Skip to main content
SpringerLink
Log in

Spectral statistics on zoll surfaces

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Berlin, Heidelberg, New York: Springer 1978

    Google Scholar 

  2. Berry, M.: Quantum chaology. Proc. R. Soc. Lond.A413, 183–198 (1987)

    Google Scholar 

  3. Berry, M., Tabor, M.: Level clustering in the regular spectrum. Proc. R. Soc. Lond.A356, 375–394 (1977)

    Google Scholar 

  4. Besse, A.: Manifolds all of whose geodesics asre closed. Ergeb. Math. Gren. Z.93, Springer (1978)

  5. Bleher, P.: The energy level spacing for two harmonic oscillators with generic ratio of frequencies. J. Stat. Phys.63, 261–283 (1991)

    Google Scholar 

  6. Bohigas, O., Giannoni, M.J.: Chaotic motion and random matrix theories. Springer Lect. Notes in Physics209 (1984)

  7. Bohigas, O., Giannoni, M.J., Schmit, C.: Spectral fluctuations of classically chaotic systems. Springer Lect. Notes in Physics209 (1984)

  8. Brummelhuis, R., Uribe, A.: A trace formula for Schrödinger operators. Commun. Math. Phys.136, 567–584 (1991)

    Google Scholar 

  9. Brummelhuis, R., Paul, T., Uribe, A.: Spectral Estimates at a critical level

  10. Colin de Verdièere Y.: Sur le spectre des operateurs a bicharactéristiques toutes periodiques. Comment. Math. Helv.54, 508–522 (1979)

    Google Scholar 

  11. Colin de Verdière, Y.: Quasi-modes sur les variétés Riemanniennes. Inv. Math.43, 15–52 (1977)

    Google Scholar 

  12. Colin de Verdière, Y.: Le spectre conjoint d'operateurs pseudodifferentiels qui commutent: Le cas integrable. Math. Z.171, 51–73 (1980)

    Google Scholar 

  13. Connes, A.: A survey of foliation and operator algebras. Proc. Symp. Pure Math.38 (1982)

  14. Dyson, F.: Statistical properties of the energy levels of complex systems I–III. J. Math. Phys.3, 140–175 (1962)

    Google Scholar 

  15. Duistermaat, J.: Oscillatory integrals, Lagrange immersions, and unfolding of singularities. Commun. Pure and Appl. Math.XXVII, 207–281 (1974)

    Google Scholar 

  16. Duistermaat, J.J., Guillemin, V.: The spectrum of positive elliptic operators and periodic bicharacteristics. Inv. Math.29, 39–79 (1975)

    Google Scholar 

  17. Guillemin, V.: Band asymptotics in two dimensions. Adv. Math.42, 248–282 (1981)

    Google Scholar 

  18. Guillemin, V., Sternberg, S.: Symplectic tecdhniques in Physics. Oxford: Oxford University Press 1983

    Google Scholar 

  19. Guillemin, V., Sternberg, S.: On some problems in integral geometry and some related problems in microlocal analysis. Am. J. Math.101, 915–955 (1979)

    Google Scholar 

  20. Guillemin, V., Sternberg, S.: Homogeneous quantization and multiplicities of group representations. J. Funct. Anal.47, 344–380 (1982)

    Google Scholar 

  21. Guillemin, V., Sternberg, S.: Geometric Asymptotics. Math. Surveys No.14, Providence R.I.: Am. Math. Soc., 1977

    Google Scholar 

  22. Guillemin, V., Uribe, A.: Circular symmetry and the trace formula. Invent. Math.96, 385–423 (1989)

    Google Scholar 

  23. Guillemin, V., Uribe, A.: Reduction and the trace formula. J. Diff. Geom.32, 315–347 (1990)

    Google Scholar 

  24. Major, P.: Poisson law for the number of lattice points in a random strip with finite area. Preprint (1991)

  25. Mehta, M.: Random Matrices. 2nd ed. New York: Academic Press 1991

    Google Scholar 

  26. Paul, T., Uribe, A.: Sur la formule semi-classique des traces. C.R. Acad. Sci Paris313 I, 217–222 (1991)

    Google Scholar 

  27. Sinai, Ya.G.: Poisson distribution for a geometrical problem. Adv. in Soviet Math. Publications of the A.M.S., 1991

  28. Weinstein, A.: Asymptotics of eigenvalue clusters for the Laplacian plus a potential. Duke Math. J.44, 883–892 (1977)

    Google Scholar 

  29. Weinstein, A.: Fourier integral operators, quantization, and the spectra of a Riemannian manifold. Colloques Internationaux C.N.R.S.237, Géométrie symplectique et physique. (1976)

  30. Wigner, E.: Random matrices in Physics. S.I.A.M.9, 1–23 (1967)

    Google Scholar 

  31. Zelditch, S.: Kuznecov sum formulae and Szegö limit formulae on manifolds. Comm. P.D.E.17 (1&2), 221–260 (1992)

    Google Scholar 

  32. Zelditch, S.: Spectrum and geodesic flow. In: Advanced Studies in Pure Math.21, 1–19 (1992)

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, University of Michigan, 48109, Ann Arbor, MI, USA

    Alejandro Uribe

  2. Institute for Advanced Study, 08540, Princeton, NJ, USA

    Alejandro Uribe

  3. Mathematics Department, Johns Hopkins University, 21218, Baltimore, MD, USA

    Steven Zelditch

Authors
  1. Alejandro Uribe
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Steven Zelditch
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by Ya.G. Sinai

Research supported by NSF grants DMS-8907997 and DMS 8610730

Research partially supported by NSF grant DMS-9103124

Rights and permissions

Reprints and permissions

About this article

Cite this article

Uribe, A., Zelditch, S. Spectral statistics on zoll surfaces. Commun.Math. Phys. 154, 313–346 (1993). https://doi.org/10.1007/BF02097000

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02097000

Keywords

Search

Navigation