pith. sign in

arxiv: 2605.19019 · v1 · pith:5LFCJPS4new · submitted 2026-05-18 · 🪐 quant-ph · math-ph· math.MP· nlin.CD

Semiclassical periodic-orbit theory for quantum spectra

Sebastian M\"uller , Martin Sieber This is my paper

Pith reviewed 2026-05-20 10:33 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MPnlin.CD
keywords Gutzwiller trace formulaperiodic orbitsquantum chaossemiclassical approximationrandom matrix theoryenergy level statisticsFeynman path integraldensity of states
0
0 comments X

The pith

Gutzwiller's trace formula approximates quantum energy levels in chaotic systems by summing contributions from classical periodic orbits.

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

This paper derives Gutzwiller's trace formula starting from the Feynman path integral to connect classical periodic orbits with the density of quantum energy levels in chaotic systems. A sympathetic reader cares because the resulting expression accounts for why those levels exhibit universal statistics matching random matrix theory instead of the regular spacings seen in integrable systems. The derivation isolates the dominant orbit contributions in the semiclassical limit and shows how their interference produces the observed spectral features. It then surveys how the same framework explains level correlations and provides an overview of extensions in related work.

Core claim

Gutzwiller's trace formula provides semiclassical approximations for quantum energy levels in classically chaotic systems by linking them to classical periodic orbits. The formula is obtained from the Feynman path integral by stationary-phase evaluation around the periodic orbits, yielding an expression for the density of states as a sum over orbit contributions. This semiclassical link explains the universal features in the distribution of quantum energy levels that are described by random matrix theory.

What carries the argument

Gutzwiller's trace formula, which expresses the oscillating part of the quantum density of states as a sum over contributions from classical periodic orbits.

If this is right

  • The formula yields approximate quantum spectra directly from classical periodic orbit data without solving the Schrödinger equation.
  • It reproduces the spectral statistics of random matrix theory for chaotic systems, including level repulsion and rigidity.
  • Related semiclassical methods for transport and scattering follow from the same stationary-phase treatment of the path integral.

Where Pith is reading between the lines

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

  • Similar orbit-sum techniques could be tested for their accuracy in mixed phase-space systems that are neither fully chaotic nor integrable.
  • The path-integral starting point suggests possible adaptations to time-dependent driving or open quantum systems where orbits escape.

Load-bearing premise

The semiclassical approximation remains valid when starting from the Feynman path integral for systems with chaotic classical dynamics.

What would settle it

Numerical comparison of the trace formula's predicted level density and pair correlations against exact diagonalization results for a concrete chaotic billiard such as the stadium shape.

Figures

Figures reproduced from arXiv: 2605.19019 by Martin Sieber, Sebastian M\"uller.

Figure 10
Figure 10. Figure 10: (The beginning of the final stretch is then connected to the end of the first stretch.) Now, a structure is characterized by [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
read the original abstract

Gutzwiller's trace formula has a central place in quantum chaos because it provides semiclassical approximations for quantum energy levels in classically chaotic systems by linking them to classical periodic orbits. In this didactic article, we discuss a derivation of the trace formula starting from the Feynman path integral. We then describe how the trace formula is used to explain universal features in the distribution of the quantum energy levels that are described by random matrix theory, and we give an overview of related work.

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

0 major / 2 minor

Summary. The manuscript is a didactic presentation of the standard derivation of Gutzwiller's trace formula for the semiclassical density of states in classically chaotic systems, beginning from the Feynman path integral, followed by its application to explain universal spectral statistics via random matrix theory and a brief overview of related literature.

Significance. If the exposition is accurate and complete, the paper offers a clear pedagogical account of established results in quantum chaos. Its primary value lies in providing an accessible entry point to the trace formula and its RMT connections rather than introducing novel theorems, parameter-free derivations, or new falsifiable predictions.

minor comments (2)
  1. [Abstract] The abstract states that the article 'discuss[es] a derivation' but does not indicate the intended level of detail or target readership (e.g., graduate students versus researchers); adding one sentence on scope would improve clarity for potential readers.
  2. [Related work section] In the overview of related work, several key references to extensions of the trace formula (e.g., to systems with mixed phase space or to higher-order ħ corrections) appear to be cited only in passing; expanding the discussion with one or two additional sentences per topic would strengthen the survey without altering the didactic focus.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee accurately summarizes the paper as a didactic derivation of Gutzwiller's trace formula from the path integral, together with its application to random matrix theory statistics and an overview of related literature. We agree that the primary value is pedagogical rather than the introduction of new theorems.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript is a didactic exposition that derives Gutzwiller's trace formula from the Feynman path integral as an independent starting point in quantum mechanics, then connects the resulting semiclassical approximation to random-matrix-theory statistics of energy levels. No load-bearing step reduces by construction to a fitted parameter, self-defined quantity, or self-citation chain; the central claims are standard results whose validity rests on the path-integral foundation and prior literature rather than on internal redefinitions or predictions that are statistically forced by the inputs. The paper advances no novel theorems or falsifiable predictions whose correctness would require external verification beyond the exposition itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard assumptions of semiclassical quantum mechanics and classical chaos theory; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Validity of the semiclassical approximation when applied to the Feynman path integral for chaotic systems
    Invoked to obtain the trace formula from the path integral.

pith-pipeline@v0.9.0 · 5598 in / 922 out tokens · 47823 ms · 2026-05-20T10:33:24.886929+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

112 extracted references · 112 canonical work pages · 7 internal anchors

  1. [1]

    Einstein,Zum Quantensatz von Sommerfeld und Epstein,Verh

    A. Einstein,Zum Quantensatz von Sommerfeld und Epstein,Verh. Dtsch. Phys. Ges.19(1917) 82

    work page 1917

  2. [2]

    Brillouin,La m ´ecanique ondulatoire de Schr¨odinger: une m ´ethode g´en´erale de r´esolution par approximations successives,C

    L. Brillouin,La m ´ecanique ondulatoire de Schr¨odinger: une m ´ethode g´en´erale de r´esolution par approximations successives,C. R. Hebd. Seances Acad. Sci. Paris183(1926) 24

    work page 1926

  3. [3]

    Keller,Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems,Ann

    J.B. Keller,Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems,Ann. Phys. (N.Y .)4(1958) 180

    work page 1958

  4. [4]

    Gutzwiller,Energy spectrum according to classical mechanics,J

    M.C. Gutzwiller,Energy spectrum according to classical mechanics,J. Math. Phys.11(1970) 1791

    work page 1970

  5. [5]

    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

  6. [6]

    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

  7. [7]

    Haake, S

    F . Haake, S. Gnutzmann and M. Ku´s,Quantum Signatures of Chaos, vol. 54 ofSpringer Series in Synergetics, Springer International Publishing, Cham, 4 ed. (2018)

    work page 2018

  8. [8]

    St ¨ockmann,Quantum Chaos: An Introduction, Cambridge University Press, Cambridge (1999), 10.1017/CBO9780511524622

    H.-J. St ¨ockmann,Quantum Chaos: An Introduction, Cambridge University Press (1999), 10.1017/CBO9780511524622

    work page doi:10.1017/cbo9780511524622 1999

  9. [9]

    Wimberger,Nonlinear Dynamics and Quantum Chaos: An Introduction, Graduate Texts in Physics, Springer, Cham, 2 ed

    S. Wimberger,Nonlinear Dynamics and Quantum Chaos: An Introduction, Graduate Texts in Physics, Springer, Cham, 2 ed. (2022)

    work page 2022

  10. [10]

    Brack and R.K

    M. Brack and R.K. Bhaduri,Semiclassical Physics, vol. 96 ofFrontiers in Physics, Addison-Wesley, Reading, MA (1997)

    work page 1997

  11. [11]

    Cvitanovi´c, R

    P . Cvitanovi´c, R. Artuso, R. Mainieri, G. Tanner and G. Vattay,Chaos: Classical and Quantum, ChaosBook.org, 17.6.8 ed. (2025)

    work page 2025

  12. [12]

    Gutzwiller,Chaos in Classical and Quantum Mechanics, vol

    M.C. Gutzwiller,Chaos in Classical and Quantum Mechanics, vol. 1 ofInterdisciplinary Applied Mathematics, Springer, New Y ork (1990)

    work page 1990

  13. [13]

    M ¨uller and M

    S. M ¨uller and M. Sieber,Quantum chaos and quantum graphs, inThe Oxford Handbook of Random Matrix Theory, G. Akemann, J. Baik and P .D. Francesco, eds., Oxford University Press (2011)

    work page 2011

  14. [14]

    Quantum chaotic systems: a random-matrix approach

    M. Kieburg,Quantum chaotic systems: a random matrix approach, inthis volume(2026), https://arxiv.org/pdf/2604.12141v1 [2604.12141]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  15. [15]

    Hamiltonian Chaos

    S. Tomsovic,Hamiltonian chaos, inthis volume(2026), https://arxiv.org/pdf/2604.12976v1 [2604.12976]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  16. [16]

    Hannay and A.M.O

    J. Hannay and A.M.O. de Almeida,Periodic orbits and a correlation function for the semiclassical density of states,J. Phys. A: Math. Gen.17(1984) 3429

    work page 1984

  17. [17]

    Berry,Semiclassical theory of spectral rigidity,Proc

    M.V. Berry,Semiclassical theory of spectral rigidity,Proc. R. Soc. Lond. A400(1985) 229

    work page 1985

  18. [18]

    Sieber and K

    M. Sieber and K. Richter,Correlations between periodic orbits and their r ˆole in spectral statistics,Phys. Scr.2001(2001) 128

    work page 2001

  19. [19]

    M ¨uller, S

    S. M ¨uller, S. Heusler, A. Altland, P . Braun and F . Haake,Periodic-orbit theory of universal level correlations in quantum chaos,New J. Phys.11(2009) 103025

    work page 2009

  20. [20]

    Feynman,Space-time approach to non-relativistic quantum mechanics,Rev

    R.P . Feynman,Space-time approach to non-relativistic quantum mechanics,Rev. Mod. Phys.20(1948) 367

    work page 1948

  21. [21]

    Altland and B

    A. Altland and B. Simons,Condensed Matter Field Theory, Cambridge University Press, Cambridge, 3 ed. (2023). Semiclassical periodic-orbit theory for quantum spectra29

    work page 2023

  22. [22]

    Morette,On the definition and approximation of Feynman’s path integrals,Phys

    C. Morette,On the definition and approximation of Feynman’s path integrals,Phys. Rev.81(1951) 848

    work page 1951

  23. [23]

    Vleck,The correspondence principle in the statistical interpretation of quantum mechanics,Proc

    J.H.V. Vleck,The correspondence principle in the statistical interpretation of quantum mechanics,Proc. Natl. Acad. Sci. USA14(1928) 178

    work page 1928

  24. [24]

    Sturm,M ´emoire sur les ´equations diff´erentielles lin´eaires du second ordre,J

    J.C.F . Sturm,M ´emoire sur les ´equations diff´erentielles lin´eaires du second ordre,J. Math. Pures Appl.1(1836) 106

    work page

  25. [25]

    Gutzwiller,Phase-integral approximation in momentum space and the bound states of an atom,J

    M.C. Gutzwiller,Phase-integral approximation in momentum space and the bound states of an atom,J. Math. Phys.8(1967) 1979

    work page 1967

  26. [26]

    Dirac,The Principles of Quantum Mechanics, International Series of Monographs on Physics, Clarendon Press, Oxford, 4 ed

    P .A.M. Dirac,The Principles of Quantum Mechanics, International Series of Monographs on Physics, Clarendon Press, Oxford, 4 ed. (1958)

    work page 1958

  27. [27]

    Miller,Classical-limit quantum mechanics and the theory of molecular collisions, inAdvances in Chemical Physics, I

    W.H. Miller,Classical-limit quantum mechanics and the theory of molecular collisions, inAdvances in Chemical Physics, I. Prigogine and S.A. Rice, eds., vol. 25, (New Y ork), pp. 69–177, John Wiley & Sons (1974)

    work page 1974

  28. [28]

    Selberg,Harmonic analysis and discontinuous groups in weakly symmetric riemannian spaces with applications to Dirichlet series,J

    A. Selberg,Harmonic analysis and discontinuous groups in weakly symmetric riemannian spaces with applications to Dirichlet series,J. Indian Math. Soc., New Series20(1956) 47

    work page 1956

  29. [29]

    Gutzwiller,The anisotropic Kepler problem in two dimensions,J

    M.C. Gutzwiller,The anisotropic Kepler problem in two dimensions,J. Math. Phys.14(1973) 139

    work page 1973

  30. [30]

    Cvitanovi´c and B

    P . Cvitanovi´c and B. Eckhardt,Periodic-orbit quantization of chaotic systems,Phys. Rev. Lett.63(1989) 823

    work page 1989

  31. [31]

    Main,Use of harmonic inversion techniques in semiclassical quantization and analysis of quantum spectra,Phys

    J. Main,Use of harmonic inversion techniques in semiclassical quantization and analysis of quantum spectra,Phys. Rep.316(1999) 233

    work page 1999

  32. [32]

    Berry and M

    M.V. Berry and M. Tabor,Closed orbits and the regular bound spectrum,Proc. R. Soc. Lond. A349(1976) 101

    work page 1976

  33. [33]

    Berry and M

    M.V. Berry and M. Tabor,Level clustering in the regular spectrum,Proc. R. Soc. Lond. A356(1977) 375

    work page 1977

  34. [34]

    Schomerus and M

    H. Schomerus and M. Sieber,Bifurcations of periodic orbits and uniform approximations,J. Phys. A: Math. Gen.30(1997) 4537

    work page 1997

  35. [35]

    Balian and C

    R. Balian and C. Bloch,Distribution of eigenfrequencies for the wave equation in a finite domain i: three-dimensional problem with smooth boundary surface,Ann. Phys.60(1970) 401

    work page 1970

  36. [36]

    Balian and C

    R. Balian and C. Bloch,Distribution of eigenfrequencies for the wave equation in a finite domain. iii. eigenfrequency density oscillations., Ann. Phys.69(1972) 76

    work page 1972

  37. [37]

    Berry and J.P

    M.V. Berry and J.P . Keating,A rule for quantizing chaos?,J. Phys. A: Math. Gen.23(1990) 4839

    work page 1990

  38. [38]

    Keating,Periodic orbit resummation and the quantization of chaos,Proc

    J.P . Keating,Periodic orbit resummation and the quantization of chaos,Proc. R. Soc. Lond. A436(1992) 99

    work page 1992

  39. [39]

    Berry and J.P

    M.V. Berry and J.P . Keating,A new asymptotic representation ofζ( 1 2 +it)and quantum spectral determinants,Proc. R. Soc. Lond. A437 (1992) 151

    work page 1992

  40. [40]

    Sieber,Wavefunctions, Green’s functions and expectation values in terms of spectral determinants,Nonlinearity20(2007) 2721

    M. Sieber,Wavefunctions, Green’s functions and expectation values in terms of spectral determinants,Nonlinearity20(2007) 2721

    work page 2007

  41. [41]

    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

  42. [42]

    Casati, I

    G. Casati, I. Guarneri and F . Valz-Gris,On the connection between quantization of nonintegrable systems and statistical theory of spectra,Lett. Nuovo Cim.28(1980) 279

    work page 1980

  43. [43]

    Verbaarschot,The spectrum of the QCD Dirac operator and chiral random matrix theory: The threefold way,Phys

    J.J.M. Verbaarschot,The spectrum of the QCD Dirac operator and chiral random matrix theory: The threefold way,Phys. Rev. Lett.72 (1994) 2531

    work page 1994

  44. [44]

    Altland and M.R

    A. Altland and M.R. Zirnbauer,Nonstandard symmetry classes in mesoscopic normal-superconducting hybrid structures,Phys. Rev. B 55(1997) 1142

    work page 1997

  45. [45]

    Argaman, F .-M

    N. Argaman, F .-M. Dittes, E. Doron, J.P . Keating, A.Y . Kitaev, M. Sieber et al.,Correlations in the actions of periodic orbits derived from quantum chaos,Phys. Rev. Lett.71(1993) 4326

    work page 1993

  46. [46]

    Cohen, H

    D. Cohen, H. Primack and U. Smilansky,Quantal-classical duality and the semiclassical trace formula,Ann. Phys.264(1998) 108

    work page 1998

  47. [47]

    Smilansky and B

    U. Smilansky and B. Verdene,Action correlations and random matrix theory,J. Phys. A: Math. Gen.36(2003) 3525

    work page 2003

  48. [48]

    Sieber,Leading off-diagonal approximation for the spectral form factor for uniformly hyperbolic systems,J

    M. Sieber,Leading off-diagonal approximation for the spectral form factor for uniformly hyperbolic systems,J. Phys. A: Math. Gen.35 (2002) L613

    work page 2002

  49. [49]

    Spehner,Spectral form factor of hyperbolic systems: leading off-diagonal approximation,J

    D. Spehner,Spectral form factor of hyperbolic systems: leading off-diagonal approximation,J. Phys. A: Math. Gen.36(2003) 7269

    work page 2003

  50. [50]

    Turek and K

    M. Turek and K. Richter,Leading off-diagonal contribution to the spectral form factor of chaotic quantum systems,J. Phys. A: Math. Gen. 36(2003) L455

    work page 2003

  51. [51]

    Turek, D

    M. Turek, D. Spehner, S. M ¨uller and K. Richter,Semiclassical form factor for spectral and matrix element fluctuations of multidimensional chaotic systems,Phys. Rev. E71(2005) 016210

    work page 2005

  52. [52]

    M ¨uller,Classical basis for quantum spectral fluctuations in hyperbolic systems,Eur

    S. M ¨uller,Classical basis for quantum spectral fluctuations in hyperbolic systems,Eur. Phys. J. B34(2003) 305

    work page 2003

  53. [53]

    Heusler, S

    S. Heusler, S. M ¨uller, P . Braun and F . Haake,Universal spectral form factor for chaotic dynamics,J. Phys. A: Math. Gen.37(2004) L31

    work page 2004

  54. [54]

    M ¨uller, S

    S. M ¨uller, S. Heusler, P . Braun, F . Haake and A. Altland,Semiclassical foundation of universality in quantum chaos,Phys. Rev. Lett.93 (2004) 014103

    work page 2004

  55. [55]

    M ¨uller, S

    S. M ¨uller, S. Heusler, P . Braun, F . Haake and A. Altland,Periodic-orbit theory of universality in quantum chaos,Phys. Rev. E72(2005) 046207

    work page 2005

  56. [56]

    Heusler, S

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

    work page 2007

  57. [57]

    Keating and S

    J.P . Keating and S. M¨uller,Resummation and the semiclassical theory of spectral statistics,Proc. R. Soc. A463(2007) 3241

    work page 2007

  58. [58]

    Braun and D

    P . Braun and D. Waltner,New approach to periodic orbit theory of spectral correlations,J. Phys. A: Math. Theor.52(2019) 065101

    work page 2019

  59. [59]

    Altland, P

    A. Altland, P . Braun, F . Haake, S. Heusler, G. Knieper and S. M¨uller,Near action-degenerate periodic-orbit bunches: a skeleton of chaos, inPath Integrals – New Trends and Perspectives: Proceedings of the 9th International Conference, W. Janke and A. Pelster, eds., p. 40, World Scientific (2008)

    work page 2008

  60. [60]

    M ¨uller and M

    S. M ¨uller and M. Novaes,Semiclassical calculation of spectral correlation functions of chaotic systems,Phys. Rev. E98(2018) 052207

    work page 2018

  61. [61]

    Semiclassical theory of transport

    M. Novaes,Semiclassical theory of transport, inthis volume(2026), https://arxiv.org/pdf/2604.13193v1 [2604.13193]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  62. [62]

    Shukla,Higher order correlations in quantum chaotic spectra,Phys

    P . Shukla,Higher order correlations in quantum chaotic spectra,Phys. Rev. E55(1997) 3886

    work page 1997

  63. [63]

    Nagao and S

    T. Nagao and S. M ¨uller,The n-level spectral correlations for chaotic systems,J. Phys. A: Math. Theor42(2009) 375102

    work page 2009

  64. [64]

    M ¨uller and M

    S. M ¨uller and M. Novaes,Full perturbative calculation of spectral correlation functions for chaotic systems in the unitary symmetry class, Phys. Rev. E98(2018) 052208

    work page 2018

  65. [65]

    Bolte and S

    J. Bolte and S. Keppeler,Semiclassical time evolution and trace formula for relativistic spin-1/2 particles,Phys. Rev. Lett.81(1998) 1987

    work page 1998

  66. [66]

    Bolte and S

    J. Bolte and S. Keppeler,Semiclassical form factor for chaotic systems with spin- 1 2 ,J. Phys. A: Math. Gen.32(1999) 8863

    work page 1999

  67. [67]

    Bolte and J

    J. Bolte and J. Harrison,The spectral form factor for quantum graphs with spin-orbit coupling, inQuantum Graphs and Their Applications, G. Berkolaiko, R. Carlson, S.A. Fulling and P . Kuchment, eds., pp. 51–64, AMS (2006), DOI

    work page 2006

  68. [68]

    Braun,Beyond the Heisenberg time: semiclassical treatment of spectral correlations in chaotic systems with spin- 1 2 ,J

    P . Braun,Beyond the Heisenberg time: semiclassical treatment of spectral correlations in chaotic systems with spin- 1 2 ,J. Phys. A: Math. Theor.45(2012) 045102

    work page 2012

  69. [69]

    Quantum graph models of quantum chaos: an introduction and some recent applications

    G. Berkolaiko and S. Gnutzmann,An introduction to quantum graphs and current applications, inthis volume(2026), https://arxiv.org/pdf/2604.12690v1 [2604.12690]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  70. [70]

    Kottos and U

    T. Kottos and U. Smilansky,Periodic orbit theory and spectral statistics for quantum graphs,Ann. Phys.274(1999) 76

    work page 1999

  71. [71]

    Berkolaiko, H

    G. Berkolaiko, H. Schanz and R.S. Whitney,Form factor for a family of quantum graphs: An expansion to third order,J. Phys. A: Math. 30Semiclassical periodic-orbit theory for quantum spectra Gen.36(2003) 8373

    work page 2003

  72. [72]

    Berkolaiko,Form factor for large quantum graphs: evaluating orbits with time reversal,Waves Random Media14(2004) S7

    G. Berkolaiko,Form factor for large quantum graphs: evaluating orbits with time reversal,Waves Random Media14(2004) S7

    work page 2004

  73. [73]

    Gnutzmann and A

    S. Gnutzmann and A. Altland,Universal spectral statistics in quantum graphs,Phys. Rev. Lett.93(2004) 194101

    work page 2004

  74. [74]

    Gnutzmann and A

    S. Gnutzmann and A. Altland,Spectral correlations of individual quantum graphs,Phys. Rev. E72(2005) 056215

    work page 2005

  75. [75]

    Pluha ˇr and H.A

    Z. Pluha ˇr and H.A. Weidenm¨uller,Universal quantum graphs,Phys. Rev. Lett.112(2014) 144102

    work page 2014

  76. [76]

    Pluha ˇr and H.A

    Z. Pluha ˇr and H.A. Weidenm¨uller,Quantum graphs and random-matrix theory,J. Phys. A: Math. Theor48(2015) 275102

    work page 2015

  77. [77]

    Quantum chaos in many-body systems of indistinguishable particles

    J.-D. Urbina and K. Richter,Quantum chaos in many-body systems of indistinguishable particles, inthis volume(2026), https://arxiv.org/pdf/2604.12745v1 [2604.12745]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  78. [78]

    The role of classical periodic orbits in quantum many-body systems

    D. Waltner and B. Gutkin,The role of classical periodic orbits in quantum many-body systems, inthis volume(2026), https://arxiv.org/pdf/2604.14015v1 [2604.14015]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  79. [79]

    Engl, J.D

    T. Engl, J.D. Urbina and K. Richter,Periodic mean-field solutions and the spectra of discrete bosonic fields: Trace formula for Bose-Hubbard models,Phys. Rev. E.92(2015) 062907

    work page 2015

  80. [80]

    Dubertrand and S

    R. Dubertrand and S. M ¨uller,Spectral statistics of chaotic many-body systems,New J. Phys.18(2016) 033009

    work page 2016

Showing first 80 references.