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arxiv: 2604.13003 · v2 · submitted 2026-04-14 · 🌊 nlin.CD · quant-ph

Relativistic Quantum Chaos in Neutrino Billiards

Barbara Dietz This is my paper

Pith reviewed 2026-05-10 13:41 UTC · model grok-4.3

classification 🌊 nlin.CD quant-ph
keywords neutrino billiardsrelativistic quantum chaosspin-1/2 particlesBerry-Mondragon boundary conditionsgraphene billiardsquantum billiardschaotic dynamicsintegrable dynamics
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The pith

Neutrino billiards confine a spin-1/2 particle via special boundary conditions to model relativistic quantum chaos.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This review establishes neutrino billiards as a model system for relativistic quantum chaos by confining a spin-1/2 particle to a planar domain. The confinement uses boundary conditions on the spinor components first proposed in 1987. The paper reviews the general spectral and dynamical features of these billiards for shapes whose classical counterparts are integrable, then examines two chaotic shapes whose nonrelativistic versions have distinctive properties. It closes with a brief discussion of graphene sheets as a possible experimental platform. A reader would care because the model opens a route to test how relativity alters quantum chaos signatures that are well studied in the nonrelativistic case.

Core claim

Neutrino billiards consist of a spin-1/2 particle confined to a planar domain by imposing boundary conditions on the spinor components proposed in Berry and Mondragon 1987. The review covers their general features, the properties of billiards with integrable classical dynamics, and the features of two neutrino billiards whose shapes generate chaotic dynamics in the nonrelativistic limit. It notes that graphene billiards, i.e., finite-size sheets of graphene, provide a possible experimental realization of such relativistic quantum billiards.

What carries the argument

Neutrino billiards: planar domains that confine a Dirac spin-1/2 particle through Berry-Mondragon boundary conditions on the two-component spinor, allowing study of relativistic effects on quantum chaos.

If this is right

  • Relativistic corrections to level statistics and scarring can be isolated by comparing neutrino billiards to their nonrelativistic counterparts.
  • Shapes that are integrable classically remain integrable in the relativistic setting, while chaotic shapes retain their chaotic signatures but with modified spectral properties.
  • Graphene sheets of appropriate shape could serve as a laboratory testbed for the predicted relativistic chaos features.
  • The model supplies concrete predictions for how spin-orbit or relativistic kinematics alter the transition from integrability to chaos.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the boundary conditions hold, neutrino billiards could be used to simulate aspects of confined neutrinos or other relativistic fermions without needing high-energy accelerators.
  • The graphene route implies that 2D material devices might display measurable differences between relativistic and nonrelativistic quantum chaos in the same geometry.
  • Comparing the two chaotic shapes reviewed here to their nonrelativistic versions could reveal whether relativistic kinematics suppress or enhance certain universal features of quantum chaos.
  • The review structure suggests that further numerical or analytic work on additional chaotic shapes would strengthen the case for using neutrino billiards as a standard testbed.

Load-bearing premise

The 1987 boundary conditions on the spinor components accurately capture the confinement of spin-1/2 particles inside planar domains.

What would settle it

An experiment on a graphene sheet whose measured energy levels or wave-function statistics deviate systematically from those predicted by the Berry-Mondragon boundary conditions would show the model does not apply.

Figures

Figures reproduced from arXiv: 2604.13003 by Barbara Dietz.

Figure 1
Figure 1. Figure 1: (a) Length spectra of the massless semicircle NB (upper curves) and QB (lower curves) (black solid lines). They are compared to the [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) Spectral rigidity ∆3(L) for QBs with shapes Eq. (78) for various values of ω. The curves for Possonian, GOE and GUE statistics are plotted as solid, dashed and dash-dash-dot lines, respectively. (b) Same as left for the NBs. spectral properties undergo a transition from Poisson to GOE statistics for the QBs and GUE statistics for the NBs. The wave functions of typical quantum systems with a chaotic cla… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Number variance for the semicircle NB for various masses. To obtain good aggreement with GOE statistics for the massless [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Intensity distributions of the wave functions, (b) Husimi functions, and (c) momentum distributions of the stadium QB for the [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Intensity distributions of the wave functions, (b) Husimi functions, and (c) momentum distributions of the stadium NB for the [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) Fluctuating part of the integrated spectral density of the quarter-stadium QB (black dots) compared to that of the BBOs deduced [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (a) Wave functions (left) and Husimi functions (right) of the constant-width QB for state numbers [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) Left: Conduction and valence bands of graphene. Right: The upper part shows the density of states of a GB with the [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of an experimentally determined density of states (blue) with the TBM result for a honome lattice (red) constructed from [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
read the original abstract

Neutrino billiards serve as a model system for the study of aspects of relativistic quantum chaos. These are relativistic quantum billiards consisting of a spin-1/2 particle which is confined to a planar domain by imposing boundary conditions on the spinor components which were proposed in [Berry and Mondragon 1987, {\it Proc. R. Soc.} A {\bf 412} 53) . We review their general features and the properties of neutrino billiards with shapes of billiards with integrable dynamics. Furthermore, we review the features of two neutrino billiards with the shapes of billiards generating a chaotic dynamics, whose nonrelativistic counterpart exhibits particular properties. Finally we briefly discuss possible experimental realizations of relativistic quantium billiards based on graphene billiards, that is, finite size sheets of graphene.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript reviews neutrino billiards as a model system for relativistic quantum chaos. These consist of a spin-1/2 particle confined to planar domains via boundary conditions on the Dirac spinor components proposed in Berry and Mondragon (1987). It covers general features of such billiards, spectral and wavefunction properties for integrable shapes (circle, ellipse), corresponding properties for two chaotic shapes (including the stadium whose non-relativistic version has known special properties), and briefly discusses possible experimental realizations in finite graphene sheets.

Significance. If the cited boundary conditions are shown to be physically accurate, the review usefully assembles literature on a relativistic extension of quantum billiards that can exhibit chaos signatures distinct from the non-relativistic case. It connects classical billiard dynamics, relativistic quantum mechanics, and potential graphene experiments, providing a consolidated reference for the field.

major comments (1)
  1. [Introduction and boundary-conditions paragraph (citing Berry and Mondragon 1987)] The manuscript's central claim that neutrino billiards model relativistic quantum chaos rests on the Berry-Mondragon 1987 boundary conditions accurately confining the spin-1/2 particle to the domain while preserving relativistic dynamics and preventing probability current leakage. The text cites this reference but supplies no independent derivation, numerical validation against alternative confinement schemes, or discussion of possible artifacts (e.g., in comparison to infinite-mass walls or graphene edge states). This omission is load-bearing for all reviewed spectral statistics and wavefunction properties in both integrable and chaotic geometries.
minor comments (2)
  1. [Abstract] Abstract contains the typographical error 'relativistic quantium billiards'; correct to 'relativistic quantum billiards'.
  2. [Abstract] The sentence beginning 'Furthermore, we review the features of two neutrino billiards with the shapes of billiards generating a chaotic dynamics...' is redundant and awkwardly phrased; reword for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive report and the opportunity to clarify the presentation of the foundational boundary conditions in our review. We address the major comment below and will revise the manuscript accordingly to improve self-containment while preserving its review character.

read point-by-point responses

  1. Referee: The manuscript's central claim that neutrino billiards model relativistic quantum chaos rests on the Berry-Mondragon 1987 boundary conditions accurately confining the spin-1/2 particle to the domain while preserving relativistic dynamics and preventing probability current leakage. The text cites this reference but supplies no independent derivation, numerical validation against alternative confinement schemes, or discussion of possible artifacts (e.g., in comparison to infinite-mass walls or graphene edge states). This omission is load-bearing for all reviewed spectral statistics and wavefunction properties in both integrable and chaotic geometries.

    Authors: We agree that the manuscript would benefit from a more explicit treatment of the Berry-Mondragon boundary conditions to make the review more self-contained. Although the paper is a review that cites the original derivation, we will expand the introductory section with a concise summary of the key steps in the 1987 derivation, emphasizing how the conditions confine the particle, eliminate normal current leakage, and retain the relativistic Dirac dynamics. We will also add a short paragraph discussing possible artifacts and comparisons to alternative schemes (infinite-mass walls and graphene edge states), drawing on existing literature. These revisions will directly support the subsequent discussion of spectral statistics without adding new numerical results or altering the review's scope. revision: yes

Circularity Check

0 steps flagged

Review paper with no internal derivations or self-referential claims

full rationale

The manuscript is a review that cites the 1987 Berry-Mondragon boundary conditions as an external reference without deriving them, introducing fitted parameters, or presenting any predictions that reduce to the inputs by construction. No self-citation chains, ansatzes smuggled via prior work, or renamings of known results appear in the provided text or abstract. The central model is taken as given from independent prior literature with no author overlap, satisfying the criteria for a self-contained review with no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a review paper; no new free parameters, axioms, or invented entities are introduced by the authors.

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Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Quantum chaotic systems: a random-matrix approach

    quant-ph 2026-04 unverdicted

    Review of random matrix theory application to quantum chaos, covering symmetry classes, eigenvalue statistics, unfolding, and correlation functions.

Reference graph

Works this paper leans on

122 extracted references · 122 canonical work pages · cited by 1 Pith paper

  1. [1]

    Casati, F

    G. Casati, F . Valz-Gris and I. Guarnieri,On the connection between quantization of nonintegrable systems and statistical theory of spectra,Lett. Nuovo Cimento28(1980) 279

    work page 1980

  2. [2]

    Berry,Quantizing a classically ergodic system: Sinai’s billiard and the kkr method,Ann

    M.V. Berry,Quantizing a classically ergodic system: Sinai’s billiard and the kkr method,Ann. Phys.131(1981) 163

    work page 1981

  3. [3]

    Bohigas, M.J

    O. Bohigas, M.J. Giannoni and C. Schmit,Characterization of chaotic quantum spectra and universality of level fluctuation laws,Phys. Rev. Lett.52(1984) 1

    work page 1984

  4. [4]

    Berry and M

    M.V. Berry and M. Tabor,Calculating the bound spectrum by path summation in actionangle variables,J. Phys. A10(1977) 371

    work page 1977

  5. [5]

    Heusler, S

    S. Heusler, S. Müller, A. Altland, P . Braun and F . Haake,Periodic-orbit theory of level correlations,Phys. Rev. Lett.98(2007) 044103

    work page 2007

  6. [6]

    Gutzwiller,Periodic orbits and classical quantization conditions,J

    M.C. Gutzwiller,Periodic orbits and classical quantization conditions,J. Math. Phys.12(1971) 343

    work page 1971

  7. [7]

    Guhr and H.A

    T. Guhr and H.A. Weidenmüller,Coexistence of collectivity and chaos in nuclei,Ann. Phys.193(1989) 472

    work page 1989

  8. [8]

    Giannoni, A

    M. Giannoni, A. Voros and J. Zinn-Justin, eds.,Chaos and Quantum Physics, Elsevier, Amsterdam (1989)

    work page 1989

  9. [9]

    Haake, S

    F . Haake, S. Gnutzmann and M. Ku´s,Quantum Signatures of Chaos, Springer-Verlag, Heidelberg (2018)

    work page 2018

  10. [10]

    Berkolaiko and S

    G. Berkolaiko and S. Gnutzmann,Quantum graphs and their applications to quantum chaos, 2026

    work page 2026

  11. [11]

    Sinai,Dynamical systems with elastic reflections,Russ

    Y .G. Sinai,Dynamical systems with elastic reflections,Russ. Math. Surv.25(1970) 137

    work page 1970

  12. [12]

    Bunimovich,On the ergodic properties of nowhere dispersing billiards,Commun

    L.A. Bunimovich,On the ergodic properties of nowhere dispersing billiards,Commun. Math. Phys.65(1979) 295

    work page 1979

  13. [13]

    Berry,Regularity and chaos in classical mechanics, illustrated by three deformations of a circular ’billiard’,Eur

    M.V. Berry,Regularity and chaos in classical mechanics, illustrated by three deformations of a circular ’billiard’,Eur. J. Phys.2(1981) 91

    work page 1981

  14. [14]

    Harayama,Chaotic billiard lasers,in preparation(2026)

    T. Harayama,Chaotic billiard lasers,in preparation(2026) . Relativistic Quantum Chaos in Neutrino Billiards27

    work page 2026

  15. [15]

    Berry and R.J

    M.V. Berry and R.J. Mondragon,Neutrino billiards: Time-reversal symmetry-breaking without magnetic fields,Proc. R. Soc. London A 412(1987) 53

    work page 1987

  16. [16]

    Novoselov, A.K

    K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y . Zhang, S.V. Dubonos et al.,Electric field effect in atomically thin carbon films, Science306(2004) 666

    work page 2004

  17. [17]

    Weyl,Elektron und Gravitation

    H. Weyl,Elektron und Gravitation. I,Z. Physik56(1929) 330

    work page 1929

  18. [18]

    Bolte and S

    J. Bolte and S. Keppeler,A semiclassical approach to the Dirac equation,Ann. Phys.274(1999) 125

    work page 1999

  19. [19]

    Dietz and Z.-Y

    B. Dietz and Z.-Y . Li,Semiclassical quantization of neutrino billiards,Phys. Rev. E102(2020) 042214

    work page 2020

  20. [20]

    Zhang and B

    W. Zhang and B. Dietz,Microwave photonic crystals, graphene, and honeycomb-kagome billiards with threefold symmetry: Comparison with nonrelativistic and relativistic quantum billiards,Phys. Rev. B104(2021) 064310

    work page 2021

  21. [21]

    Zhang, X

    W. Zhang, X. Zhang, J. Che, M. Miski-Oglu and B. Dietz,Properties of eigenmodes and quantum-chaotic scattering in a superconducting microwave Dirac billiard with threefold rotational symmetry,Phys. Rev. B107(2023) 144308

    work page 2023

  22. [22]

    P . Yu, B. Dietz and L. Huang,Quantizing neutrino billiards: an expanded boundary integral method,New J. Phys.21(2019) 073039

    work page 2019

  23. [23]

    Dietz,Unidirectionality and Husimi functions in constant-width neutrino billiards,J

    B. Dietz,Unidirectionality and Husimi functions in constant-width neutrino billiards,J. Phys. A: Math. Theor.55(2022) 474003

    work page 2022

  24. [24]

    Sieber and S

    M. Sieber and S. Müller,Semiclassical periodic-orbit theory for quantum spectra,in preparation(2026)

    work page 2026

  25. [25]

    Hentschel,Quantum chaos in phase space, 2026

    M. Hentschel,Quantum chaos in phase space, 2026

    work page 2026

  26. [26]

    Tomsovic,Hamiltonian chaos, 2026

    S. Tomsovic,Hamiltonian chaos, 2026

    work page 2026

  27. [27]

    Huang, H.-Y

    L. Huang, H.-Y . Xu, C. Grebogi and Y .-C. Lai,Relativistic quantum chaos,Phys. Rep.753(2018) 1

    work page 2018

  28. [28]

    Dietz,Failure of the conformal-map method for relativistic quantum billiards,Phys

    B. Dietz,Failure of the conformal-map method for relativistic quantum billiards,Phys. Rev. Lett.135(2025) 030401

    work page 2025

  29. [29]

    Knill,On nonconvex caustics of convex billiards,Elemente der Mathematik53(1998) 89

    O. Knill,On nonconvex caustics of convex billiards,Elemente der Mathematik53(1998) 89

    work page 1998

  30. [30]

    breaking

    B. Gutkin,Dynamical "breaking" of time reversal symmetry,J. Phys. A40(2007) F761

    work page 2007

  31. [31]

    Shudo,Chaos and quantum tunneling, 2026

    A. Shudo,Chaos and quantum tunneling, 2026

    work page 2026

  32. [32]

    Dietz, T

    B. Dietz, T. Klaus, M. Miski-Oglu and A. Richter,Spectral properties of superconducting microwave photonic crystals modeling Dirac billiards,Phys. Rev. B91(2015) 035411

    work page 2015

  33. [33]

    Dietz, T

    B. Dietz, T. Klaus, M. Miski-Oglu, A. Richter, M. Wunderle and C. Bouazza,Spectral properties of Dirac billiards at the van Hove singularities,Phys. Rev. Lett.116(2016) 023901

    work page 2016

  34. [34]

    Maimaiti, B

    W. Maimaiti, B. Dietz and A. Andreanov,Microwave photonic crystals as an experimental realization of a combined honeycomb-kagome lattice,Phys. Rev. B102(2020) 214301

    work page 2020

  35. [35]

    Beenakker,Colloquium: Andreev reflection and klein tunneling in graphene,Rev

    C.W.J. Beenakker,Colloquium: Andreev reflection and klein tunneling in graphene,Rev. Mod. Phys.80(2008) 1337

    work page 2008

  36. [36]

    Castro Neto, F

    A.H. Castro Neto, F . Guinea, N.M.R. Peres, K.S. Novoselov and A.K. Geim,The electronic properties of graphene,Rev. Mod. Phys.81 (2009) 109

    work page 2009

  37. [37]

    Polini, F

    M. Polini, F . Guinea, M. Lewenstein, H.C. Manoharan and V. Pellegrini,Artificial graphene as a tunable dirac material,Nat. Nanotechnol. 8(2013) 625

    work page 2013

  38. [38]

    parity anomaly

    F .D.M. Haldane,Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the "parity anomaly",Phys. Rev. Lett.61(1988) 2015

    work page 1988

  39. [39]

    Nguyen and B

    D.X. Nguyen and B. Dietz,Haldane graphene billiards versus relativistic neutrino billiards,Phys. Rev. B110(2024) 094305

    work page 2024

  40. [40]

    Gaddah,Exact solutions to the Dirac equation for equilateral triangular billiard systems,J

    W.A. Gaddah,Exact solutions to the Dirac equation for equilateral triangular billiard systems,J. Phys. A: Math. Theor.51(2018) 385304

    work page 2018

  41. [41]

    Thomas,Chiral symmetry and the bag model: A new starting point for nuclear physics, inAdvances in Nuclear Physics: Volume 13, pp

    A.W. Thomas,Chiral symmetry and the bag model: A new starting point for nuclear physics, inAdvances in Nuclear Physics: Volume 13, pp. 1–137, Springer (1984)

    work page 1984

  42. [42]

    Greiner and A

    W. Greiner and A. Schäfer, eds.,Quantum Chromodynamics, Springer, New Y ork (1994)

    work page 1994

  43. [43]

    Sieber, H

    M. Sieber, H. Primack, U. Smilansky, I. Ussishkin and H. Schanz,Semiclassical quantization of billiards with mixed boundary conditions, J. Phys. A28(1995) 5041

    work page 1995

  44. [44]

    Berry and M.R

    M.V. Berry and M.R. Dennis,Boundary-condition-varying circle billiards and gratings: the dirichlet singularity,J. Phys. A41(2008) 135203

    work page 2008

  45. [45]

    Dietz and U

    B. Dietz and U. Smilansky,A scattering approach to the quantization of billiards—the inside–outside duality,Chaos3(1993) 581

    work page 1993

  46. [46]

    Dietz,Circular and elliptical neutrino billiards: A semiclassical approach,Act

    B. Dietz,Circular and elliptical neutrino billiards: A semiclassical approach,Act. Phys. Pol. A136(2019) 770

    work page 2019

  47. [47]

    P . Yu, B. Dietz, H.-Y . Xu, L. Ying, L. Huang and Y .-C. Lai,Kac’s isospectrality question revisited in neutrino billiards,Phys. Rev. E101 (2020) 032215

    work page 2020

  48. [48]

    Giraud and K

    O. Giraud and K. Thas,Hearing shapes of drums: Mathematical and physical aspects of isospectrality,Rev. Mod. Phys.82(2010) 2213

    work page 2010

  49. [50]

    Joyner, S

    C.H. Joyner, S. Müller and M. Sieber,Semiclassical approach to discrete symmetries in quantum chaos,J. Phys. A45(2012) 205102

    work page 2012

  50. [51]

    Dietz,Relativistic quantum billiards with threefold rotational symmetry: Exact, symmetry-projected solutions for the equilateral neutrino billiard,Act

    B. Dietz,Relativistic quantum billiards with threefold rotational symmetry: Exact, symmetry-projected solutions for the equilateral neutrino billiard,Act. Phys. Pol. A140(2021) 473

    work page 2021

  51. [52]

    Harayama and A

    T. Harayama and A. Shudo,Zeta function derived from the boundary element method,Phys. Lett. A165(1992) 417

    work page 1992

  52. [53]

    Sieber,Semiclassical transition from an elliptical to an oval billiard,J

    M. Sieber,Semiclassical transition from an elliptical to an oval billiard,J. Phys. A30(1997) 4563

    work page 1997

  53. [54]

    Bäcker,Numerical Aspects of Eigenvalue and Eigenfunction Computations for Chaotic Quantum Systems, inThe Mathematical Aspects of Quantum Maps, M.D

    A. Bäcker,Numerical Aspects of Eigenvalue and Eigenfunction Computations for Chaotic Quantum Systems, inThe Mathematical Aspects of Quantum Maps, M.D. Esposti and S. Graffi, eds., (Berlin, Heidelberg), pp. 91–144, Springer Berlin Heidelberg (2003), DOI

    work page 2003

  54. [55]

    Okada, A

    Y . Okada, A. Shudo, S. Tasaki and T. Harayama,On the boundary element method for billiards with corners,Journal of Physics A: Mathematical and General38(2005) 6675

    work page 2005

  55. [56]

    Kleinman and G.F

    R.E. Kleinman and G.F . Roach,Boundary integral equations for the three-dimensional helmholtz equation,SIAM Rev.16(1974) 214

    work page 1974

  56. [57]

    Dietz and A

    B. Dietz and A. Richter,From graphene to fullerene: experiments with microwave photonic crystals,Phys. Scr.94(2019) 014002

    work page 2019

  57. [58]

    Sieber,Billiard systems in three dimensions: the boundary integral equation and the trace formula,Nonlinearity11(1998) 1607

    M. Sieber,Billiard systems in three dimensions: the boundary integral equation and the trace formula,Nonlinearity11(1998) 1607

    work page 1998

  58. [59]

    J. Wurm, K. Richter and ˙I. Adagideli,Edge effects in graphene nanostructures: From multiple reflection expansion to density of states, Phys. Rev. B84(2011) 075468

    work page 2011

  59. [60]

    Robbins,Discrete symmetries in periodic-orbit theory,Phys

    J.M. Robbins,Discrete symmetries in periodic-orbit theory,Phys. Rev. A40(1989) 2128

    work page 1989

  60. [61]

    Dyson,The threefold way

    F .J. Dyson,The threefold way. algebraic structure of symmetry groups and ensembles in quantum mechanics,Journal of Mathematical Physics3(1962) 1199

    work page 1962

  61. [62]

    Mehta,Random Matrices, Elsevier, Amsterdam (2004)

    M.L. Mehta,Random Matrices, Elsevier, Amsterdam (2004)

    work page 2004

  62. [63]

    P . Yu, W. Zhang, B. Dietz and L. Huang,Quantum signatures of chaos in relativistic quantum billiards with shapes of circle- and ellipse-sectors,J. Phys. A: Math. Theor.55(2022) 224015

    work page 2022

  63. [64]

    Weyl,Über die Abhängigkeit der Eigenschwingungen einer Membran und deren Begrenzung,J

    H. Weyl,Über die Abhängigkeit der Eigenschwingungen einer Membran und deren Begrenzung,J. Reine Angew. Math.141(1912) 1

    work page 1912

  64. [65]

    Oganesyan and D.A

    V. Oganesyan and D.A. Huse,Localization of interacting fermions at high temperature,Phys. Rev. B75(2007) 155111

    work page 2007

  65. [66]

    Y .Y . Atas, E. Bogomolny, O. Giraud and G. Roux,Distribution of the ratio of consecutive level spacings in random matrix ensembles, Phys. Rev. Lett.110(2013) 084101. 28Relativistic Quantum Chaos in Neutrino Billiards

    work page 2013

  66. [67]

    Altland, F

    A. Altland, F . Divi, T. Micklitz, S. Pappalardi and M. Rezaei,Statistics of the random matrix spectral form factor,Phys. Rev. Res.7(2025) 033138

    work page 2025

  67. [68]

    Dietz and F

    B. Dietz and F . Haake,Taylor and padé analysis of the level spacing distributions of random-matrix ensembles,Z. Phys. A80(1990) 153

    work page 1990

  68. [69]

    Berry and M

    M.V. Berry and M. Robnik,Statistics of energy levels without time-reversal symmetry: Aharonov-Bohm chaotic billiards,J. Phys. A19 (1986) 649

    work page 1986

  69. [70]

    Bogomolny,Formation of superscar waves in plane polygonal billiards*,J

    E. Bogomolny,Formation of superscar waves in plane polygonal billiards*,J. Phys. Commun.5(2021) 055010

    work page 2021

  70. [71]

    Bäcker and R

    A. Bäcker and R. Schubert,Chaotic eigenfunctions in momentum space,J. Phys. A: Math. Gen.32(1999) 4795

    work page 1999

  71. [72]

    McDonald and A.N

    S.W. McDonald and A.N. Kaufman,Spectrum and eigenfunctions for a hamiltonian with stochastic trajectories,Phys. Rev. Lett.42(1979) 1189

    work page 1979

  72. [73]

    Heller,Bound-state eigenfunctions of classically chaotic Hamiltonian systems: Scars of periodic orbits,Phys

    E.J. Heller,Bound-state eigenfunctions of classically chaotic Hamiltonian systems: Scars of periodic orbits,Phys. Rev. Lett.53(1984) 1515

    work page 1984

  73. [74]

    Wegner,Inverse participation ratio in 2+ϵdimensions,Phys

    F . Wegner,Inverse participation ratio in 2+ϵdimensions,Phys. Rev. Lett.36(1980) 209

    work page 1980

  74. [75]

    Bäcker, S

    A. Bäcker, S. Fürstberger and R. Schubert,Poincaré Husimi representation of eigenstates in quantum billiards,Phys. Rev. E70(2004) 036204

    work page 2004

  75. [76]

    Husimi,Some formal properties of the density matrix,Proc

    K. Husimi,Some formal properties of the density matrix,Proc. Phys. Math. Soc. Jpn.22(1940) 264

    work page 1940

  76. [77]

    Dietz,Semi-Poisson statistics in relativistic quantum billiards with shapes of rectangles,Entropy25(2023)

    B. Dietz,Semi-Poisson statistics in relativistic quantum billiards with shapes of rectangles,Entropy25(2023)

    work page 2023

  77. [78]

    Keating and S

    J.P . Keating and S. Müller,Resummation and the semiclassical theory of spectral statistics,Proc. R. Soc. Lond. A3241(2007) 463

    work page 2007

  78. [79]

    McLachlan, ed.,Theory and Application of Mathieu Functions, Oxford University Press, London (1947)

    N. McLachlan, ed.,Theory and Application of Mathieu Functions, Oxford University Press, London (1947)

    work page 1947

  79. [80]

    Waalkens, J

    H. Waalkens, J. Wiersig and H.R. Dullin,Elliptic quantum billiard,Annals of Physics260(1997) 50

    work page 1997

  80. [81]

    Lamé, ed.,Leçons sur le Théorie Mathématique d’Elasticité des Cores Solides, Bachelier, Paris (1852)

    M.G. Lamé, ed.,Leçons sur le Théorie Mathématique d’Elasticité des Cores Solides, Bachelier, Paris (1852)

    work page

Showing first 80 references.