatives in ∂. +To bound the integrations we again follow the strategy of the proof of the Γ2-sum of +Lemma 3.2, Section 3.1.3. 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; xj1). +This we bound by |∂γ(1) +N | ≤ Cρ1+#∂/3. The integrations can then be performed exactly as in +39 + +Section 3.1.3. We conclude the bound +˙ +k +� +ℓ=1 +� +e∈Tℓ +|ge| +���∂γ(1) +N (x1; xj1) +��� +� +(i,j)∈τ +i,j̸=1 +���γ(1) +N (xi; xj) +��� dx3 . . . dx� nℓ+2 +≤ ρ1+#∂/3 � +Ca3 log(b/a) +�� nℓ−k � +Cs(log N)3�k . +Again, as in the proof of Lemma 3.2 we have by Cayley’s formula that #Tn = nn−2 ≤ Cnn! +and by [GMR21, Appendix D.5] that #A({1},{2},n1,...,nk) ≤ (k + 2)!4 +� nℓ+2 ≤ C(k2 + 1)k!4 +� nℓ. +Following the same arguments as for Equation (4.10) 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 +� +(4.19) +uniformly in x1, x2. +Next consider the first term of Equation (4.17). The argument is almost the same, only +we have to distinguish between which γ(1) +N -factor(s) the derivatives in ∂ hits. We consider the +case ∂ = ∂µ +x1∂ν +x1. The other cases are similar. Suppose that the γ(1) +N -edge on the path (in the +anchored tree) from {1} to {2} is γ(1) +N (x1; xj1) and the γ(1) +N -factor not on the path is γ(1) +N (xj2; x1). +We distinguish between three cases: +1. If both derivatives are on γ(1) +N (x1; xj1) we may bound this exactly as above. +2. If one derivative is on γ(1) +N (x1; xj1) (say ∂ν +x1) and one derivative (say ∂µ +x1) is on γ(1) +N (xj2; x1) +we bound |∂ν +x1γ(1) +N (x1; xj1)| ≤ Cρ4/3. +Then the argument is similar, only now one of +the γ(1) +N -integrations is with ∂µγ(1) +N +instead. Thus, in the computation leading to Equa- +tion (4.19) we should replace one factor Csρ1/3(log N)3 with +´ +|∂µγ(1) +N | dx. +3. If both derivatives are on γ(1) +N (xj2; x1) then analogously we bound |γ(1) +N (x1; xj1)| ≤ Cρ and +in the computation leading to Equation (4.19) we should replace one factor Csρ2/3(log N)3 +with +´ +|∂µ∂νγ(1) +N | dx. +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. 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� +(4.20) +40 + +uniformly in x1, x2. 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. +Here we have +Lemma 4.9. The polyhedron P from Definition 2.7 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. +The proof of Lemma 4.9 is a long and technical computation, which we give in Appendix B. +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. +By combining this with Equations (4.19) and (4.20) 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 +� +(4.21) +uniformly in x1, x2. Combining Lemmas 4.6 and 4.8 and Equations (4.12), (4.14) and (4.21) +and using that for any real number t > 0 and integer n ≥ 1 we may bound t ≲ tn + 1 this +shows the desired. +Remark 4.10 (Treating more diagrams as small). One can improve the error bound in The- +orem 1.3 slightly by treating more diagrams as small. +This is similar to what is done in +[BCGOPS22] for the dilute Bose gas. We sketch the overall idea. +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. Then choose as “large” the diagrams for which the bound +in Equation (4.10) gives contributions to the energy density much smaller than ρ5/3(a3ρ)2−ε. +This happens for k0 > K for some large K ∼ ε−1. 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. In +total we would then get an error of size Oε(ρ5/3(a3ρ)2−ε) in Theorem 1.3. +41 + +4.3 +Subleading 3-particle diagrams (proof of Lemma 4.2) +Proof of Lemma 4.2. Recall the formula for ρ(3) +Jas of Theorem 3.4. In this formula there are terms +like ρ � +p≥1 +1 +p! +� +(π,G)∈L2p Γ2 +π,G(x2, x3). We have ρ = ρ(1) +Jas(x1) = � +p′≥1 +1 +p′! +� +(π′,G′)∈L1 +p′ Γ1 +π′,G′(x1) by +translation invariance. 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}. 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. 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. With this, the formula +for ρ(3) +Jas of Theorem 3.4 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! +� +(π,G)∈ ˜L3p +Γ3 +π,G +� +. +We split the diagrams in ˜L3 +p into two groups, large and small similarly to the proof of Lemma 4.1. +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. (This k is exactly the same k as in the proof of +Lemma 3.2.) 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. (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 3.1.2 and 3.1.3.) We may still think +of ng +n∗ +g as the “number of added vertices”. As for Equation (4.10) we have (for p = 2k0+ng0) +������������ +1 +p! +� +(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. +(4.22) +The main difference compared to Equation (4.10) is that we here allow diagrams that are not +linked. +This doesn’t matter, since when we compute the integrals (as in Section 3.1.3) 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 3.1.3. We split the diagrams into two (exhaustive) groups: +1. Small diagrams with 1 ≤ k ≤ 2, ng = 0, n∗ +g = 0, and +2. Large diagrams as the rest, i.e. with +(A) k ≥ 3, or +42 + +(B) ng + n∗ +g ≥ 1. +As in Section 4.2, the splitting is motivated by counting powers in Equation (4.22). Note that +for p ≥ 1 the diagrams with k = 0, ng = 0, n∗ +g = 0 are not present. We write +� +p≥1 +1 +p! +� +(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. Exactly +as in Equation (4.12) we may bound, using Equation (4.22) +|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 +� +. +For the small diagrams we have +Lemma 4.11. We have +|ξ3 +small| ≤ Ca3ρ4 log(b/a) +uniformly in x1, x2, x3. +As with Lemma 4.8, the proof is simply a computation, which we give in Appendix A.2. We +conclude the desired. +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. 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. +5.1 +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, +(5.1) +see Section 2.1 and recall Definition 1.6. Thus, it is the same as the s-wave scattering function +in 4 dimensions. In particular it satisfies the bound +Lemma 5.1 ([LY01, Lemma A.1], Lemma 2.2). The scattering function satisfies +� +1 − a2 +|x|2 +� ++ ≤ +f0(x) ≤ 1 for all x and |∇f0(x)| ≤ 2a2 +|x|3 for |x| > a. +As for the 3-dimensional setting we consider the trial state +ψN(x1, . . . , xN) = +1 +√CN +� +i 0 such that if sa2ρ log(b/a)(log N)2 < c, then +the formulas in Equation (3.3) hold (with ρ(n) +Jas and Γn +π,G interpreted as appropriate in the two- +dimensional setting). +The analogues of Lemmas 4.1 and 4.2 read +Lemma 5.8. 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! +� +(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 +�� +. +45 + +The proof is again similar to the 3-dimensional case replacing the bounds on +´ +|g| and +´ +|γ(1) +N | +as in Equation (5.2) above. Apart from this, there are two main changes. The first is in the +proof of the analogue of Lemma 4.6, namely Equation (A.1), where one bounds +´ +|x|2|1 − f 2|. +In two dimensions this bound is, using Lemma 5.1, +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. +The other main difference is for the analogue of Lemma 4.9. Here the 2-dimensional analogue +reads +Lemma 5.9. The polygon P from Definition 5.2 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. +The proof is similar to that of Lemma 4.9 given in Appendix B only one skips Appendix B.2 +and notes that R = LkF +2π ∼ N1/2. +Putting together the formulas in Theorem 5.7 with the bounds in Lemmas 5.5, 5.6 and 5.8 +we easily find the analogue of Equation (4.1). We then need to bound a few terms. Following +the type of arguments of Section 4, namely Equations (4.2), (4.3), (4.4), (4.5) and (4.6) and +using Lemma 5.1 we get the bounds +ˆ � +|∇f(x)|2 + 1 +2v(x)f(x)2 +� +|x|n dx ≤ + + + + + +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. +Plugging this into the analogue of Equation (4.1) we get the analogue of Equation (4.7), +⟨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 +�� +. +(5.3) +As above, we can choose L ∼ a(a2ρ)−10 still ensuring that LkF +2π is rational. (More precisely one +chooses L ∼ a(kFa)−20, since ρ is defined in terms of L.) Then N ∼ (a2ρ)−19. Choose moreover +b = a(a2ρ)−β, +s ∼ (a2ρ)−α| log(a2ρ)|−γ. +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�� +(5.4) +for a2ρ small enough. Note that for this choice of s, N we have s ∼ N4/133(log N)10/7. Thus +any Q with N3/2 ≪ Q ≤ CNC satisfies the condition Q−1/2 ≤ Cs−2 of Definition 5.2. +The extension to the thermodynamic limit of Section 4.1 is readily generalized. We thus +conclude the proof of Theorem 1.7. +5.2 +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, +(5.5) +see Section 2.1 and recall Definition 1.8. Thus, it is the same as the s-wave scattering function +in 3 dimensions. In particular it satisfies the bound +Lemma 5.10 ([LY01, Lemma A.1], Lemma 2.2). The scattering function satisfies +� +1 − +a +|x| +� ++ ≤ +f0(x) ≤ 1 for all x and |∂f0(x)| ≤ +a +|x|2 for |x| > a. +Before giving the proof of Theorem 1.9 we first compare our definition of the scattering length +to that of [ARS22]. In [ARS22] the following definition is given. +Definition 5.11 ([ARS22, Section 1.3]). 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. +The value of aodd is independent of R > R0 so aodd is well-defined. We claim that +Proposition 5.12. The p-wave scattering length a defined in Definition 1.8 and the odd-wave +scattering length aodd defined in Definition 5.11 agree, i.e. a = aodd. +Proof. 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. Thus, we have +4 +R − aodd += inf +�ˆ R +−R +� +2|∂h|2 + v|h|2� +dx : h(x) = −h(−x), h(R) = 1 +� +. +Any h we write as h(x) = xf(x) +R . 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 +� +. +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 +� +. +Taking R → ∞ in this we recover the definition of a. We conclude that a = aodd. +Concerning the assumption on v that +´ �1 +2vf 2 +0 + |∂f0|2� +dx < ∞ we have the following two +propositions. +Proposition 5.13. 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. Then +´ � 1 +2vf 2 +0 + |∂f0|2� +dx < +∞, where f0 denotes the p-wave scattering function. +Proof. Let [x1, x2] be an interval where v(x) = ∞ for x1 ≤ x ≤ x2 and note that f0(x) = 0 for +all |x| ≤ x2. 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 +< ∞. +Proposition 5.14. Suppose that v ≥ 0 is even, compactly supported and smooth. +Then +´ � 1 +2vf 2 +0 + |∂f0|2� +dx < ∞, where f0 denotes the p-wave scattering function. +Proof. For smooth v also the scattering function f0 is smooth. Recall the scattering equation +(5.5). 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. +The function f0 is smooth and even. Thus for small x we have f(x) = f(0) + O(|x|2), hence +the integral converges around 0. By the decay of +1 +x2 the integral converges at ∞. We conclude +the desired. +We now give the proof of Theorem 1.9. We consider the trial state given in Equation (2.1) +where f is a rescaled scattering function +f(x) = +� +1 +1−a/bf0(|x|) +|x| ≤ b, +1 +|x| ≥ b +and +DN(x1, . . . , xN) = det[uk(xi)]1≤i≤N +k∈BF +, +uk(x) = +1 +L1/2eikx, +N = #BF. +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. In this case we +have (see [KL18, Lemma 3.2] or Lemma B.11) +Lemma 5.15. 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. +48 + +As for the 2-dimensional setting one easily generalizes the computation of the kinetic energy in +Lemma 2.13 and the calculation of the 2-particle reduced density for a Slater determinant in +Lemma 2.14. That is, +Lemma 5.16. 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) +� +. +Lemma 5.17. 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) +� +. +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. Here we have the bounds (using Lem- +mas 5.10 and 5.15) +ˆ +Λ +|g| ≲ a log(b/a), +ˆ +Λ +|γ(1) +N | ≲ log N. +(5.6) +Then, the 1-dimensional analogue of Theorem 3.4 reads +Theorem 5.18. There exists a constant c > 0 such that if aρ log(b/a) log N < c, then the +formulas in Equation (3.3) hold (with ρ(n) +Jas and Γn +π,G interpreted as appropriate for the 1- +dimensional setting.) +For the analogues of Lemmas 4.1 and 4.2 we have to a bit more careful. 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. (Pointwise +we only have the bound +��Γ3 +π,G +�� ≤ Caρ4 log(b/a) log N (see Section 4.3 and Appendix A.2) for +any subleading diagram (π, G), i.e. for (π, G) ∈ ˜L3 +p with p ≥ 1.) +Remark 5.19 (Why this was not a problem for dimensions d = 2, 3). 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 (4.22). This bound should be compared to the energy +density of the leading interaction term of order adρ2+2/d. Considering just the power of ρ, we +see that such terms are subleading compared to the interaction term for d ̸= 1. +Similarly the argument for Γ2 is also slightly different compared to that of Lemma 4.1. We +have the bounds +Lemma 5.20. There exists a constant c > 0 such that if aρ log(b/a) log N < c, then +������ +∞ +� +p=1 +1 +p! +� +(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�� +. +49 + +The proof is similar to that of Lemmas 4.1 and 4.2. We postpone it to the end of this section. +Note here that the N-dependence is not just via logarithmic factors. Thus, we need to be more +careful in choosing the size of the smaller boxes when applying the box method arguments of +Section 4.1. With this we get the analogue of Equation (4.1) 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�� +� +. +(5.7) +For the 2-body error terms we may follow the type of arguments of Section 4, namely Equa- +tions (4.2), (4.3), (4.4), (4.5) and (4.6) exactly as for the 2-dimensional case. +By using +Lemma 5.10 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 ≤ + + + + + +C, +n = 0, +Ca log(b/a), +n = 1, +Cab, +n = 2. +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.e. that 1/a0 < ∞. For the 3-body terms we may +do as for the 3-dimensional case, Section 4. For the first term we bound |x1 − x3| ≤ 2b in the +support of ∂f12∂f23 and f13 ≤ 1. For the other terms we bound f13 ≤ 1. By the translation +50 + +invariance one integration gives a volume (i.e. length) factor L. 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� +. +≤ CNa2ρ4 � +b4ρ4 + (log(b/a))2(log N)2 + bρ +� +b4ρ4 + log(b/a) +�� +. +We conclude the analogue of Equation (4.7) 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) +��� +. +(5.8) +We need to be careful how we choose N (i.e. how we choose L), since the error depends on N +not just via logarithmic terms. 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). 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� +. +(5.9) +Using the box method similarly as in Section 4.1 we also have to be careful with how we choose +the parameter d. As in Equation (4.9) 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� +. +Here we change notation from N to n and choose d = a(aρ)−δ. To get the error smaller than +desired, we see that we need to choose δ > 3/2. 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}. +51 + +Then, also ˜ρ = ρ (1 + O((aρ)γ)) so ρ = ˜ρ(1 + O((a˜ρ)γ)). Optimising in α, β, δ we see that for +α = 12 +5 , +β = 13 +17, +δ = 32 +17 +(5.10) +we get γ = 30/17, i.e. +e(˜ρ) ≤ π2 +3 ˜ρ3 + 2π2 +3 a˜ρ4 � +1 + O +� +(a˜ρ)13/17�� +. +This concludes the proof of Theorem 1.9. +It remains to give the +Proof of Lemma 5.20. Note first that, completely analogously to Equations (4.10) and (4.22), +we have +1 +p! +����������� +� +(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! +������������ +� +(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. +(5.11) +We will use this to split the diagrams of L2 +p and ˜L3 +p into groups. We split diagrams in L2 +p into +three (exhaustive) groups: +1. 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. +2. 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. Large diagrams with k + ng + n∗ +g ≥ 3. +We then split +∞ +� +p=1 +1 +p! +� +(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. +We will then do a Taylor expansion of ξsmall,0 but not of the other terms. +We split diagrams in ˜L3 +p into three (exhaustive) groups: +52 + +1. Small diagrams with k + ng + n∗ +g = 1 and {1}, {2} and {3} in 3 different clusters. (Then +ng = 0.) +2. Small diagrams with k + ng + n∗ +g = 1 and {1}, {2} and {3} in < 3 different clusters. +(Then k = ng = 0.) +3. Large diagrams with k + ng + n∗ +g ≥ 2. +We then split +∞ +� +p=1 +1 +p! +� +(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. +Again, we do a Taylor expansion of ξ3 +small,0 but not of the other terms. For simplicity we will +only compute the derivatives ∂2 +x1. 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 (5.10). (By the symmetry, we could bound ξsmall,0 by bounding its 6th derivative ∂2 +x1∂2 +x2∂2 +x3 +instead.) To keep the result symmetric in x1, x2, x3 we will symmetrize the result afterwards. +We have immediately by Equation (5.11) that +|ξ≥1| ≤ Ca3ρ5(log(b/a))3(log N)3, +��ξ3 +≥1 +�� ≤ Ca2ρ5(log(b/a))2(log N)2. +(5.12) +Similarly as in the proof of Lemma 4.1 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. +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) +�� . +For the diagrams in ξsmall,0 and ξ3 +small,0 we have similarly to Lemma 4.8 that +��∂2 +x1ξsmall,0 +�� ≤ Caρ5 log(b/a), +��∂2 +x1ξ3 +small,0 +�� ≤ Caρ6 log(b/a) +(5.13) +uniformly in x1, x2, x3. +For the diagrams in ξsmall,≥1 and ξ3 +small,≥1 the analysis is somewhat +similar to the proof of Lemma 4.6. We have +Lemma 5.21. 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� +, +(5.14) +��ξ3 +small,≥1 +�� ≤ Cab4ρ10 � +|x1 − x2|2 + |x1 − x3|2 + |x2 − x3|2� +(5.15) +uniformly in x1, x2, x3. +We give the proof of Lemma 5.21 in Appendix A.3. +Combining Lemma 5.21 and Equa- +tions (5.12) and (5.13) concludes the proof of Lemma 5.20. +Acknowledgements +A.B.L. would like to thank Johannes Agerskov and Jan Philip Solovej for valuable discus- +sions. We thank Alessandro Giuliani for helpful discussions and for pointing out the reference +[GMR21]. +Funding from the European Union’s Horizon 2020 research and innovation pro- +gramme under the ERC grant agreement No 694227 is acknowledged. +53 + +A +Small diagrams +In this appendix we compute the contributions of all the small diagrams of Lemmas 4.6, 4.8, +4.11 and 5.21. We first consider those of Lemmas 4.6 and 4.8. +A.1 +Small 2-particle diagrams (proof of Lemmas 4.6 and 4.8) +Recall from the proof of Lemma 4.1, Section 4.2 that +ξsmall,0 + ξsmall,≥1 = +∞ +� +p=1 +1 +p! +� +(π,G)∈L2 +p +(π,G) small +Γ2 +π,G. +The criterion for being small is defined in the proof of Lemma 4.1 around Equation (4.11), +and will be recalled below. The diagrams are split into types (A), (B) and (C) according their +underlying graphs G as in the proof of Lemma 4.1. We further split the type (B) into two types +(B1) and (B2). 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.e. connected to {1}. The diagrams +of type (B2) are those diagrams for which the extra vertex {3} is in the cluster containing {2}, +i.e. connected to {2}. That is, the different types are as follows. See also Figure A.1. +(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. +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.1: g-graphs of small diagrams of different types. For each diagram only +the graph G is drawn. The relevant diagrams come with permutations π such that +the diagrams are linked. +We first give the +Proof of Lemma 4.6. Consider first all diagrams of type (C) of smallest size, i.e. with g-graph +G0 = +1 +2 +54 + +Since this graph is connected, all π ∈ S3 give rise to a linked diagram (π, G0). By Wick’s rule, +the π-sum then gives the factor ρ(3). 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. +Recall the bound ρ(3) ≤ Cρ5|x1 − x2|2|x1 − x3|2|x2 − x3|2 from Lemma 2.15. Now we bound +|g23| ≤ 1 and |x2 − x3| ≤ b in the support of g23. Thus +������ +� +π∈S3:(π,G0)∈L2 +1 +Γ2 +π,G0 +������ +≤ Cb2ρ5|x1 − x2|2 +ˆ � +1 − f(x)2� +|x|2 dx. +Recalling Lemma 2.2 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. +(A.1) +We conclude that all diagrams of smallest size contribute ≤ Ca3b4ρ5|x1 − x2|2. +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; x4)γ(1) +N (x4; x3)γ(1) +N (x3; x1)γ(1) +N (x2; x5)γ(1) +N (x5; x2)g13g23g45 dx3 dx4 dx5 += −1 +L15 +� +k1,...,k5∈PF +˚ +eik1(x1−x4)eik2(x4−x3)eik3(x3−x1)eik4(x2−x5)eik5(x5−x2)g13g23g45 dx3 dx4 dx5 += −1 +L15 +� +k1,...,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,...,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. Bounding |g23| ≤ 1 and |ˆg(k)| ≤ +´ +|g| ≤ a3 log(b/a) we get that +��Γ2 +π,G +�� ≤ Ca6ρ4(log(b/a))2. +One may do a similar computation for all the remaining diagrams. 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′). By bounding as above we conclude that the contribution of small diagrams +of type (C) is bounded as desired. +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.2: Exemplary diagrams of types (A), (B1) and (B2). The dashed lines +denote g-edges, and the arrows denote (directed) edges of the permutation. +Proof of Lemma 4.8. 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. +We consider the examples in Figure A.2. +The contribution of the diagram in Figure A.2a to ∂µ +x1∂ν +x1ξsmall is +1 +2∂µ +x1∂ν +x1Γ2 +π,G += −1 +2L12 +� +k1,...,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,...,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,...,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). The same type of +computation is valid for all other diagrams of type (A). +Consider now the diagram in Figure A.2c of type (B2). 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). Similarly, all other diagrams of type (B2) may be bounded using the +same method as for types (A). +56 + +Finally, we consider the diagram in Figure A.2b of type (B1). 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)). +All larger diagrams of type (B1) may be bounded similarly. We conclude the desired. +A.2 +Small 3-particle diagrams (proof of Lemma 4.11) +We now give the +Proof of Lemma 4.11. Recall that +ξ3 +small = +∞ +� +p=1 +1 +p! +� +(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. +As in the proof of Lemmas 4.6 and 4.8 in Appendix A.1 we +compute the value of a few examples and explain how to compute the value of the remaining +diagrams. We consider the examples of Figure A.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.3: Exemplary small diagrams in ˜L3 +2 +The contribution of the diagram in Figure A.3a is +Γ3 +π,G = −1 +L15 +� +k1,...,k5∈PF +¨ +eik1(x1−x4)eik2(x4−x2)eik3(x2−x1)eik4(x3−x5)eik5(x5−x3)g45 dx4 dx5 += −1 +L12 +� +k1,...,k5∈PF +ei(k1−k3)x1ei(k3−k2)x2ei(k4−k5)x3χ(k2−k1=k4−k5)ˆg(k5 − k4) += O(a3ρ4 log(b/a)). +57 + +Similarly, the contribution of the diagram in Figure A.3b is +Γ3 +π,G = −1 +L15 +� +k1,...,k5∈PF +¨ +eik1(x1−x4)eik2(x4−x5)eik3(x5−x1)eik4(x2−x3)eik5(x3−x2)g45 dx4 dx5 += −1 +L12 +� +k1,...,k5∈PF +ei(k1−k3)x1ei(k4−k5)x2ei(k5−k4)x3χ(k2−k1=k2−k3)ˆg(k3 − k2) += O(a3ρ4 log(b/a)). +One may follow this kind of computation for any diagram. The central property we used is +that the internal vertices are all in the same linked component as some external vertex. 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′). We conclude the desired. +A.3 +Small diagrams in 1 dimension (proof of Lemma 5.21) +We now give the +Proof of Lemma 5.21. We first give the proof of Equation (5.14). We split the two cases (A) +and (B) of small diagrams further. They are given as follows. +(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. +(B) {1} and {2} in the same cluster and 1 ≤ k + ng + n∗ +g ≤ 2, +(B1) k = 0, +(B2) k = 1. +See also Figure A.4. +1 +2 +(a) Type (A1) +1 +2 +(b) Type (A2) +1 +2 +(c) Type (B1) +1 +2 +(d) Type (B2) +Figure A.4: g-graphs of small diagrams of different types. For each diagram only +the graph G is drawn. The relevant diagrams come with permutations π such the +the diagrams are linked. The diagrams of type (A1) and (B1) may have some of +the drawn g-edges not present, but the same connected components. +Moreover, +the diagrams of type (B1) may have one of the internal vertices drawn not present +(indicated by a ◦). With the modification of the drawings described here these are +all small diagrams. +We will consider some examples of diagrams. Namely those drawn in Figure A.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.2) +58 + +All other diagrams can be treated in a similar fashion. For the argument we will need a different +formula for ρt. Recall the definition in Equation (3.4). We may write the characteristic function +as +χ((π, ∪Gℓ) linked) = 1 − χ((π, ∪Gℓ) not linked). +That is, +ρ(A1,...,Ak) +t += +� +π∈S∪Aℓ +(−1)π � +j∈∪Aℓ +γ(1) +N (xj; xπ(j)) − +� +π∈S∪Aℓ +(−1)πχ((π,∪Gℓ) not linked) +� +j∈∪Aℓ +γ(1) +N (xj; xπ(j)). +For our case we only need to consider cases where there are at most two clusters. If there is +just one cluster then ρ(A) +t += ρ(|A|)((xj)j∈A). So suppose we have two clusters A1, A2. 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. Thus, +ρ(A1,A2) +t +((xj)j∈A1∪A2) = +� +π∈SA1∪A2 +(−1)π +� +j∈A1∪A2 +γ(1) +N (xj; xπ(j)) +− +� +π1∈SA1 +(−1)π1 � +j∈A1 +γ(1) +N (xj; xπ1(j)) +� +π2∈SA2 +(−1)π2 � +j∈A2 +γ(1) +N (xj; xπ2(j)) += ρ(|A1|+|A2|)((xj)j∈A1∪A2) − ρ(|A1|)((xj)j∈A1)ρ(|A2|)((xj)j∈A2). +(A.3) +We now consider the diagrams in Figures A.4a, A.4b and A.4d and (A.2). 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. +(A.4) +Using Equation (A.3) and (the 1-dimensional versions of) Lemmas 2.14 and 2.15 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, . . . , 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, . . . , 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, . . . , x5) + ρ(3)(x1, x2, x3)ρ(2)(x4, x5) +≤ Cρ13|x1 − x2|2|x1 − x3|2|x2 − x3|2|x4 − x5|2. +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 +. +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 +. +For the diagram of type (B1) we get by bounding g23|x2−x3|2 ≤ b2 (as in the proof of Lemma 4.6) +�������� +� +π∈S3:(π,G0)∈L2 +1 +type (B1) +Γ2 +π,G0 +�������� +≤ Cb2ρ9|x1 − x2|2 +ˆ +|g(x)||x|2 dx. +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 +. +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. +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. Combining these bounds +we conclude the proof of Equation (5.14). +To prove Equation (5.15) 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). 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. +Summing this over all the permutations of {1, 2, 3} we conclude the proof of Equation (5.15). +60 + +B +Derivative Lebesgue constants (proof of Lemma 4.9) +In this appendix we give the proof of Lemma 4.9. We recall the statement in slightly different +notation for convenience. +Lemma 4.9. The polyhedron P from Definition 2.7 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 . +Recall that by construction R ∼ N1/3 is rational. +The proof follows quite closely the argument in [KL18]. In particular the structure is that +of induction. The 3-dimensional integral is bounded one dimension at a time. We start by +introducing some notation from [KL18]. +Notation B.1. For any real number x we will write [x] for either ⌊x⌋ or ⌈x⌉. Similarly we will +write ⟨x⟩ = x − [x], i.e. ⟨x⟩ is either the fractional part {x} = x − ⌊x⌋ or x − ⌈x⌉. For any +computation we do below, the definition of [x] is fixed, but the computations hold with either +choice. +Additionally for a d-dimensional vector x = (x1, . . . , xd) we write x( ˜d) = (x1, . . . , x +˜d) for the +first ˜d ≤ d components. +We emphasize that expressions like k2, x3, . . . do not denote squares or cubes of numbers +k, x, but instead refer to coordinates of vectors k, x. The instances where we do want to denote +a square, cube or higher power should be clear. +By potentially relabelling the coordinates it suffices to consider the cases µ = 1, µ = ν = 1 +and µ = 1, ν = 2. +(Alternatively, by appealing to Lemma 2.11 and choosing Q ≳ N4 in +Definition 2.7 we have a symmetry of coordinates up to error-terms which are subleading +compared to Lemma 4.9.) Hence define +t1(k) = k1, +t2(k) = k1k1 = (k1)2, +t3(k) = k1k2. +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. +As in the proof of Lemma 2.12 we write RP as a union of O(s) closed tetrahedra. We also recall +that Rz /∈ Z3. As in the proof of Lemma 2.12 we get by the inclusion exclusion principle O(s) +terms with tetrahedra of lower dimension (triangles or line segments). 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. +They are the equations of the planes +bounding the tetrahedron T. Since any k ∈ T has integer coordinates we can replace Λj by +⌊Λj⌋ and λj by ⌈λj⌉. It will be convenient to not distinguish between ⌊·⌋ and ⌈·⌉ and use +instead the notation [·] introduced in Notation B.1. 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.1) +61 + +where we allow [·] to be different in any of the 6 instances it appears. +Sums over lower-dimensional tetrahedra can be written as differences of sums over 3- +dimensional tetrahedra (with potentially different meanings of [·]). We will thus only consider +3-dimensional tetrahedra. That is, for a tetrahedron T of the form Equation (B.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. +(B.2) +Gluing together tetrahedra as in Lemma 2.12 we conclude the desired bound, Lemma 4.9. The +remainder of this section gives the proof of Equation (B.2). +B.1 +Reduction to simpler tetrahedron +We first reduce to the case of a simpler tetrahedron T. Consider what happens by shifting all +k’s by some fixed lattice vector κ ∈ Z3 with |κ| ≤ CR. 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. +A similar computation holds for t1, t3. We may bound |κ| ≤ CR and thus we may assume that +T ⊂ [0, CR]3. (Recall that +´ +[0,2π]3 +��� +k∈T∩Z3 eikx�� dx ≤ C(log R)3 by [KL18, Theorem 4.1], see +the proof of Lemma 2.12.) +For any tetrahedron of the form (B.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)] + + +[Λ3(k1,k2)] +� +k3=0 +− +[λ3(k1,k2)−1] +� +k3=0 + + , +where the λj’s and Λj’s are the equations of the planes bounding the tetrahedron T, i.e. piece- +wise affine functions. As in Equation (B.1) each instance of [·] may be either of the definitions +of Notation B.1. 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.e. they are affine functions. When we do this +splitting, we have to choose (in each new tetrahedron) which definition of [·] to use for the new +bounding plane. 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. These boundary terms are sums over lower-dimensional +tetrahedra, and may thus be bounded by sums over 3-dimensional ones as above. +Remark B.2. One may similarly let the k1- and k2-sums go from 0 by writing e.g. +[Λ2(k1)] +� +k2=[λ2(k1)] += +[Λ2(k1)] +� +k2=0 +− +[λ2(k1)−1] +� +k2=0 +. +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). This is why we don’t do this. +The terms with Λ3 and λ3 − 1 may be treated the same way, so we just look at the one with +Λ3. 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. +62 + +B.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. +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. This +follows from the construction of P in Definition 2.7, since forming the edges between pairs of +close points constructs a triangulation of P. +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 2.7, we have +σ /∈ Q, +γ ∈ Q, +αj ∈ Z, +|αj| ≤ C +� +Q, +j = 1, 2, 3. +We might have that αj = 0. 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. Hence α3 ̸= 0. By choice of L, we have that R is rational, +and so Rσγ /∈ Q. (The choice of L such that R is rational, is exactly so that Rσγ /∈ Q.) 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. +(B.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.4) +where m(2) = (m1, m2) is defined in Equation (B.3). We shall prove the following bound. +Lemma B.3. 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. +63 + +As a first step, consider the case where both α1 = α2 = 0 in Equation (B.3). Then the k3-sum +and x3-integral in Lemma B.3 factors out. Using [KL18, Lemma 3.2] to evaluate the k3-sum +and x3-integral we conclude the desired. Hence we can assume that at most one of α1, α2 is 0. +(This will be relevant for Lemma B.8, but only then.) +A simple calculation shows that [KL18, Lemma 3.1] +Dj +3(x) = Gj +3(x) + F j +3(x), +j = 1, 2, 3. +(B.5) +By a straightforward modification of the argument in [KL18, Lemma 3.3] (including the factor +tj) we have +Lemma B.4 ([KL18, Lemma 3.3]). 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). +We thus want to bound the integral of F j +3. Again, by a straightforward modification of the +argument in [KL18, Lemma 3.7] (including the factor tj) we have +Lemma B.5 ([KL18, Lemma 3.7]). For any j = 1, 2, 3 we have +ˆ +[0,2π]3 +��F j +3 (x) +�� dx ≲ +∞ +� +r=1 +(2π)r +r! +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). +To bound the right hand side of Lemma B.5 we bound either definition of ⟨·⟩ by the fractional +part {·}. This follows the strategy in [KL18]. In analogy with [KL18, Lemma 3.6] we have +Lemma B.6 ([KL18, Lemma 3.6]). 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. +Proof. If ⟨·⟩ = {·} this is clear. Hence suppose that ⟨x⟩ = x − ⌈x⌉. Then +⟨x⟩ = {x} − 1 + +� +1 +if x ∈ Z +0 +otherwise. +By construction Λ3(k1, k2) /∈ Z for k1, k2 ∈ Z. Thus, ⟨Λ3(k1, k2)⟩ = {Λ3(k1, k2)} − 1. 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. +64 + +We now bound the second summand of Lemma B.6 similarly to [KL18, Lemmas 3.8 and 3.9]. +We first define ˜Λ3, a rational approximation of Λ3. Recall the definition of Λ3 in Equation (B.3). +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∞ +. +We will choose Q∞ = Q3α3. Define then +˜Λ3(k(2)) = Λ3(k(2)) − γ3 = p +q − Q1α1 +Q3α3 +k1 − Q2α2 +Q3α3 +k2. +(B.6) +Note that this takes values in +1 +qQ∞Z for integers k1, k2. +In particular (for integers k1, k2) +{˜Λ3(k(2))} ∈ {0, +1 +qQ∞, . . . , qQ∞−1 +qQ∞ }. 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. +(B.7) +We claim that +Lemma B.7. 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 3.8 and 3.9] in a few key location, so we give it +here. +Proof. Using Equation (B.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. +All the mixed terms have at least one power of γ3 + {˜Λ3(k(2))} = γ3. (Indeed, in the mixed +terms we have ˜Λ3(k(2)) ∈ Z so {˜Λ3(k(2))} = 0.) 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 2.7. Similarly expanding the first summand, all the terms with +at least one power of γ3 may be bounded the same way. 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.8) +where the error is O(2r) uniform in x(2). For the first summand we have by a simple modification +of [KL18, Lemma 3.8] (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). +This importantly uses that {˜Λ3(k(2))} ∈ {0, +1 +qQ∞, . . . , 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 +3.8]. +We have q ≤ Q∞ = Q3α3 ≤ CQ3/2. We conclude the desired. +Next we bound the second term in Lemma B.7, where ˜Λ3 is integer. If there are no valid choices +of k1, k2 for which ˜Λ3(k1, k2) is an integer, then this term is clearly zero. Otherwise we have +the following. +Lemma B.8. 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. 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, . . . , h}}. 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. +(B.9) +The proof is an exercise in elementary number theory analysing the set I0. +Proof. Define k(2) +0 +to be any point in the (non-empty) set I0. Recall Equation (B.6), and that +���˜Λ3(k1, k2) +��� ≤ CR for any k(2) ∈ I0. (This follows since the relevant tetrahedron is contained +in [0, CR]3.) By redefining αj as αj/ gcd(α1, α2, α3) we may assume that α1, α2, α3 have no +shared prime factors. (This only decreases their values, so that still |αj| ≤ C√Q.) 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. +66 + +Solving the general problem. +We first consider the general problem of finding all k1, k2 ∈ Z +for which ˜Λ3(k1, k2) is an integer. This set has the form k(2) +0 +Γ for some two-dimensional lattice +Γ. We now find spanning lattice vectors of Γ. +Define αij = gcd(αi, αj) for i ̸= j. (Note that the αj’s are not necessarily pairwise coprime, +only all 3 αj’s have no shared factor by the reduction above. Also, since α1 and α2 are not +both 0, we have α12 ̸= 0 is well-defined.) 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. One should note here that κ0 is not +“short”. Indeed |κ0| ≳ Q since both Q1, Q2 ≳ Q, see Definition 2.7, and α1, α2 are not both +0. We now look for the lattice vector in Γ giving the smallest possible (integer) increase of ˜Λ3. +This lattice vector together with κ0 spans Γ. Note that +δ˜Λ3(κ) := ˜Λ3(k(2) +0 ++ κ) − ˜Λ3(k(2) +0 ) = −Q1α1κ1 − Q2α2κ2 +Q3α3 +. +(B.10) +Suppose first that either α1 = 0 or α2 = 0, say α2 = 0. Now, Q1 ̸= Q3 and |αj| ≤ C√Q so Qj +is not a factor of αi for any i = 1, 3, j = 1, 2, 3. Thus, gcd(Q3α3, Q1α1) = gcd(α1, α3) = 1 since +α2 = 0. 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. Thus there is at most one k(2) +0 +∈ I0 and the lemma is clear. +Suppose then that α1 ̸= 0, α2 ̸= 0. Varying κ ∈ Z2 we have by B´ezout’s lemma that the +numerator in Equation (B.10) assumes as values all multiples of gcd(Q1α1, Q2α2). We have +gcd(Q1α1, Q2α2) = gcd(α1, α2) = α12. For the ratio δ˜Λ3(κ) to be an integer we need that the +numerator is some multiple of Q3α3. 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. Thus, the smallest integer +increase of ˜Λ3 is α12 ≥ 1 and this happens along some lattice vector κ1. Immediately then +Γ ⊃ {aκ1 + bκ0 : a, b ∈ Z}. 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. In summary then +Γ = {aκ1 + bκ0 : a, b ∈ Z}, +˜Λ3(k(2) +0 ++ aκ1 + bκ0) = ˜Λ3(k(2) +0 ) + aα12, +a, b ∈ Z. +(B.11) +Moreover +I0 = +� +k(2) +0 ++ Γ +� +∩ +� +(k1, k2) ∈ Z2 : [λ1] ≤ k1 ≤ [Λ1], [λ2(k1)] ≤ k2 ≤ [Λ2(k1)] +� +. +Finding the candidate for κ. +We now find the candidate for the κ in the lemma. 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. For such κ we have (for sufficiently large N) that +a ̸= 0 as |κ0| ≳ Q ≫ R and any such κ has |κ| ≤ CR. Let κ2 = a2κ1 + b2κ0 be the κ such +that k(2) +0 ++ κ ∈ I0 with minimal value of |a2|. (κ2 is unique up to potentially a sign if both +k(2) +0 +− κ2 ∈ I0 and k(2) +0 ++ κ2 ∈ I0.) It follows from Equation (B.11) that |a2| ≤ CR/α12 ≤ CR +since |δ˜Λ3(κ2)| ≤ CR as the tetrahedron is contained in [0, CR]3. +If b2 = 0 then a2 = ±1, else if b2 ̸= 0 then gcd(a2, b2) = 1. 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|. +67 + +Characterizing all allowed κ’s. +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, . . . , h}}, +for some h ∈ Z, h ≥ 0. +(B.12) +(The intuition for the remainder of the argument is as follows. +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. Since |κ0| ≫ R and either κ1 = κ2 or |κ1| ≫ R, this is impossible.) +To prove Equation (B.12) we first introduce the following notation. We view a lattice vector +κ ∈ Z2 as a vector κ ∈ R2 and write κ∥ for its component parallel to κ0. Note that κ∥ need not +have integer coordinates. Define the constant A such that κ∥ +1 = Aκ0. (Note that A need not +be an integer.) Let 0 ̸= κ = aκ1 + bκ0 ∈ Γ with k(2) +0 ++ κ ∈ I0. We have +κ∥ = aκ∥ +1 + bκ0 = (aA + b)κ0. +Thus, since |κ0| ≳ Q, |κ| ≲ CR and |a| ≥ 1 (since κ ̸= 0) we have +�� b +a + A +�� ≤ CR +Q . +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. +But ba2 − b2a is an integer. Hence (for N sufficiently large) we have ba2 = b2a. Now, if b2 = 0 +then b = 0 and so a2 = ±1 is a divisor of a so κ = ±aκ2. If b2 ̸= 0 then gcd(a2, b2) = 1 and +thus a2 is again a divisor of a and a/a2 = b/b2. Then κ = +a +a2κ2 is a multiple of κ2. This shows +the desired. +Integral form. +To prove Equation (B.9) we do the following. Define e2 = κ2/|κ2| as the unit +vector parallel to κ2 and e⊥ +2 as the unit vector perpendicular to κ2. Then define the domain +S0 := +� +x(2) ∈ R2 : +��x(2) · e2 +�� ≤ 4π, +��x(2) · e⊥ +2 +�� ≤ 4π +� +and note that [0, 2π]2 ⊂ S0. Thus, using Equation (B.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). +The integrand is constant in the e⊥ +2 -direction, and 2π-periodic in the e2-direction. 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. +This concludes the proof. +Combining Lemmas B.5, B.6, B.7 and B.8 the r- and ν-sums in Lemmas B.5 and B.6 are readily +bounded because of the factor 1/r! from Lemma B.5. 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.8. If the set I0 from Lemma B.8 is empty, then the +bound is valid without the last term. In particular it is valid with any k(2) +0 +∈ [0, CR]2, (non- +zero) κ = κ(2) ∈ Z2 and h = 0. Thus, by Lemma B.4 and Equation (B.5) we prove the desired +bound, Lemma B.3. +68 + +B.3 +Reduction from d = 2 to d = 1 +For j = 1, 2 we will do one more step reducing the dimension. The argument is basically the +same as for going from dimension d = 3 to d = 2 in Appendix B.2. We sketch the main +differences. +As we did in Appendix B.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 +. +Remark B.9. 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.2). +Indeed, we already computed the sum over the relevant k3. 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.2. +Additionally, by splitting the k1-sum into at most 2 parts, we may assume that Λ2 is just the +equation for a line. Here again one needs to be careful with what to do with the boundary +terms. This gives some sums over 1-dimensional tetrahedra (i.e. line segments), which we can +write as differences of sums over 2-dimensional tetrahedra exactly as for the 3-dimensional case. +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 +� +. +We claim the following inductive bound. +Lemma B.10. 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. +Proof. 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 . +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. +Here we might have that α1 = 0 or α2 = 0. +If α2 = 0 this line is parallel to the k2-axis and so does not give rise to a bound for the +k2-sum. Thus α2 ̸= 0. If α1 = 0 the sum in Dj +2(x) and integral thereof factorizes, and hence by +69 + +[KL18, Lemma 3.2] we have that +ˆ +[0,2π]2 +��Dj +2(x(2)) +�� dx(2) ≤ C log R +ˆ 2π +0 +��� ˜Dj +1(x1) +��� dx1. +Hence, this case yields the desired inductive bound, Lemma B.10. Suppose then α1, α2 ̸= 0. +Then +Λ2(k1) = n2 − m1k1 = n2 − Q1α1 +Q2α2 +k1, +n2 /∈ Q, +|αj| ≤ CQ1/4, +j = 1, 2. +Lemmas B.4, B.5 and B.6 are readily adapted and proven as before. +The adaptation of +Lemma B.7 is then mostly analogous. 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. +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. +��������� +[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. 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. We thus conclude the desired bound. +B.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.3 and B.10. For ˜Dj +1 we may assume that the lower bound of the summations are at 0 +by the same procedure as in Appendix B.1. 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. For instance, +t2 +� +k(2) +0 ++ τκ +� += (k1 +0)2 + 2k1 +0κ1τ + (κ1)2τ 2. +Thus, bounding both the integrals amounts to bounding the following: +Lemma B.11. Let M ≥ 2 be an integer. 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. +70 + +Proof. The bound (1) is elementary, see also [KL18, Lemma 3.2]. 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 +� +. +(B.13) +Consider now the integrals (2) and (3). By symmetry of complex conjugation +´ 2π +0 += 2 +´ π +0 . We +split the integrals according according to whether x ≤ 1/M or x ≥ 1/M. 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. +For x ≥ 1/M we use Equation (B.13) and note that |eix − 1| ≥ cx for x ≤ π. 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. +This concludes the proof. +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 . +Substituting y = |κ|x, using Lemma B.11 and recalling that h|κ| ≲ R and |κ| ≥ 1 by Lemma B.8 +we may bound this by R2 log R. An analoguous bound holds for j = 1. This takes care of all +the one-dimensional integrals. In combination with Lemmas B.3 and B.10 we get the bounds +for j = 1, 2 of Equation (B.2). It remains to consider the two-dimensional integral for j = 3. +71 + +B.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.3. +We +first reduce to the case of a simpler tetrahedron (triangle). By shifting the sums by a fixed +κ = (κ1, 0) ∈ Z2 and using the bounds in Lemma B.11 to evaluate the extra contributions of the +shift, we may assume that the k1-sum starts at 0. By splitting the k2-sum as in Appendix B.1 +we may assume that that k2-sum also starts at 0. 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. Recall that |Λ1| ≤ CR and for any 0 ≤ k ≤ [Λ1] +we have |Λ2(k)| ≤ CR. +The analysis given here is in spirit the same as given in Appendices B.2, B.3 and B.4. It is +sufficiently different that we find it easier to do the arguments separately. We shall show the +following. +Lemma B.12. We have the following bound +¨ +[0,2π]2 +������ +[Λ1] +� +k=0 +[Λ2(k)] +� +ℓ=0 +kℓeikxeiℓy +������ +dx dy ≤ CR2(log R)2 log Q. +Combining then Lemmas B.3, B.10, B.11 and B.12 and choosing Q some sufficiently large +power of N as required in Definition 2.7 we conclude the proof of Equation (B.2) and thus of +Lemma 4.9. It remains to give the proof of Lemma B.12. +Proof. Denote M = [Λ1] and recall Λ2(k) = n2−m1k = n2− Q1α1 +Q2α2k. First note that by mapping +ℓ �→ [Λ2(k)] − ℓ we may assume that m1 ≥ 0 . If m1 = 0 the sum factors, and so does the +integral into two one-dimensional sums/integrals. These may be bounded using Lemma B.11. +In this case we get the bound ≤ CR2(log R)2 as desired. Hence assume that m1 > 0. 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 +. +The first sum factors into one-dimensional integrals which we may bound using Lemma B.11 +again. The second we may shift by a constant ℓ (again then using Lemma B.11 to evaluate the +contribution of the shift) and assume that the lower limit of the ℓ-sum is 0. The upper limit +then becomes [Λ(k)], where +Λ(k) = n2 − ([n2 − m1M] + 1) − m1k := n − mk . +Geometrically, this means that the domain of the (k, ℓ)-sum is a triangle with two sides along +the axes. 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. +72 + +By the symmetries of translation invariance and complex conjugation we may integrate over +the domain [−π, π] × [0, π] instead. We evaluate the ℓ-sum using Equation (B.13). Recall that +[Λ(k)] = Λ(k) − ⟨Λ(k)⟩. 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). +The third summand (III) may be calculated as +−eiy +(eiy − 1)2 +� +⟨Λ(k)⟩ +� +ei[Λ(k)]y − 1 +� +iy + O(y2) +� +. +The factor +−eiy +(eiy−1)2 may be bounded by 1/y2. For this term we split the y-integral according to +whether y ≤ 1/n or y ≥ 1/n. For y ≤ 1/n we expand additionally ei[Λ(k)]y − 1 = O(ny). 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. +For y ≥ 1/n we bound ei[Λ(k)]y − 1 = O(1). 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. +For the second summand (II) we again split the integral according to whether y ≤ 1/n or +y ≥ 1/n. 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). +Hence this contributes the term +ˆ π +−π +dx +ˆ 1/n +0 +dy +����� +M +� +k=0 +keikxO(Λ(k)2y) +����� ≲ 1 +nM2n ≲ R2. +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! +⟨Λ(k)⟩ν � +eiΛ(k)y(Λ(k)eiy − Λ(k) − 1) + 1 +� +. +73 + +Again we bound the factor +eiy +(eiy−1)2 as 1/y2. We treat each summand similarly as in Lemmas B.6 +and B.7 (or rather, the 2-dimensional version of these as used in Appendix B.3.) Completely +analogously to Lemma B.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.e. +either ⟨·⟩ = {·} or ⟨·⟩ = · − ⌈·⌉). +Also the application of +Lemma B.7 is analogous to its use in Appendix B.3. There is at most one k such that ˜Λ(k) ∈ Z +for the appropriate rational approximation ˜Λ of Λ. 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. 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. +The rest of the argument in Lemma B.7 is the same. 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.e. +¨ ����� +� +k +keikx(II) +����� dx dy ≲ log Q +¨ ����� +� +k +keikx(I) +����� dx dy + R2 log R. +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. +(B.14) +In order to evaluate the integral on the right-hand side, we split the integration domain into 5 +regions, see Figure B.1. +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}. +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. +(I1). 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.1: Decomposition of the domain [−π, π] × [0, π] into different regions. +(or rather the numerator) to second order in y. Using that Λ(k) = O(n) we get that (∗) ≲ M2n2. +Thus the integral gives +I1 ≲ +ˆ 2/M +−2/M +dx +ˆ 2/n +0 +dyM2n2 ≲ Mn ≲ R2. +(I2). +We expand eiy = 1 + O(y) in (∗). +Then (∗) ≲ +M2n +y ++ M2 +y2 . +The integral is then +I2 ≲ R2 log R. +(I3, I4, I5). For the remaining integrals we use the explicit formula for Λ(k) = n−mk. 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.15) +where we introduced D(z) = �M +k=0 eikz = ei(M+1)z−1 +eiz−1 +. From Equation (B.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 ++ . . . + 1 +z4. +(B.16) +(I3). We have y ≤ 1/n and |x| ≥ 2/M. We expand Equation (B.15) to second order in y. +Expanding first the exponentials and then derivatives of D where needed we get +(B.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)|. +75 + +Now we use the bounds Equation (B.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. Thus +I3 ≲ 1 +n +ˆ π +1/M +dz +� +n2|∂D(z)| + nm|∂2D(z)| + m2|∂3D(z)| +� +≲ R2 log R. +(I4). We expand the exponentials eiy = 1 + O(y). Then +|(B.15)| ≤ |∂D(x)| +y2 ++ n|∂D(x − my)| +y ++ |∂D(x − my)| +y2 ++ m|∂2D(x − my)| +y +. +Using the bounds Equation (B.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. +(I5). Again, expanding the exponentials eiy = 1 + O(y) we have as for I4 that +|(B.15)| ≤ |∂D(x)| +y2 ++ n|∂D(x − my)| +y ++ |∂D(x − my)| +y2 ++ m|∂2D(x − my)| +y +. +We use the bounds +|∂D(z)| = +����� +M +� +k=0 +keikz +����� ≤ M2, +|∂2D(z)| = +����� +M +� +k=0 +k2eikz +����� ≤ M3. +Thus +I5 ≲ 1 +M +ˆ π +1/n +M2 +y2 + nM2 + mM3 +y +dy ≲ R2 log R. +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. +Together with Equation (B.14) this concludes the proof. +References +[ARS22] +J. Agerskov, R. Reuvers, and J. P. Solovej. “Ground state energy of dilute Bose gases in 1D”. +2022. doi: 10.48550/arXiv.2203.17183. +[BCGOPS22] +G. Basti, S. Cenatiempo, A. Giuliani, A. Olgiati, G. 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