Advertisement

The Berry-Tabor Conjecture

  • Jens Marklof
Part of the Progress in Mathematics book series (PM, volume 202)

Abstract

One of the central observations ofquantum chaologyis that statistical properties of quantum spectra exhibit surprisingly universal features, which seem to mirror the chaotic or regular dynamical properties of the underlying classical limit. I will report on recent studies of simple regular systems, where some of the observed phenomena can be established rigorously. The results discussed are intimately related to the distribution of values of quadratic forms, and in particular to a quantitative version of the Oppenheim conjecture.

Keywords

Pair Correlation Pair Correlation Function Geodesic Flow Quantitative Version Diophantine Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. V. Berry and M. Tabor, Level clustering in the regular spectrumProc. Roy. Soc.A 356 (1977) 375–394.zbMATHCrossRefGoogle Scholar
  2. 2.
    M. V. Berry, Quantizing a classically ergodic system: Sinai’s billiard and the KKR methodAnn. Phys.131 (1981) 163–216.CrossRefGoogle Scholar
  3. 3.
    O. Bohigas, M.-J. Giannoni and C Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation lawsPhys. Rev. Lett.52 (1984) 1–4.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Z. Cheng and J. L. Lebowitz, Statistics of energy levels in integrable quantum systems, Phys. Rev. A 44 (1991) 3399–3402.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Z. Cheng, J. L. Lebowitz and P. Major, On the number of lattice points between two enlarged and randomly shifted copies of an ovalProbab. Theory Related Fields100 (1994) 253–268.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Y. Colin de Verdière, Quasi-modes sur les variétés RiemanniennesInvent. Math.43 (1977) 15–52.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Y. Colin de Verdière, Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiqueComment. Math. Helvetici54 (1979) 508–522.zbMATHCrossRefGoogle Scholar
  8. 8.
    J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristicsInvent. Math.29 (1975) 39–79.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    A. Eskin, G. Margulis and S. Mozes, Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann.of Math.147 (1998) 93–141.MathSciNetzbMATHCrossRefGoogle Scholar
  10. A. Eskin, G. Margulis and S. Mozes, Quadratic forms of signature (2, 2) and eigen-value spacings on rectangular 2-tori, preprint.Google Scholar
  11. 11.
    P. Major, Poisson law for the number of lattice points in a random strip with finite areaProb. Theo. Rel. Fields92 (1992) 423–464.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    J. Marklof, Limit theorems for theta sumsDuke Math. J.97 (1999) 127–153.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    J. Marklof, Theta. sums, Eisenstein series, and the semiclassical dynamics of a precessing spin, in: D. Hejhal et al. (eds.)Emerging Applications of Number TheoryIMA Vol. Math. Appl. 109 (Springer, New York, 1999) 405–450.Google Scholar
  14. 14.
    J. Marklof, Spectral form factors of rectangle billiardsComm. Math. Phys.199 (1998) 169–202.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    J. Marklof, Pair correlation densities of inhomogeneous quadratic forms, preprint 2000.Google Scholar
  16. 16.
    S. W. McDonald and A. N. Kaufman, Spectrum and eigenfunctions for a Hamiltonian with stochastic trajectoriesPhys. Rev. Lett.42 (1979) 1189–1191.CrossRefGoogle Scholar
  17. 17.
    Z. Rudnick and P. Sarnak, The pair correlation function for fractional parts of polynomialsComm. Math. Phys.194 (1998) 61–70.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Z. Rudnick, P. Sarnak and A. Zaharescu, The distribution of spacings between the fractional parts of ant, preprint November 1999.Google Scholar
  19. 19.
    Z. Rudnick and A. Zaharescu, A metric result on the pair correlation of fractional parts of sequencesActa Arith.LXXXIX (1999) 283–293.Google Scholar
  20. 20.
    [] Z. Rudnick and A. Zaharescu, The distribution of spacings between fractional parts of lacunary sequences, preprint December 1999.Google Scholar
  21. P. Sarnak, Values at integers of binary quadratic forms, Harmonic Analysis and Number Theory (Montreal, PQ, 1996), 181–203, CMS Conf. Proc. 21, Amer. Math. Soc., Providence, RI, 1997.Google Scholar
  22. 22.
    Ya. G. Sinai, Poisson distribution in a geometrical problem, Adv.Soy. Math. AMS Publ. 3 (1991) 199–215.MathSciNetGoogle Scholar
  23. 23.
    A. Uribe and S. Zelditch, Spectral statistics on Zoll surfacesComm. Math. Phys.154 (1993) 313–346.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    A. Weinstein, Asymptotics of eigenvalue clusters for the Laplacian plus a potentialDuke Math. J.44 (1977) 883–892.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    J. M. VanderKam, Values at integers of homogeneous polynomialsDuke Math. J.97 (1999) 379–412.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    J. M. VanderKam, Pair correlation of four-dimensional flat toriDuke Math. J.97 (1999) 413–438.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    J. M. VanderKam, Correlations of eigenvalues on multi-dimensional flat toriComm. Math. Phys.210 (2000) 203–223.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    S. Zelditch, Level spacings for integrable quantum maps in genus zeroCommun. Math. Phys.196 (1998) 289–329.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2001

Authors and Affiliations

  • Jens Marklof
    • 1
  1. 1.School of MathematicsUniversity of BristolBristol UK

Personalised recommendations