diff --git "a/89E3T4oBgHgl3EQfqwpo/content/tmp_files/load_file.txt" "b/89E3T4oBgHgl3EQfqwpo/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/89E3T4oBgHgl3EQfqwpo/content/tmp_files/load_file.txt" @@ -0,0 +1,783 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf,len=782 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='04654v1 [nlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='CD] 11 Jan 2023 Classical and Quantum Elliptical Billiards: Mixed Phase Space and Short Correlations in Singlets and Doublets T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Ara´ujo Lima1, ∗ and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' do Carmo2, † 1Departamento de F´ısica, Universidade Federal Rural de Pernambuco, Recife, PE 52171-900, Brazil 2Instituto Federal de Alagoas, Piranhas, AL 57460-000, Brazil (Dated: January 13, 2023) Billiards are flat cavities where a particle is free to move between elastic collisions with the bound- ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In chaos theory these systems are simple prototypes, their conservative dynamics of a billiard may vary from regular to chaotic, depending only on the border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The results reported here seek to shed light on the quantization of classically chaotic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' We present numerical results on classical and quantum properties in two bi-parametric families of Billiards, Elliptical Stadium Bil- liard (ESB) and Elliptical-C3 Billiards (E-C3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Both are elliptical perturbations of chaotic billiards with originally circular sectors on their borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Our numerical calculations show evidence that the elliptical families can present a mixed classical phase space, identified by a parameter ρc < 1, which we use to guide our analysis of quantum spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' We explored the short correlations through nearest neighbor spacing distribution p(s), which showed that in the mixed region of the classical phase space, p(s) is well described by the Berry-Robnik-Brody (BRB) distributions for the ESB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In agreement with the expected from the so-called ergodic parameter α = tH/tT, the ratio between the Heisenberg time and the classical diffusive-like transport time signals the possibility of quantum dynamical localization when α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For the E-C3B family, the eigenstates can be split into singlets and doublets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' BRB describes p(s) for singlets as the previous family in the mixed region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' However, the p(s) for doublets are described by new distributions recently introduced in the literature but only tested in a few cases for ρc < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' We observed that as ρc decreases, the p(s)’s tend to move away simultaneously from the GOE (singlets) and GUE (doublets) distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Keywords: Billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' GOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' GUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' INTRODUCTION The idea that molecules may be behind Thermody- namics (grounded in Statistical Mechanics) was one of the tremendous scientific advances of the 19th century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In particular, these particles, constituents of gases, are asso- ciated with the concept of ergodicity, then called molecu- lar chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The word ergodic came from the Greek ergon (work) and odos (trajectory) and was used by Boltzmann to represent the hypothetical visit to all points of the phase space by a particle of that gas with random micro- scopic dynamic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The introduction of the proba- bility in theory that came to be called Statistical Mechan- ics of Equilibrium passed by a long probationary regime, with more convincing results occurring only in the first decades of the 20th century [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The so-called Ergodic Hypothesis only gained the rigor of a theorem with the work of the Russian mathematician Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Sinai in the 60s- 70s for an ideal gas of only two particles [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' A system is chaotic if two neighboring trajectories in the phase space separate exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Suppose the distance in phase space between such trajectories is proportional to eλt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The λ parameter is called the Lyapunov exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In re- ality, λ represents the greatest of Lyapunov’s exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Therefore, the existence of at least one positive Lyapunov exponent characterizes a chaotic system [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Billiards sys- tems are prototypes in the study of chaos and describe ∗ Corresponding author: tiago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='araujol@ufrpe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='br † ricardo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='carmo@ifal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='br the free movement of a point particle in a closed domain Ω with elastic reflections on the boundary ∂Ω of the do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The nature of this conservative dynamical system depends exclusively on the shape of the border ∂Ω, vary- ing from entirely regular (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=', ellipses and annular con- centric regions) to completely chaotic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=', Sinai billiard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Without loss of generality, we consider that the particle has mass m = 1 and velocity of module |v| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' A dis- crete dynamics well describes this 2-dimensional motion in time on variables (ℓ, φ), the fraction of perimeter of ∂Ω, and the incidence angle where a collision happens parametrizes the discrete-time generally [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' A primor- dial example that deserves to be mentioned here is the Bunimovich stadium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' This billiard can present λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Its shape consists of two semicircles joined by two finite- size segments 2t, forming a stadium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' It is chaotic for any t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In a pure circular billiard, collisions keep the angular momentum in relation to its center (focus) con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The Bunimovich stadium does not present this property in its dynamics, known as a defocusing [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' It is hugely relevant to this work because the Elliptical Sta- dium Billiard is a perturbation of it, resulting in classical dynamics with mixed phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Quantum mechanics has been one of the best-tested physical theories since its emergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The theory makes excellent predictions not only for the atom of hydro- gen, which is classically integrable, as well as the he- lium atom, which is classically not integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Nothing is more natural than whether there is an effect analogous to chaos in quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The term quantum chaos is generally understood as studying the quantum behav- 2 ior of classically chaotic systems [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' One commonly used means of studying these systems is to statistically char- acterize spectral properties in the semiclassical regime and compare them with results from the random matri- ces theory [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In billiards, obtaining the energy spectrum is an essen- tial step for analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The problem is to solve the time- independent Schr¨odinger equation with null potential in the planar region Ω with Dirichlet boundary conditions at ∂Ω: � ∇2ϕn(r) = −k2 nϕ(r), r ∈ Ω ϕn(r) = 0, r ∈ ∂Ω, (1) expression is also known as the Helmholtz Equation [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Where k2 n = 2mEn/ℏ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In order to characterize univer- sality, one must first unfold the energy spectrum {En} so that a unit means (⟨sn⟩ = 1) nearest neighbor spacing (nns) sn = En+1 − En is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' This approach be- came relevant after two important conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Namely, the Berry-Tabor (BT) conjecture [9] and the Bohigas- Giannoni-Schmit (BGS) conjecture [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The BT con- jecture states that, in the semiclassical limit, the statis- tical properties of the energy spectrum of a classically integrable system must correspond to the prediction of uncorrelated randomly distributed energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' As a result, the semiclassical nns distribution p(s) must obey Poisson: pP(s) = exp(−s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (2) On the other hand, according to the BGS conjecture, in the case of a classically chaotic system, the spectral prop- erties must follow the universal statistics of the eigen- values of Gaussian random matrices [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Several recent works have improved the turnover of the BGS conjecture in a theorem [11–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' These proofs still have controver- sies and limitations pointed out by some authors [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The terminology ”BGS conjecture” fits the current arti- cle for quantized billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' More recently, in [17], the conjecture was extended to purely ergodic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' If one disregards spin, in the presence (absence) of time- reversal symmetry, p(s) must correspond to that of the GOE, Gaussian Orthogonal Ensemble (GUE, Gaussian Unitary Ensemble): � pGOE(s) = (π/2)s exp(−πs2/4), pGUE(s) = (32/π2)s2 exp(−4s2/π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (3) Based on these assumptions, Leyvraz, Schmit, and Selig- man (LSS) [18] predicted and tested numerically that chaotic billiards with only a three-fold (C3) symmetry (without reflection symmetry) have doublets with spec- tral statistics of the GUE type, although billiards are by time reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' LSS considered a billiard consisting of three straight segments of an equilateral triangle with rounded corners by two circumferences of different radii, here called Circular-C3 Billiard (C-C3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In particular, LSS showed results for a double ratio between the radii where there is a satisfactory agreement for p(s) with the GUE statistics, for a total of approximately 800 dou- blets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Later, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Dembowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' used microwave bil- liards with C3 symmetry to check experimentally the re- sult predicted by LSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Besides, they showed that singlets follow GOE [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The BT [20], BGS [21–23] and LSS conjectures have been investigated in the literature, but there have been comparatively fewer studies on the LSS findings [24–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Until now, little has been said about the situations of C3 symmetric billiards with mixed classical phase space, where chaotic sea and stable KAM-islands coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Here, we propose to shed light on the quantum properties of bil- liards with mixed classical phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For this, we per- form numerical calculations on the energy spectra of two bi-parametric families of billiards with elliptical sectors on their boundaries and analyze the short correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The first one is the Elliptical Stadium Billiard (ESB), a perturbation of the Bunimovich Stadium, whose mixed classical phase space was studied in [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In sequence, we introduce a perturbation of the C-C3B, replacing the circumferences with ellipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Recently, billiards with el- liptical borders have been studied in other contexts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=', in singular potentials [30], in relativistic limits [31], and flows that move around chaotic cores [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' We start the analysis by presenting the billiards, discussing their classical dynamics, showing some mixed phase spaces, and calculating the fraction of the chaotic sea on these phase spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Finally, we follow with the quantized bil- liards’ spectral properties, investigating the nns distribu- tion p(s) with formulas for intermediate quantum statis- tics derived for the doublets recently [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' THE BI-PARAMETRIC BILLIARDS FAMILIES AND CLASSICAL DYNAMICS The billiards systems studied in this work belong to two bi-parametric families, the Elliptical Stadium Bil- liards (ESB) and Elliptical-C3 Billiards (E-C3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The first one consists of a perturbation of the Bunimovich Stadium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' It comprises two half-ellipses (major semi-axis a and minor semi-axis 1) that bracket a rectangular sec- tor of thickness 2t and height 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' [28] showed that in the region a ∈ (1, √ 2) and t ∈ (0, ∞) are possible to find chaotic dynamics or a mixed phase space depending on the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In [29] is presented a critical be- havior of the billiard dynamics near a transition curve, t(a) = √ a2 − 1 for the interval a ∈ (1, � 4/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Based on these previous works, we focus our analysis on this last interval and t ∈ (0, 1/ √ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The E-C3B is based on C-C3B, but ellipses instead of circumferences curve the corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The larger (smaller) ellipse has Ae (ae) and Be (be) semi-axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In all cases described here, the relations Ae = 2ae and Be = 2be are maintained, with ae and be in the range (0, √ 3/6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The LSS billiard is reproduced with ae = be = √ 3/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Here, by our knowledge, we present for the first time a perturbation on the C-C3B resulting in a system that shows a mixed phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 3 A Fundamental Domain (FD) is a neighborhood in Ω that contains only one image for any point in the sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Besides the boundary of the ∂Ω, there are addi- tional boundaries between adjacent FDs, which are the symmetry lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Classically, billiard dynamics can always be reduced to a FD by assuming specular reflections at the symmetry lines [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For this, we use the FD of each billiard in our calculations on the classical dynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 1, we graph billiards in families indicating the parameters and their respective FDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' ESB E-C3B 1 a t 1 a t Symmetry Lines Additional Boundaries Original Boundaries �3/3 �3/3 ae be Ae Be 120º 120º 120º 120º (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (a) Original Boundaries of Elliptical Stadium Billiard and Elliptical-C3 Billiard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For ESB, the symmetry lines are referent to reflections on vertical and horizontal axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' While E-C3B are referent to 120◦ rotational axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (b) Fundamental Domains of ESB and E-C3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The symmetry lines are replaced by additional boundaries forming the planar region where we analyze these billiards’ classical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The global dynamical properties of the ESB with unit mass and velocity may be characterized through colli- sions of orbits with the vertical side of its FD shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' An additional part of the boundary dictates this edge and does not change with the variation of parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The reduced phase space is then a rectangle defined by the vertical position y, where a collision occurs at dis- crete time n, and the tangent component of the velocity in a collision, vy, with 0 < y < 1 and −1 < vy < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The small gray dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 2 show the phase plane for some values of parameters (a, t) after n = 105 colli- sions from the initial conditions (ICs), clearly exhibiting a mixed (regular-irregular) characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' We plot one example of a stable trajectory in red for each one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Quan- titative characterization of these mixed-phase spaces can be made through the chaotic (regular) fraction ρc (ρr) of each phase portrait with ρc + ρr = 1 and 0 ⩽ ρc ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The phase plane is partitioned into Nc small disjoint cells to measure these quantities [27, 29, 35–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For a given orbit, let N(n) be the number of different cells in the phase space, which are visited up to n impacts in the cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The relative measure r(n) is defined as the fraction of visited cells averaged over a set of ICs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=', r(n) = ⟨N(n)⟩/Nc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' So the chaotic fraction of the phase space is obtained via ρc = lim n→∞ lim Nc→∞ r(n), (4) for ICs in the chaotic sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In our numerical approach, we consider Nc = 106, n = 2 · 107, and averages in 20 random ICs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For the billiards with mixed phase space in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 2, ρ(a) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='991884 and ρ(b) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='857184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 3 shows a numerical diagram of ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The ergodic property (ρc = 1) is numerically guaranteed in black regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' This diagram also supports previous works [28, 29], where a critical transition from a mixed phase space to a fully ergodic was found to cross a critical line t(a) = √ a2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' y y vy (a) a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='04 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='15 (b) a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='04 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='01 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Upper Panels: ESB boundaries for some values of pa- rameters (a, t) with stable trajectories in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Lower Panels: corresponding phase portraits for 105 collisions with the ver- tical boundary from the IC (y0, vy0) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0) (small gray dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The red plots correspond to the trajectories in the up- per panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The mixed phase spaces present ρ(a) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='991884 and ρ(b) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='857184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The E-C3B’s classical dynamical properties will be studied in the same way but are characterized through the collisions of the orbits with the horizontal side of its FD shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 1, which does not change with the vari- ation of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The reduced phase space is then a rectangle defined by the horizontal position x, and the tangent component of the velocity in a collision, vx, with 0 < x < √ 3/3 and −1 < vx < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The small gray dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 4 show the phase plane for some values of pa- rameters (ae, be) after n = 2 · 107 collisions from the ICs, clearly exhibiting mixed (regular-irregular) characteris- tic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The values of chaotic fraction are ρ(a) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='935152 and ρ(b) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='800792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 3 shows a numerical diagram of ρc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The ergodic property (ρc = 1) is numerically guaranteed in black regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' This map 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='04 ae be FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Left Panel: diagram of the chaotic fraction of the phase space ρc for the ESB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The tiny green line is the critical line t(a) = √ a2 − 1 studied in [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Right Panel: same diagram for the E-C3B showing a distinguished phase space behavior depending on the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The ergodic property (ρc = 1) is numerically guaranteed in black regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' These maps will guide us in exploring quantum properties, where these values will be relevant parameters to our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' will guide us in exploring quantum properties described in the next section, where these values will be relevant parameters to our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' G x x (a) ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2784 (b) ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2886088 vx be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='256 be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='281522 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Upper Panels: E-C3B boundaries for some values of parameters (ae, be) with stable trajectories in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Lower Panels: corresponding phase portraits for 2·107 collisions with the horizontal boundary from the IC (x0, vx0) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0) (small gray dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The red plots correspond to the trajectories in the upper panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The mixed phase spaces present ρ(a) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='935152 and ρ(b) c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='800792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' QUANTIZATION AND EIGENVALUES SHORT CORRELATIONS All Energy spectra {En} of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (1) were calculated with an algorithm based on the scaling method intro- duced by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Vergini and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Saraceno (VS) in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' This approach allows us to access high-lying energy eigenval- ues that have been unfolded to obtain a unit mean spac- ing (⟨sn⟩ = 1) for each billiard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Our results are based on sets of approximately 70,000 eigenvalues for a given pair of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' According to [6], there is possibly no more intensely studied spectral statistics more than p(s), the density of probability of finding two levels nearest neighbor spaced by s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The Singlets Case Initially, we focused on results for ESB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Some pro- poses have been made to describe these distributions for systems whose present mixed-phase space on its classi- cal counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Here we focus on two of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' They result in intermediate formulas between Poisson and GOE statistics through parameters variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Firstly, we cite the purely phenomenologic approach by Brody [39], where an exponent ν is gradually varied to obtain a smooth change between the integrable (ν = 0) and chaotic (ν = 1) cases: pB(s) = aν(ν + 1)sν exp � −aνs(ν+1)� , (5) where aν = � Γ � ν+2 ν+1 ��ν+1 and Γ(x) is the Gamma func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The second distribution cited here is the Berry- Robnik-Brody (BRB), a proposal that takes under con- sideration the chaotic (regular) fraction of the classical phase space ρc (ρr) [40]: pBRB(s) = exp(−ρrs) \uf8f1 \uf8f2 \uf8f3 ρ2 r (β + 1)Γ � β+2 β+1 �Q � 1 β + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' aβ(ρcs)β+1 � + [2ρrρc + (β + 1)aβρβ+2 c sβ] exp[−aβ(ρcs)β+1] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (6) As in the Brody distribution, aβ = � Γ � β+2 β+1 ��β+1 and Q(κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' x) is the Incomplete Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' This dis- tribution can go through other distributions varying the free parameters ρc and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For β = 0, pBRB(s) = pP(s) and for β = 1 it recovers the distribution of Berry-Robnik (BR) [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' If ρc = 0, pBRB(s) = pP(s) again, while for ρc = 1, pBRB = pB(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The nns for ESB were previously studied in [22] with around 3,000 eigenvalues of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' We use the VS method to obtain around 65,000 eigenvalues beyond the first 5,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The BRB distribution can fit all p(s) ob- tained for all parameters tested on ESB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' We have two independent parameters for this distribution, ρc, and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' However, we fixed ρc at the value obtained in the diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The upper panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 5 shows representa- tive results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The chaotic case presents β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='020, the GOE distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The mixed (0 < ρc < 1) present β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='978 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='018 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='191 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='014, intermediate distributions between Poisson and GOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' These results go in the direction of the quantum localization, previously studied in other billiards systems [42, 43] and discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 �� � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 ��� p(s) p(s) s s s ( ESB E-C3B E-C3B E-C3B ESB ESB (b) \x0e \x0f \x10 \x11 (e) \x12\x13 \x14 ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2784 be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='256 ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='25 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='287 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='15 t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='01 ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2886088 be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='281522 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Representative results for BRB distributions fits for p(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Upper Panels: results on ESB with a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='04 and some values of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The chaotic case t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='287 (ρc = 1) presents β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='020, the GOE distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The mixed cases t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='15 and t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='01 (0 < ρc < 1) present β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='978 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='018 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='191 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='014 respectively, intermediate distributions between Poisson and GOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Lower Panels: results on E-C3B with some values of (ae, be).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The chaotic case, (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='25) (ρc = 1) presents β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='097, the GOE distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The mixed cases, (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2784, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='256) and (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2886088, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='281522) (0 < ρc < 1) present β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='999 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='057 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='203 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='073 respectively, in the range of intermediate distributions between Poisson and GOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The fits with the Brody formula and BRB distribution are indistinguishable in both billiards families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Quantum dynamical localization corresponds to a pe- culiar quantum distribution of the linear or angular mo- mentum peaked at zero, with walls that decay exponen- tially, differently from the classical results, which pre- dicts, for a chaotic or disordered system, a diffusive trans- port [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The phenomenon can be reviewed in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' An interesting feature of the quantum dynamical localization is that it allows us to estimate the conditions under which the comparison with the standard random matrix theory is adequate or, in other words, whether an energy eigen- values data set belongs to the deep semiclassical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' We follow closely [42] in the short description below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The key idea is to express the ergodic parameter α = tH/tT, where tH is the (quantum) Heisenberg time, and tT is the (classical) transport time, in terms of accessible magni- tudes, such as the (quantum) energy E and the (classical) number of collisions off the billiard border, NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' From [42] the ratio is expressed as α = kL πNT , (7) where L is the perimeter of the boundary and k2 ∼ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The condition for quantum dynamical localization in a given energy spectrum, α ⩽ 1, can then be written as k ⩽ kc = πNT/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' To estimate NT, we consider an en- semble of orbits initially directed perpendicularly to ∂Ω and follow its random spreading as a function of the dis- crete time n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The symbols in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 6 illustrate the results for the mean square momentum ⟨p2⟩ as a function of n in a monolog scale (averaged in sets of 103 randomly chosen ICs) for members of two billiards family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Sat- uration of ⟨p2⟩ occurs at different times NT depending on parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For the ESB family, all calculated spec- tra have kmax ≲ kc as the largest eigenvalue, equivalent to the 70,000th level at least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' These facts are in agree- ment with the intermediate statistics well fitted with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (6) as in [23, 27, 40, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The same occurs for the sin- glets in the E-C3B family, where the condition kmax ≲ kc is equivalent to the 70,000th level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The representative results are in the lower panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The chaotic case presents β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='097, the GOE distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The mixed (0 < ρc < 1) present β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='999 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='057 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='203 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='073, in the range of intermediate distribu- tions between Poisson and GOE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In the next section, we discuss the doublets subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The Doublets Case Consider a classically chaotic system with time- reversal (TR) invariance and a point-group (PG) symme- try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' If the TR and the PG operations do not commute, non-self-conjugate invariant subspaces of the PG must exhibit GUE spectral fluctuations instead of GOE ones [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For example, consider a billiard in the xy plane with the C3 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Such a billiard has eigenfunctions ϕm (m = −1, 0, +1), such that ϕ0 is symmetric and repeats itself after a rotation of 2π/3 about the symmetry axis, whereas ϕ±1 will be repeated only after three consecutive rotations of 2π/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In other words, if R(2π/3) is the rota- tion operator for an angle of 2π/3, one has R(2π/3)ϕm = exp(i 2π 3 m)ϕm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Let Θ be the time reversal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Θ is an antiunitary operator that commutes with the Hamil- tonian H, which has eigenvalue Em, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=', Hϕm = Emϕm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' It follows that HΘϕm = ΘHϕm = EmΘϕm (Θϕm is also an eigenfunction of H with the same eigenvalue Em).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Are 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 \x15 \x16\x17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 0 \x18 \x19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='4 log10 n Elliptical-C3 Billiard Elliptical Stadium Billiard 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Calculated mean square of the momentum as a func- tion of the discrete time n in a monolog scale (number of collisions of the particle off the billiard boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Lines are guides for the eyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Upper panel: results for members of the ESB family with a = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The red dots are for t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='287 and present saturation at NT ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Blue dots are for t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='15, and saturation at NT ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='103, and black dots are for t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='01 presenting NT ≃ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Lower Panel: same calcula- tions for the E-C3B, the red dots are for (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='25) and present saturation at NT ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Blue dots are for (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2784, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='256) and saturation at NT ≃ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='102, and, black dots are for (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2886088, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='281522) presenting NT ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' ϕm and Θϕm the same eigenstate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For this subspace one may write Θϕm = (−1)mϕ−m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Thus, Θϕ0 = ϕ0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=', ϕ0 is a singlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The top panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 7 show cases of the probability density |varphi0|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' On the other hand, ϕ1 and ϕ−1 must correspond to distinct states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' One refers to this doublet state as a Kramers degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The mid- dle panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 7 show the real and imaginary parts of the member ϕ1 of a doublet, say (ϕ1, ϕ−1), in the same billiard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The probability density |ϕ1|2 recovers the C3 symmetry (rightmost middle panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The bottom panels in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 7 show the same state under the application under rotation operator R(2π/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' A complex conjugation of the shown state obtains the other mem- ber ϕ−1 of the doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Since these degenerate states are not TR invariant, they must follow the GUE of ran- dom matrices, providing the billiard is classically chaotic, according to the LSS results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For the E-C3B, the degenerate states remain invariant to TR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' However, the spectral distribution will be changed for cases where the classical dynamics is not completely chaotic (ρc < 1), with a p(s) resultant that deviates from the GUE case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Thus, it is necessary to use new inter- mediate formulas to study the distribution of doublets in billiards with mixed classical phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The following formulas we derived in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Following the same steps in [39] led to the eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (5), a Brody-like formula for the transition between the Poisson and GUE distributions is FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Top panels: Density plots of squared eigenfunctions corresponding to singlet states in the LSS billiard, exhibiting the underlying C3 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In the color scale, |ϕ1,a|2 is the maximum probability in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Middle panels: Real and imaginary parts of a member ϕ1 of a doublet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In the color scale, ±ϕ1,a is the minimum and maximum of the wave func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The probability density recovers the C3 symmetry (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Bottom panels: Same state in the middle under the application of the rotation operator R(2π/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' obtained, namely, pB,2(s) = (η + 1)b2 ηs2η exp � −bηsη+1� , (8) where bη = � Γ �2η + 1 η + 1 ��−(η+1) , (9) and 0 ⩽ η ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' For η = 0, pB,2(s) reduces to the Poisson distribution, whereas for η = 1, the Wigner distribution for the GUE is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In [40], the dynamical local- ization of chaotic eigenstates was taken into account and their coupling with the regular ones through tunneling effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The so-called BRB distribution previously dis- cussed in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Following this, the formula that corresponds to the Poisson ↔ GUE crossover is pBRB,2(s)eρrs = ρrρcb 1 γ+1 γ (2 − ρrs) Q �1 + 2γ 1 + γ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' bγ (ρcs)γ+1 � + � ρ2 r � 1 + bγργ+1 c sγ+1� + (1 + γ) � ργ+1 2 bγsγ�2 � e−bγ(ρcs)γ+1, (10) where bγ is defined as in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (9) and Q(κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' x) is the incom- plete Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Here, pBRB,2(s) = pP(s) if ρr = 1 or if γ = 0, and pBRB,2(s) = pB,2(s) if ρr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In [27], the above formula was widely tested only in the regime of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='507 full ergodicity (polygonal cases) and in a single case with ρc < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Here, we detail a non-polygonal billiards family that produces a wide variability of ρc values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In these cases, pBRB,2 well-fitted distributions of nns for ρc < 1 for all investigated cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The representative results are in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' As in the previous section, the doublets sub- space is in the region of the spectrum such that k ≲ kc, equivalent to 60,000th level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 E-C3B E-C3B E-C3B (a) (b) \x1a\x1b ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2784 be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='256 ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='25 ae = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2886088 be = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='281522 p(s) p(s) p(s) s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Representative results for BRB-like distributions, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (10), fits for p(s) in doublets subspace for same members of E- C3B family of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In panel (a), the chaotic case (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='25) (ρc = 1) presents γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='960 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='050, in the range of a GUE distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In panels (b) and (c), the mixed cases (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2784, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='256) and (ae, be) = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='2886088, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='281522) (0 < ρc < 1) present γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='000 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='032 and γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='13 respectively, in the range of intermediate distributions between Poisson and GUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Fits with Brody-like formula (8), and BRB-like distribution (10), are indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' CONCLUSIONS ANS PERSPECTIVES This paper presents numerical results on classical dy- namics and quantization in two bi-parametric billiard families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The ESB comprises two ellipses of minor semi- axe unitary, major semi-axe a, and a rectangular region of length 2t [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The other family, introduced here as E- C3B, presents the C3 symmetry [18, 25, 27] and is formed by an equilateral triangle with rounded corners by two ellipses with semi-axis Ae = 2ae and Be = 2be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' First, we investigate the classical dynamics of these billiards where we built detailed diagrams for the chaotic fraction (ρc) of their phase spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' After that, we investigated the nns distributions p(s) for these systems, a measure of short correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In the asymmetric ESB family, the param- eters space region (a, t) where the classical phase space is mixed (regular and chaotic regions coexist), all found statistics present intermediated results between Poisson and GOE distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The BRB distribution [40], eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (6), very well fitted all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' These results perfectly agree with the expected from the ergodic parameter α that signals the possibility of quantum dynamical local- ization when α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' All sets of eigenvalues used as data are in a range of energy that satisfies this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In the E-C3B family, the eigenstates can be split into sin- glets and doublets subspaces due the symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The first subspace presents similar results to the previous family, reinforcing the agreement with the expected energy range set with α < 1 [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The doublets subspace, whose for the chaotic cases is expected a GUE distribution shows the more relevant result in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' All found statistics present intermediated results between Poisson and GUE distributions for the parameter space (ae, be) where the classical phase space is mixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' A BRB-like formula [27], eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (10), well fitted all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' This formula was tested for ρc < 1 and α < 1 in just a few cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Particularly in the E-C3B family, the minimum value of the chaotic fraction of the classical phase space is ρc ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' This limitation can be avoided if we set free the conditions Ae = 2ae and Be = 2be, used here to follow closer to the C-C3B introduced by LSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In this perspective, a phase diagram analog to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' 3 even more intricate is generated, possible further explorations of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The parameter β in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (6) was extensively compared with other localization metrics, including analyses involv- ing Husimi functions, calculations of the entropy localiza- tion measure [42], and normalized inverse participation ratio [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' How the new distribution, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (10), uses the same arguments to include the parameter γ is merito- rious in a future comparison between this quantity and other localization metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Another theme meritorious of investigation is the level statistics in an energy range that α ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The BR formulas are expected to provide a good description of the deep semiclassical regime [41], an excellent agreement has been found with numerical experiments in a billiard for which the eigenvalues set is around 1,500,000th level [42], an impressive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' The BR-like formula in [27] should be tested in a range of high energy in the doublets subspace to close the com- parisons between the short correlations in the singlets sets and doublets subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In addition, our results indi- cate an intriguing correlation between singlets and dou- blets spectra for the E-C3B family, producing p(s)’s that move away from the GOE and GUE distributions as ρc decreases, thus requiring a further investigation of the ob- 8 served effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' In this perspective, a range opens up to in- vestigate the correlation of spectra of different subspaces [26, 34, 46–48] in billiards that present only rotational symmetries greater than three, which will give the pos- sibility of performing other tests with the new formulas (8) and (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' ACKNOWLEDGMENTS Useful discussions with F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' de Aguiar and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Terto are gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' This work has been sup- ported by the Brazilian Agencies CNPq, CAPES and FACEPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' Dorfman, An Introduction to Chaos in Nonequilib- rium Statiscal Mechanis, 1st ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' (Cambridge University Press, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E3T4oBgHgl3EQfqwpo/content/2301.04654v1.pdf'} +page_content=' [2] Y.' metadata={'source': 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