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arxiv: 2604.14015 · v1 · submitted 2026-04-15 · 🪐 quant-ph · nlin.CD

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

Daniel Waltner , Boris Gutkin This is my paper

Pith reviewed 2026-05-10 12:38 UTC · model grok-4.3

classification 🪐 quant-ph nlin.CD
keywords duality relationclassical periodic orbitsquantum many-body systemsspectral statisticskicked spin chaincoupled cat mapssemiclassical methods
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The pith

A duality relation extracts classical periodic orbits from the quantum spectra of many-body systems such as kicked spin chains.

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

The paper applies a recently developed duality relation to extend semiclassical analysis beyond few-particle systems into the many-body regime. This relation connects the quantum spectrum to a sum over classical periodic orbits, allowing the orbits to be recovered from energy-level data without enumerating them directly. For the kicked spin chain the authors demonstrate the extraction explicitly. They then turn to coupled cat maps to study spectral statistics in chaotic many-body systems and to examine the combined limit of large semiclassical parameter and large particle number.

Core claim

The duality relation is used to extract classical periodic orbits from the quantum spectrum of the kicked spin chain. For coupled cat maps the spectral statistics of chaotic many-body systems are analyzed, including the double limit of large semiclassical parameter and large particle number.

What carries the argument

The duality relation that equates the quantum spectrum to a trace formula over classical periodic orbits, bypassing direct summation over the exponentially large set of orbits.

If this is right

  • Classical periodic orbits can be read out from the quantum spectrum of a kicked spin chain.
  • Spectral statistics of chaotic many-body systems can be computed via the duality for coupled cat maps.
  • The simultaneous limit of large semiclassical parameter and large particle number is well-defined for the statistics of these systems.

Where Pith is reading between the lines

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

  • Numerical recovery of orbits from spectra could become feasible for system sizes where explicit orbit listing is impossible.
  • The same duality might be tested on other many-body maps or lattices to see whether orbit contributions remain extractable.
  • If the double limit behaves regularly, it may supply a practical route to semiclassical approximations for larger quantum spin chains.

Load-bearing premise

The duality relation extends directly to many-body systems without being invalidated by the exponential growth of classical orbits or quantum Hilbert-space dimension.

What would settle it

Calculate the quantum spectrum of a small kicked spin chain, apply the duality to recover candidate orbits, and compare them with the independently enumerated classical periodic orbits of the corresponding map; mismatch would falsify the extraction claim.

read the original abstract

Semiclassical methods have been applied very successfully to describe the nontrivial transition from the quantum to the classical regime in $\textit{single}$-particle or at least $\textit{few}$-particle systems. Challenges on the way to an extension to $\textit{many}$-body systems result from the exponential proliferation of the number of classical orbits in chaotic systems and the exponential growth of the quantum Hilbert-space dimension with the particle number. To circumvent these problems, we apply here our recently developed duality relation. Considering the kicked spin chain as example for a many-body system, we show how the duality relation can be used to extract the classical orbits from the quantum spectrum. For coupled cat maps, we analyze the spectral statistics of chaotic many-body systems and discuss the double limit of large semiclassical parameter and large particle number.

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

2 major / 2 minor

Summary. The paper claims that a recently developed duality relation can be used to extract classical periodic orbits from the quantum spectrum of many-body systems, using the kicked spin chain as an example, and to analyze spectral statistics for coupled cat maps in the double limit of large semiclassical parameter and large particle number, thereby addressing the challenges of exponential orbit proliferation and Hilbert space growth.

Significance. If the duality applies without major limitations, this work offers a promising framework for semiclassical analysis in many-body quantum systems. It credits the use of the duality to sidestep direct enumeration of orbits, which could lead to new insights into quantum chaos and spectral properties in the many-body regime. The discussion of the double limit is particularly noteworthy as it tackles the joint semiclassical and thermodynamic limits.

major comments (2)
  1. [Kicked spin chain example] The extraction of classical orbits from the quantum spectrum via the duality needs to be supported by explicit calculations or numerical comparisons showing agreement with known orbits, especially given the exponential proliferation mentioned; without this, the claim that it circumvents the problem remains unverified.
  2. [Coupled cat maps] The analysis of spectral statistics in the double limit should include specific results, such as level spacing distributions or other statistics, to demonstrate the behavior in the large semiclassical parameter and large particle number limit.
minor comments (2)
  1. The abstract is well-written but could include a brief mention of the duality relation's form for better context.
  2. Ensure all references to prior work on the duality are clearly cited to allow readers to follow the development.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which will help improve the presentation of our results. We address each major comment below.

read point-by-point responses

  1. Referee: The extraction of classical orbits from the quantum spectrum via the duality needs to be supported by explicit calculations or numerical comparisons showing agreement with known orbits, especially given the exponential proliferation mentioned; without this, the claim that it circumvents the problem remains unverified.

    Authors: We thank the referee for this observation. The manuscript derives the explicit duality formula that extracts the classical periodic orbits directly from the quantum spectrum of the kicked spin chain, thereby avoiding explicit enumeration. However, we agree that a concrete numerical verification would make the claim more compelling. In the revised version we will add a specific example with numerical data comparing the orbits obtained via the duality to those known from classical analysis. revision: yes

  2. Referee: The analysis of spectral statistics in the double limit should include specific results, such as level spacing distributions or other statistics, to demonstrate the behavior in the large semiclassical parameter and large particle number limit.

    Authors: We agree that explicit statistical measures would strengthen the discussion of the double limit. The manuscript provides an analytical treatment of the spectral statistics for coupled cat maps in this regime. To address the comment we will include concrete results, such as level-spacing distributions computed for large values of the semiclassical parameter and particle number, in the revised manuscript. revision: yes

Circularity Check

1 steps flagged

Moderate self-citation load-bearing on prior duality relation

specific steps
  1. self citation load bearing [Abstract]
    "To circumvent these problems, we apply here our recently developed duality relation. Considering the kicked spin chain as example for a many-body system, we show how the duality relation can be used to extract the classical orbits from the quantum spectrum. For coupled cat maps, we analyze the spectral statistics of chaotic many-body systems and discuss the double limit of large semiclassical parameter and large particle number."

    The demonstration that the duality extracts classical orbits from the quantum spectrum and enables spectral statistics analysis in the many-body limit is achieved by direct application of the authors' own prior duality relation. This makes the claimed extraction and analysis dependent on the self-cited result rather than an independent derivation within the present work.

full rationale

The paper's core strategy relies on applying a duality relation developed in the authors' recent prior work to map quantum spectra onto classical orbits in many-body systems. This is explicitly stated in the abstract as the method to circumvent exponential proliferation issues. While the application to new examples (kicked spin chain, coupled cat maps) adds content, the central claim of extracting orbits and analyzing statistics in the double limit is justified primarily by invoking this self-developed relation without re-derivation or external validation shown here. No other circular patterns (self-definition, fitted predictions, or ansatz smuggling) are present in the provided text. This yields a moderate circularity score rather than zero, as the load-bearing step reduces to self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests primarily on the applicability of the authors' prior duality relation to the chosen many-body models; no new free parameters, invented entities, or additional axioms are indicated in the abstract.

axioms (1)
  • domain assumption The duality relation holds for the kicked spin chain and coupled cat maps without significant corrections from many-body effects.
    Invoked when applying the relation to extract orbits and analyze statistics, as the paper does not re-derive the duality here.

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Forward citations

Cited by 2 Pith papers

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

  1. Semiclassical periodic-orbit theory for quantum spectra

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    Didactic derivation of Gutzwiller's trace formula from the path integral, with overview of its use in explaining random matrix theory statistics for quantum energy levels.

  2. 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

100 extracted references · 100 canonical work pages · cited by 2 Pith papers

  1. [1]

    St ¨ockmann,Quantum chaos - an introduction, Cambridge University Press (1999)

    H.J. St ¨ockmann,Quantum chaos - an introduction, Cambridge University Press (1999)

    work page 1999

  2. [2]

    Haake, S

    F . Haake, S. Gnutzmann and M. Ku´s,Quantum Signatures of Chaos, Springer (2019)

    work page 2019

  3. [3]

    Gutzwiller,Chaos in Classical and Quantum Mechanics, Springer, New Y ork (1990)

    M. Gutzwiller,Chaos in Classical and Quantum Mechanics, Springer, New Y ork (1990)

    work page 1990

  4. [4]

    Einstein,Zum Quantensatz von Sommerfeld und Epstein,Verhandlungen der Deutschen Physikalischen Gesellschaft19(1917) 82

    A. Einstein,Zum Quantensatz von Sommerfeld und Epstein,Verhandlungen der Deutschen Physikalischen Gesellschaft19(1917) 82

    work page 1917

  5. [5]

    Wintgen,Connection between long-range correlations in quantum spectra and classical periodic orbits,Phys

    D. Wintgen,Connection between long-range correlations in quantum spectra and classical periodic orbits,Phys. Rev. Lett.55(1987) 1589

    work page 1987

  6. [6]

    Wintgen, K

    D. Wintgen, K. Richter and G. Tanner,The semiclassical helium atom,Chaos2(1992) 19

    work page 1992

  7. [7]

    St ¨ockmann and J

    H.J. St ¨ockmann and J. Stein,”Quantum” chaos in billiards studied by microwave absorption,Phys. Rev. Lett.64(1990) 2215

    work page 1990

  8. [8]

    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

  9. [9]

    Berry,Semiclassical theory of spectral rigidity,Proc

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

    work page 1985

  10. [10]

    Sieber and K

    M. Sieber and K. Richter,Correlations between periodic orbits and their role in spectral statistics,Phys. ScriptaT90(2001) 128

    work page 2001

  11. [11]

    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

  12. [12]

    T. Guhr, A. M ¨uller-Groeling and H. Weidenm¨uller,Random matrix theories in quantum physics: Common concepts,Phys. Rep.299 (1998) 190

    work page 1998

  13. [13]

    Baldwin and G

    G. Baldwin and G. Klaiber,Photo-fission in heavy elements,Phys. Rev.71(1947) 3

    work page 1947

  14. [14]

    Iudice and F

    N.L. Iudice and F . Palumbo,New isovector collective modes in deformed nuclei,Phys. Rev. Lett.41(1978) 1532

    work page 1978

  15. [15]

    Bohle, A

    D. Bohle, A. Richter, W. Steffen, A.E.L. Dieperink, N.L. Iudice, F . Palumbo et al.,New magnetic dipole excitation mode studied in the heavy deformed nucleus 156Gd by inelastic electron scattering,Phys. Lett. B137(1984) 27

    work page 1984

  16. [16]

    Simula,Collective dynamics of vortices in trapped Bose-Einstein condensates,Phys

    T. Simula,Collective dynamics of vortices in trapped Bose-Einstein condensates,Phys. Rev. A87(2013) 023630

    work page 2013

  17. [17]

    Butts and D.S

    D.A. Butts and D.S. Rokhsar,Predicted signatures of rotating Bose-Einstein condensates,Nature397(1999) 327

    work page 1999

  18. [18]

    Knobel, W

    M. Knobel, W. Nunes, L. Socolovsky, E.D. Biasi, J. Vargas and J. Denardin,Superparamagnetism and other magnetic features in granular materials,Journal of Nanoscience and Nanotechnology8(2008) 2836

    work page 2008

  19. [19]

    Georges, G

    A. Georges, G. Kotliar, W. Krauth and M.J. Rozenberg,Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions,Rev. Mod. Phys.68(1996) 13. The role of classical periodic orbits in quantum many-body systems27

    work page 1996

  20. [20]

    Verstraete, V

    F . Verstraete, V. Murg and J. Cirac,Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems,Advances in Physics57(2008) 143

    work page 2008

  21. [21]

    Perez-Garcia, F

    D. Perez-Garcia, F . Verstraete, M.M. Wolf and J.I. Cirac,Matrix product state representations,Quantum Info. Comput.7(2007) 401

    work page 2007

  22. [22]

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

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

    work page 1928

  23. [23]

    Berry and M

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

    work page 1977

  24. [24]

    Ribeiro, M.A.M

    A. Ribeiro, M.A.M. de Aguiar and A.F .R. de Toledo Piza,Trace formula for systems with spin from the coherent state propagator,J. Math. Phys.48(2007) 112103

    work page 2007

  25. [25]

    Waltner, P

    D. Waltner, P . Braun, M. Akila and T. Guhr,Trace formula for interacting spins,J. Phys. A50(2017) 085304

    work page 2017

  26. [26]

    Keating,The Riemann zeta function and quantum chaology, inQuantum Chaos, Eds.: G

    J.P . Keating,The Riemann zeta function and quantum chaology, inQuantum Chaos, Eds.: G. Casati, I. Guarneri and U. Smilansky, North Holland, Amsterdam, 1993

    work page 1993

  27. [27]

    Keppeler,Spinning Particles-Semiclassics and Spectral Statistics, Springer Tracts in Modern Physics, vol

    S. Keppeler,Spinning Particles-Semiclassics and Spectral Statistics, Springer Tracts in Modern Physics, vol. 193, Springer Berlin, Heidelberg (2003)

    work page 2003

  28. [28]

    Akila, B

    M. Akila, B. Gutkin, P . Braun, D. Waltner and T. Guhr,Semiclassical prediction of large spectral fluctuations in interacting kicked spin chains,Annals of Physics389(2018) 250

    work page 2018

  29. [29]

    Akila, D

    M. Akila, D. Waltner, B. Gutkin and T. Guhr,Particle-time duality in the kicked ising chain,J. Phys. A49(2016) 375101

    work page 2016

  30. [30]

    Akila, D

    M. Akila, D. Waltner, B. Gutkin, P . Braun and T. Guhr,Semiclassical identification of periodic orbits in a quantum many-body system, Phys. Rev. Lett.118(2017) 164101

    work page 2017

  31. [31]

    E. Lieb, T. Schultz and D. Mattis,Two soluble models of an antiferromagnetic chain,Ann. Phys.16(1961) 407

    work page 1961

  32. [32]

    Bertini, P

    B. Bertini, P . Kos and T. Prosen,Exact spectral form factor in a minimal model of many-body quantum chaos,Phys. Rev. Lett.121(2018) 264101

    work page 2018

  33. [33]

    Braun, D

    P . Braun, D. Waltner, M. Akila, B. Gutkin and T. Guhr,Transition from quantum chaos to localization in spin chains,Phys. Rev. E101 (2020) 052201

    work page 2020

  34. [34]

    Waltner and P

    D. Waltner and P . Braun,Localization in the kicked ising chain,Phys. Rev. B104(2021) 054432

    work page 2021

  35. [35]

    Bertini, P

    B. Bertini, P . Kos and T. Prosen,Entanglement spreading in a minimal model of maximal many-body quantum chaos,Phys. Rev. X9 (2019) 021033

    work page 2019

  36. [36]

    Rather, S

    S.A. Rather, S. Aravinda and A. Lakshminarayan,Creating ensembles of dual unitary and maximally entangling quantum evolutions, Phys. Rev. Lett.125(2020) 070501

    work page 2020

  37. [37]

    Giudice, G

    G. Giudice, G. Giudici, M. Sonner, J. Thoenniss, A. Lerose, D.A. Abanin et al.,Temporal entanglement, quasiparticles, and the role of interactions,Phys. Rev. Lett.128(2022) 220401

    work page 2022

  38. [38]

    Lakshminarayan and K

    A. Lakshminarayan and K. ˙Zyczkowski,Quantum chaos and quantum information: Interactions and implications, inthis volume, 2026

    work page 2026

  39. [39]

    Gutkin, P

    B. Gutkin, P . Braun, M. Akila, D. Waltner and T. Guhr,Exact local correlations in kicked chains,Phys. Rev. B102(2020) 174307

    work page 2020

  40. [40]

    Bertini, P

    B. Bertini, P . Kos and T. Prosen,Exact correlation functions for dual-unitary lattice models in 1+1 dimensions,Phys. Rev. Lett.123(2019) 210601

    work page 2019

  41. [41]

    Claeys and A

    P .W. Claeys and A. Lamacraft,Maximum velocity quantum circuits,Phys. Rev. Res.2(2020) 033032

    work page 2020

  42. [42]

    Hamazaki,Exceptional dynamical quantum phase transitions in periodically driven systems,Nature Communications12(2021) 5108

    R. Hamazaki,Exceptional dynamical quantum phase transitions in periodically driven systems,Nature Communications12(2021) 5108

    work page 2021

  43. [43]

    Fritzsch and T

    F . Fritzsch and T. Prosen,Eigenstate thermalization in dual-unitary quantum circuits: Asymptotics of spectral functions,Phys. Rev. E103 (2021) 062133

    work page 2021

  44. [44]

    Schomerus and M

    H. Schomerus and M. Sieber,Bifurcations of periodic orbits and uniform approximations,J. Phys. A30(1997) 4537

    work page 1997

  45. [45]

    de Almeida and J.H

    A.M.O. de Almeida and J.H. Hannay,Resonant periodic orbits and the semiclassical energy spectrum,J. Phys. A20(1987) 5873

    work page 1987

  46. [46]

    Manderfeld and H

    C. Manderfeld and H. Schomerus,Semiclassical singularities from bifurcating orbits,Phys. Rev. E63(2001) 066208

    work page 2001

  47. [47]

    de Aguiar, C

    M. de Aguiar, C. Malta, M. Baranger and K. Davies,Bifurcations of periodic trajectories in non-integrable hamiltonian systems with two degrees of freedom: Numerical and analytical results,Ann. Phys.180(1987) 167

    work page 1987

  48. [48]

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

    work page 1980

  49. [49]

    Argaman, F .M

    N. Argaman, F .M. Dittes, E. Doron, J. Keating, A. 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

  50. [50]

    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

  51. [51]

    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. A35(2002) L613

    work page 2002

  52. [52]

    Kuipers and M

    J. Kuipers and M. Sieber,Semiclassical expansion of parametric correlation functions of the quantum time delay,Nonlinearity20(2007) 909

    work page 2007

  53. [53]

    Nagao, P

    T. Nagao, P . Braun, S. M¨uller, K. Saito, S. Heusler and F . Haake,Semiclassical theory for parametric correlation of energy levels,J. Phys. A40(2007) 47

    work page 2007

  54. [54]

    Gutkin, D

    B. Gutkin, D. Waltner, M. Guti ´errez, J. Kuipers and K. Richter,Quantum corrections to fidelity decay in chaotic systems,Phys. Rev. E81 (2010) 036222

    work page 2010

  55. [55]

    Goussev, R.A

    A. Goussev, R.A. Jalabert, H.M. Pastawski and D.A. Wisniacki,Loschmidt echo,Scholarpedia7(2012) 11687

    work page 2012

  56. [56]

    Garcia-Mata and D.A

    I. Garcia-Mata and D.A. Wisniacki,Quantum analogues of exponential sensitivity: from Loschmidt echo to Krylov complexity, inthis volume, 2026

    work page 2026

  57. [57]

    Richter and M

    K. Richter and M. Sieber,Semiclassical theory of chaotic quantum transport,Phys. Rev. Lett.89(2002) 206801

    work page 2002

  58. [58]

    Heusler, S

    S. Heusler, S. M ¨uller, P . Braun and F . Haake,Semiclassical theory of chaotic quantum transport,Phys. Rev. Lett.96(2006) 066804

    work page 2006

  59. [59]

    M ¨uller, S

    S. M ¨uller, S. Heusler, P . Braun and F . Haake,Periodic-orbit theory of level correlations,New J. Phys.9(2007) 12

    work page 2007

  60. [60]

    Brouwer,Semiclassical theory of the ehrenfest-time dependence of quantum transport,Phys

    P .W. Brouwer,Semiclassical theory of the ehrenfest-time dependence of quantum transport,Phys. Rev. B76(2007) 165313

    work page 2007

  61. [61]

    Brouwer and S

    P .W. Brouwer and S. Rahav,Semiclassical theory of the ehrenfest time dependence of quantum transport in ballistic quantum dots, Phys. Rev. B74(2006) 075322

    work page 2006

  62. [62]

    Waltner,Semiclassical Approach to Mesoscopic Systems: Classical Trajectory Correlations and Wave Interference, Springer Tracts in Modern Physics, vol

    D. Waltner,Semiclassical Approach to Mesoscopic Systems: Classical Trajectory Correlations and Wave Interference, Springer Tracts in Modern Physics, vol. 245, Springer Berlin, Heidelberg (2012)

    work page 2012

  63. [63]

    Novaes,Semiclassical theory of transport, inthis volume, 2026

    M. Novaes,Semiclassical theory of transport, inthis volume, 2026

    work page 2026

  64. [64]

    Primack and U

    H. Primack and U. Smilansky,On the accuracy of the semiclassical trace formula,J. Phys. A31(1998) 6253

    work page 1998

  65. [65]

    Gutkin and V.A

    B. Gutkin and V.A. Osipov,Classical foundations of many-particle quantum chaos,Nonlinearity29(2016) 325

    work page 2016

  66. [66]

    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

  67. [67]

    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

  68. [68]

    Gutkin and V.A

    B. Gutkin and V.A. Osipov,Clustering of periodic orbits in chaotic systems,Nonlinearity26(2013) 177. 28The role of classical periodic orbits in quantum many-body systems

    work page 2013

  69. [69]

    Gutkin and V.A

    B. Gutkin and V.A. Osipov,Clustering of periodic orbits and ensembles of truncated unitary matrices,J. Stat. Phys.153(2013) 1049

    work page 2013

  70. [70]

    Chirikov, F .M

    B.V. Chirikov, F .M. Izrailev and D.L. Shepelyansky,Dynamical stochasticity in classical and quantum mechanics,Sov. Sci. Rev. Sect. C2 (1981) 209

    work page 1981

  71. [71]

    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

  72. [72]

    M ¨uller, S

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

    work page 2009

  73. [73]

    Waltner and K

    D. Waltner and K. Richter,Towards a semiclassical understanding of chaotic single- and many-particle quantum dynamics at post-heisenberg time scales,Phys. Rev. E100(2019) 042212

    work page 2019

  74. [74]

    Pethel, N.J

    S.D. Pethel, N.J. Corron and E. Bollt,Deconstructing spatiotemporal chaos using local symbolic dynamics,Phys. Rev. Lett.99(2007) 214101

    work page 2007

  75. [75]

    Pethel, N.J

    S.D. Pethel, N.J. Corron and E. Bollt,Symbolic dynamics of coupled map lattices,Phys. Rev. Lett.96(2006) 034105

    work page 2006

  76. [76]

    K. Kaneko,Period-doubling of kink-antikink patterns, quasi-periodicity in antiferro-like structures and spatial intermittency in coupled map lattices — toward a prelude to a ”field theory of chaos”,Prog. Theor. Phys.72(1984) 480

    work page 1984

  77. [77]

    Y .G.S. L. A. Bunimovich,Spacetime chaos in coupled map lattices,Nonlinearity1(1988) 491

    work page 1988

  78. [78]

    Hannay and M.V

    J. Hannay and M.V. Berry,Quantization of linear maps on a torus – fresnel diffraction by a periodic grating,Physica D1(1980) 267

    work page 1980

  79. [79]

    Keating,The cat maps: quantum mechanics and classical motion,Nonlinearity4(1991) 309

    J.P . Keating,The cat maps: quantum mechanics and classical motion,Nonlinearity4(1991) 309

    work page 1991

  80. [80]

    Keating,Asymptotic properties of the periodic orbits of the cat maps,Nonlinearity4(1991) 277

    J.P . Keating,Asymptotic properties of the periodic orbits of the cat maps,Nonlinearity4(1991) 277

    work page 1991

Showing first 80 references.