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Semiclassical approximation for barrier billiards

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Pith's one-line read Barrier billiards are argued to have semi-Poisson spectral statistics in the semiclassical limit, via the universal asymptotic form of their exact transfer operator, and the transfer-operator trace formula reproduces the geometric…

desk verdict A real transfer-operator derivation of the barrier billiard trace formula, plus an honest but unproven heuristic for semi-Poisson statistics; the trace part is solid, the statistics part is a conjecture. read the letter →

arxiv 2504.18834 v1 pith:7FTXIPGB submitted 2025-04-26 quant-ph

Eugene Bogomolny This is my paper · ORCID
classification quant-ph MSC 81Q5081Q2015B52 PACS 05.45.Mt03.65.Sq
keywords barrierbilliardspseudo-integrablesystemssemi-PoissonstatisticstransferoperatortraceformulaWiener-Hopffactorizationrandomunitarymatricesquantumchaos
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tackles barrier billiards, rectangular enclosures with an internal barrier, which are pseudo-integrable systems whose level statistics sit between the Poisson and random-matrix extremes. The author computes the exact quantum transfer operator in the high-energy semiclassical limit and shows that its off-diagonal structure converges to a universal form whose matrix elements fall off linearly from the diagonal. That asymptotic form is shared by a random unitary ensemble called the A-matrix, whose spectral correlations are provably semi-Poisson, so the paper argues that barrier billiards inherit the same statistics: level repulsion at small spacings and exponential falloff at large ones. A second thread derives the semiclassical trace formula for barrier billiards through the transfer operator, obtaining the periodic-orbit prefactor (-1)^K(1-2η) that matches direct geometric counting. If correct, the result turns barrier billiards into one of the few pseudo-integrable models where intermediate statistics are obtained analytically rather than numerically.

What carries the argument

The argument runs through three objects. (1) The Wiener-Hopf factor K_+(α), whose large-k asymptotic $e^{{iπ/4}}$/√(b(k+α)) turns the exact S-matrix into a paraxial form in which only transmitted waves between the alternating Dirichlet and Neumann halves survive and matrix elements decay as 1/(π(j-k+1/2)). (2) The A-matrix Σ_nm = $N^{{-1}}$ cos(π(n-m)/N) with random phases, a reduction of a Ruijsenaars-Schneider Lax matrix whose uniform eigenvalue distribution yields exact finite-N semi-Poisson correlation functions. (3) For the trace formula, the Q-matrix Q_mn = $e^{{-2π i z m}}$ f_{m-n}(y), where f comes from summing over odd indices with Bernoulli polynomials; its eigenvalues are found using discrete prolate spheroidal sequences, whose eigenvalue count asymptotically equals yR ones and (1-y)R zeros, and then a phase-shift argument shows Q^M has only 2M distinct eigenvalue powers, producing the final prefactor.

What would settle it

Compute the exact B-matrix (8) for large N with the true barrier phases φ_m, unfold the eigenphases, and compare the nearest-neighbour distribution and number variance with the semi-Poisson formulas (11); any systematic deviation that does not shrink as N grows would disprove the claim. The same computation with the A-matrix (14) at identical N isolates whether the asymptotic-form equivalence holds.

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Extended reading notes

Core claim

In the semiclassical limit k→∞, the paper shows that the exact finite transfer operator B(k) of a symmetric barrier billiard, restricted to propagating modes, approaches a universal block matrix whose only non-negligible entries are off-diagonal: s_jk = (-1)^{j+k}/(π(j-k+1/2)). The same limit is obtained for the A-matrix, a random unitary matrix built from a Lax matrix of an integrable Ruijsenaars-Schneider model and known to have semi-Poisson eigenvalue statistics. Invoking the heuristic that local spectral statistics are governed by the asymptotic falloff of matrix elements, the author concludes that barrier billiard spectra are semi-Poisson: nearest-neighbour spacing P0(s)=4s $e^{{-2s}}$, two-point correlation R2(s)=1-$e^{{-4s}}$, and compressibility χ=1/2. The paper also exhibits a third ensemble (the C-matrix) with the same limiting form but different single-matrix statistics, showing that the falloff condition alone is not sufficient; the argument for barrier billiards rests on additional structure that makes the heuristic work. In the second part, the trace of even powers of B is evaluated by a saddle-point calculation, and the resulting trace formula has exactly the geometric periodic-orbit contribution, with prefactor ε_p A_p = (-1)^K(1-2η)4ab where K=[Nh1/a] and η={Nh1/a}.

Load-bearing premise

The paper's statistical conclusion stands on an unproven premise: that two large random unitary matrices whose entries fall off from the diagonal in the same way have the same level statistics; the trace-formula prefactor additionally depends on ignoring a small phase shift in the Q-matrix eigenfunction argument.

Editorial extensions

If this is right

  • The spectral statistics of symmetric barrier billiards in the high-energy limit are described by the semi-Poisson distributions (11), independent of the barrier's position and length.
  • The barrier billiard B-matrix and the A-matrix share a common large-N limiting form, so any local spectral observable that is continuous in this limit takes the A-matrix value.
  • The trace formula obtained from the transfer operator reproduces the standard geometric periodic-orbit contribution with prefactor (-1)^K(1-2η)4ab, showing that the two independent approaches agree.
  • The new mechanism, in which saddle-point corrections reorganize into the area prefactor, extends analytical trace-formula computations to pseudo-integrable systems where geometric orbit counting is hard, including, potentially, triangular billiards.
  • The derivation covers symmetric barrier billiards; general asymmetric barrier billiards are left for future work because the formulas are more cumbersome.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One testable extension suggested by the paper's contrast with the C-matrix: if asymptotic falloff alone determined statistics, the C-matrix would also be semi-Poisson, but it is not; a systematic finite-N comparison of B-, A-, and C-matrix spectral form factors would clarify which extra structural conditions, such as being a reduction of a Lax matrix or having a block product structure, actually e
  • The trace-formula calculation, being independent of geometric orbit counting, may carry over to other pseudo-integrable billiards such as triangular billiards, where periodic-orbit families and prefactors are harder to compute geometrically; the paper hints at this in its conclusion but does not perform it.
  • The small-shift approximation in the Q-matrix eigenvalue argument is the step most likely to affect the prefactor; a rigorous bound on the error term δ in (98) would turn the trace-formula result from a calculation into a proof.
  • The paraxial S-matrix (26) is formally unitary only after summing over all integers, so edge effects at finite index could produce weak deviations from semi-Poisson at finite k, which should be visible in numerical spectra.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

Summary. The paper studies the high-energy (semiclassical) limit of the exact transfer operator for symmetric barrier billiards constructed in previous work by the author. In Section II it derives the asymptotic form of the S-matrix and the B-matrix in the paraxial approximation, arriving at a common block form (16) shared with the exactly solvable random matrix ensemble A of Eq. (14). The paper then uses this asymptotic equivalence, together with the known semi-Poisson statistics of model A, to argue heuristically that the barrier billiard B-matrix has semi-Poisson spectral statistics. In Section III the paper derives a semiclassical trace formula from traces of powers of the transfer operator, reducing the computation to the eigenvalues of a prolate-spheroidal-type matrix Q, and obtains the prefactor F_s.p. = (-1)^K(1-2η) of Eq. (114), which agrees with the independent geometric periodic-orbit calculation summarized in Eq. (56). Appendices A, B, and C contain the asymptotic evaluation of K+(α), an independent derivation of the S-matrix asymptotics from Sommerfeld diffraction theory, and the geometric computation of periodic-orbit channel widths.

Significance. If the results are correct, this is a valuable analytical contribution to pseudo-integrable quantum chaos. The trace formula derivation is the most substantial technical achievement: it shows how the transfer-operator approach, which does not directly encode classical periodic orbits, nevertheless reproduces the known geometric prefactor (56) through a nontrivial eigenvalue problem for a prolate-spheroidal-type matrix. The derivation is detailed and self-contained, and the final agreement with the independent geometric formula is a strong check. The statistical conclusion about semi-Poisson statistics is explicitly heuristic, resting on the asymptotic-equivalence conjecture stated in Section II B and on the random-phase hypothesis from [14]; the paper is honest about this, and the numerical evidence in [14,15] and Section II C is consistent with the claim. The appendices are carefully worked out and reproducible. In my view the stress-test concern about the unproved equivalence conjecture does not land as a fatal objection, because the manuscript consistently frames the statistical result as a conjecture supported by analytical arguments rather than as a proven theorem.

minor comments (6)
  1. [Section II B (paragraph after Eq. (41))] There is a duplicated article in the phrase "matrices with the the same asymptotic linear falloff of matrix elements"; this should be corrected to "the same asymptotic linear falloff".
  2. [Section II C (last sentence)] The sentence "Careful discussion of spectral properties of products of certain matrices with intermediate statistics will be given somewhere" is too vague for a published paper and should be removed or replaced with a specific pointer to a planned or existing publication.
  3. [Section II B (end of subsection)] The paper notes that the reasoning for model A applies strictly to odd N, while the physical B-matrix has dimension N=kb/π of both parities as k varies. Since the even-N case is only indirectly addressed through the block-matrix construction of Section II C, an explicit sentence explaining that the even-N case is covered by that construction (or by numerical evidence) would improve the presentation.
  4. [Section III B, around Eq. (98)] The neglect of the small shift δ = {jnR/M} is a heuristic step in the eigenvalue calculation. The paper does label it as heuristic, but because this step is load-bearing for the prefactor (114), it would be helpful to add one sentence emphasizing that the final agreement with (56) is the practical justification for this approximation.
  5. [Appendix A, after Eq. (A9)] The statement "The error is assumed to be O(1/k)" could be worded more precisely as "estimated" or "argued to be O(1/k)", especially since the numerical check in Fig. 4(a) supports the asymptotic formula.
  6. [Section IV (Conclusion)] The statistical conclusion is conditional on two separate ingredients: the asymptotic-equivalence conjecture of Section II B and the random-phase hypothesis from [14]. A short sentence in the Conclusion restating both assumptions as open problems would make the logical structure of the paper fully transparent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the semi-Poisson conclusion rests on a disclosed heuristic conjecture, and the trace formula is independently derived and cross-checked.

full rationale

The derivation chain is self-contained in the relevant sense. Section II derives the semiclassical limit of K+(alpha) from the exact infinite product (7) via Euler-Maclaurin summation in Appendix A, and independently confirms the paraxial S-matrix by a Fraunhofer re-expansion, with Eqs. (20) and (23) agreeing. The semi-Poisson claim is not forced by construction: the A-matrix (14) is a distinct integrable Lax matrix from prior published work with exact joint eigenvalue density (32), and the bridge to the barrier-billiard B-matrix is an explicitly labeled heuristic conjecture, namely that matrices with the same asymptotic linear falloff of matrix elements have the same spectral properties in the large-N limit. The paper itself discloses the limitation, noting that many unitary matrices may share the same asymptotic behavior as in (16). The trace formula in Section III is derived independently from the transfer operator, and the final result (114) is cross-checked against the geometric periodic-orbit calculation (C13) and (56). The author's statement that the answer was known in advance is a verification, not a circular input. Self-citations to [14]-[16] and [19,20] are prior published, externally checkable results; the random-phase hypothesis is an assumption rather than a restatement of the conclusion. No step reduces by definition or by fitted parameter to its own input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central results rest on the exact transfer operator from [14] and on two explicitly heuristic steps: the asymptotic equivalence conjecture connecting local spectral statistics to the universal kernel (16), and the neglect of a sub-R shift in the Q-matrix eigenfunctions. No free parameters are fitted, and no new physical entities are introduced.

assumptions (5)
  • domain assumption The exact transfer operator quantization condition det(1+B(k))=0 from [14] correctly encodes the spectrum of the symmetric barrier billiard.
    All semiclassical limits in the paper start from the B-matrix (8) and S-matrix (3), which are taken as given from the author's earlier paper [14].
  • ad hoc to paper Asymptotic equivalence conjecture: finite unitary matrices with the same N→∞ block form (16) have identical local spectral statistics.
    This is the bridge from the limit S-matrix to the proven semi-Poisson statistics of model A. It is explicitly labeled heuristic in Section II B and used again to justify model C and the product matrix ξ in Section II C.
  • ad hoc to paper In the Q-matrix eigenvalue analysis, the small shift δ = {jnR/M} in (98) can be neglected when R→∞.
    The paper writes that δ is O(1) compared to β=O(R) and 'heuristically can be ignored'; this is essential for constructing the closed system (104) that gives the Q-matrix eigenvalues ±1.
  • domain assumption The Ruijsenaars-Schneider Lax matrix results of [19,20], including the uniform eigenvalue distribution (32), are correct.
    These results establish that model A has exactly the semi-Poisson statistics stated in (37)-(38), which is the known statistical model used for comparison.
  • standard math Standard asymptotic tools (Euler-Maclaurin summation, Poisson summation, saddle-point method) apply with the stated error estimates.
    Used in Appendix A for the asymptotic of K+(α) and in Section III for the trace formula; these are standard but are assumptions about the validity of the approximations.

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Pith. "Pith review of Semiclassical approximation for barrier billiards." pith.science (2026). https://pith.science/paper/7FTXIPGB

@misc{pith2026250418834,
  author       = {Pith},
  title        = {Pith review of: Semiclassical approximation for barrier billiards},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7FTXIPGB}},
  note         = {Machine review of arXiv:2504.18834}
}
read the original abstract

Barrier billiards are simple examples of pseudo-integrable models which form an appealing but poorly investigated subclass of dynamical systems. The paper examines the semiclassical limit of the exact quantum transfer operator for barrier billiards constructed in [J. Phys. A: Math. Theor. \textbf{55}, 024001 (2022)]. The obtained asymptotic expressions are used to provide analytical arguments to support the conjecture that spectral statistical properties of barrier billiards are described by the semi-Poisson distribution and to derive the trace formulas for such billiards which in the transfer operator approach is not automatic.

Figures

Figures reproduced from arXiv: 2504.18834 by the authors.

Figure 1
Figure 1. FIG. 1. (a) General barrier billiard. On all thick lines the Dirichlet boundary conditions are [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Nearest-neighbour distribution for integrable random matrix ensembles (29) with uniform [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The nearest-neighbour distributions [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) The ratio of numerical value of [PITH_FULL_IMAGE:figures/full_fig_p027_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Angles determining the Sommerfeld diffraction coefficient. [PITH_FULL_IMAGE:figures/full_fig_p028_5.png]

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Works this paper leans on

37 extracted references · 36 canonical work pages

  1. [14]

    Balasubramanian, R

    V. Balasubramanian, R. Nath Das, J. Erdmenger, and Zhuo-Yu Xian, Chaos and integrability in triangular billiards , arXiv: 2407.11114 (2024)

    arXiv 2024

  2. [1]

    As the transfer operator is exact it can be used also for numerical computations of the billiard barrier spectrum

    Calculation of the behaviour of K+(α) at small α/k is beyond the scope of the paper. As the transfer operator is exact it can be used also for numerical computations of the billiard barrier spectrum. In such a case it is of interest to find a convenient method of calculation of K+(α). By construction, the infinite product in (7) converges and may be calcu...

    work page 2000

  3. [2]

    R2(s) =P∞ n=0Pn(s) is the two-point corre- lation function which determines the probability that two levels are at distance s (with any numbers of levels inside)

    (11) Here Pn(s) is the probability that two levels are separated by distance s and inside this interval there exist exactly n additional levels. R2(s) =P∞ n=0Pn(s) is the two-point corre- lation function which determines the probability that two levels are at distance s (with any numbers of levels inside). K(τ) is the two-point form factor defined as the ...

    work page

  4. [3]

    (33) For this value the M-matrix takes the following form Mnm = lnlm cos((θn−θm)/2), |lm|2 = (−1)(N−1)/2 Y j̸=m cot((θm−θj)/2). (34) It is easy to check that such matrix can be obtained from the above S-matrix (3) after the substitution instead of real xm complex ym = eiθm with real θm (an analog of the Wick rotation). This matrix is unitary provided θj o...

    work page

  5. [4]

    M. V. Berry and M. Tabor, Level clustering in the regular spectrum , Proc. R. Soc. Lond. A 356, 375 (1977)

    work page 1977

  6. [5]

    Bohigas, M

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

    work page 1984

  7. [6]

    M. L. Mehta, Random matrices, Third edition, Academic Press (2014)

    work page 2014

  8. [7]

    Richens and M.V

    P.J. Richens and M.V. Berry, Pseudointegrable systems in classical and quantum mechanics , Physica D: Nonlinear Phenomena 2, 495 (1981)

    work page 1981

Show all 37 references
  1. [8]

    ˙Zyczkowski, Classical and quantum billiards: integrable, nonintegrable and pseudo- integrable, Acta Physica Polonica B 23, 245 (1992)

    K. ˙Zyczkowski, Classical and quantum billiards: integrable, nonintegrable and pseudo- integrable, Acta Physica Polonica B 23, 245 (1992)

    1992

  2. [9]

    Shudo, Y

    A. Shudo, Y. Shimizu, P. ˇSeba, J. Stein, H.-J. St¨ ockmann,Statistical properties of spectra of pseudointegrable systems, Phys. Rev. A, 49, 3748 (1994)

    1994

  3. [10]

    Bogomolny, U

    E. Bogomolny, U. Gerland, and C. Schmit, Short-range plasma model for intermediate spectral statistics, Eur. Phys. J. 19, 121 (2001)

    2001

  4. [11]

    Wiersig, Spectral properties of quantized barrier billiards , Phys

    J. Wiersig, Spectral properties of quantized barrier billiards , Phys. Rev. E 65, 046217 (2002)

    2002

  5. [12]

    Bogomolny, O

    E. Bogomolny, O. Giraud, and C. Schmit,Periodic orbits contribution to the 2-point correlation form factor for pseudo-integrable systems , Comm. Math. Phys. 222, 327 (2001)

    2001

  6. [13]

    Bogomolny, Intermediate spectral statistics of rational triangular quantum billiards, Phys

    ˇCrt Lozej and E. Bogomolny, Intermediate spectral statistics of rational triangular quantum billiards, Phys. Rev. E 110, 024213 (2024)

    2024

  7. [15]

    B. L. Altshuler, I. Kh. Zharekeshev, S. Kotochigova, and B. Shklovskii, Repulsion between energy levels and the metal-insulator transition , J. Exp. Theor. Phys 67, 625 (1988)

    1988

  8. [16]

    B. I. Shklovskii, B. Shapiro, B. R. Sears, P. Lambrianides, and H. B. Shore,Statistics of spectra of disordered systems near the metal-insulator transition , Phys. Rev. B 47, 11487 (1993)

    1993

  9. [17]

    Bogomolny, Barrier billiard and random matrices , J

    E. Bogomolny, Barrier billiard and random matrices , J. Phys. A: Math. Theor. 55, 024001 (2022 )

    2022

  10. [18]

    Bogomolny, Random matrices associated with general barrier billiards , J

    E. Bogomolny, Random matrices associated with general barrier billiards , J. Phys. A: Math. Theor. 55, 254002 (2022). 33

    2022

  11. [19]

    Bogomolny, Level compressibility of certain random unitary matrices , Entropy 24, 795 (2022)

    E. Bogomolny, Level compressibility of certain random unitary matrices , Entropy 24, 795 (2022)

    2022

  12. [20]

    Noble, Methods based on the Wiener-Hopf technique , Chelsea Publishing Company, New York, N

    B. Noble, Methods based on the Wiener-Hopf technique , Chelsea Publishing Company, New York, N. Y. (1988)

    1988

  13. [21]

    Bogomolny, Semiclassical quantization of multidimensional systems , Nonlinearity 5, 805 (1992)

    E. Bogomolny, Semiclassical quantization of multidimensional systems , Nonlinearity 5, 805 (1992)

    1992

  14. [22]

    Bogomolny, O

    E. Bogomolny, O. Giraud, C. Schmit, Random matrix ensembles associated with Lax matrices, Phys. Rev. Lett. 103, 054103 (2009)

    2009

  15. [23]

    Bogomolny, O

    E. Bogomolny, O. Giraud, and C. Schmit, Integrable random matrix ensembles , Nonlinearity 24, 3179 (2011)

    2011

  16. [24]

    Ruijsenaars, Action-angle maps and scattering theory for some finite-dimensional in- tegrable systems I

    S.N.M. Ruijsenaars, Action-angle maps and scattering theory for some finite-dimensional in- tegrable systems I. The pure soliton case , Commun. Math. Phys. 115, 127 (1988)

    1988

  17. [25]

    Ruijsenaars, Action-angle maps and scattering theory for some finite-dimensional in- tegrable systems II

    S.N.M. Ruijsenaars, Action-angle maps and scattering theory for some finite-dimensional in- tegrable systems II. Solitons, antisolitons, and their bound states , PubL. RIMS, Kyoto Univ. 30, 865 (1994)

    1994

  18. [26]

    Ruijsenaars, Action-angle maps and scattering theory for some finite-dimensional in- tegrable systems, III

    S.N.M. Ruijsenaars, Action-angle maps and scattering theory for some finite-dimensional in- tegrable systems, III. Sutherland type systems and their duals , PubL. RIMS, Kyoto Univ. 31, 247 (1995)

    1995

  19. [27]

    M. C. Gutzwiller, Chaos in classical and quantum mechanics , New York: Springer-Verlag (1990)

    1990

  20. [28]

    Giraud, PhD thesis, Spectral statistics of diffraction systems , (2002)

    O. Giraud, PhD thesis, Spectral statistics of diffraction systems , (2002)

    2002

  21. [29]

    Kottos and U

    T. Kottos and U. Smilansky, Quantum graphs: a simple model for chaotic scattering , J. Phys. A: Math. Gen. 36, 3501 (2003 )

    2003

  22. [30]

    Bateman, Higher transcendental functions, vol

    H. Bateman, Higher transcendental functions, vol. I, McGraw-Hill Book Company (1953)

    1953

  23. [31]

    Slepian, Prolate spheroidal wave functions

    D. Slepian, Prolate spheroidal wave functions. Fourier analysis, and uncertainty – V: The discrete case, The Bell system technical journal, 57, 1371 (1978)

    1978

  24. [32]

    Sommerfeld, Mathematische Theorie der Diffraction , Math

    A. Sommerfeld, Mathematische Theorie der Diffraction , Math. Ann. 47, 317 (1896)

  25. [33]

    Sommerfeld, Optics: Lectures on Theoretical Physics , vol

    A. Sommerfeld, Optics: Lectures on Theoretical Physics , vol. 4, Academic Press, New York, San Francisco, London (1964)

    1964

  26. [34]

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

    E. Bogomolny, Formation of superscar waves in plane polygonal billiards , J. Phys. Commun. 5, 055010 (2021). 34

    2021

  27. [35]

    Ledoux, Concentration of measure and logarithmic Sobolev inequalities, S´ eminaire de prob- abilit´ es (Strasbourg),33, 120 (1999)

    M. Ledoux, Concentration of measure and logarithmic Sobolev inequalities, S´ eminaire de prob- abilit´ es (Strasbourg),33, 120 (1999)

    1999

  28. [36]

    Giraud, J

    O. Giraud, J. Marklof, and S. O’Keefe, Intermediate statistics in quantum maps , J. Phys. A: Math. Gen. 37, L303 (2004)

    2004

  29. [37]

    Bogomolny and C

    E. Bogomolny and C. Schmit, Spectral statistics of a quantum interval-exchange map , Phys. Rev. Lett. 93, 254102 (2004). 35

    2004

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