Communications in Mathematical Physics

, Volume 170, Issue 2, pp 375–403 | Cite as

Distribution of energy levels of quantum free particle on the Liouville surface and trace formulae

  • Pavel M. Bleher
  • Denis V. Kosygin
  • Yakov G. Sinai


We consider the Weyl asymptotic formula
$$\# \left\{ {E_n \leqq R^2 } \right\} = \frac{{Area Q}}{{4\pi }}R^2 + n(R),$$
for eigenvalues of the Laplace-Beltrami operator on a two-dimensional torusQ with a Liouville metric which is in a sense the most general case of an integrable metric. We prove that if the surfaceQ is non-degenerate then the remainder termn(R) has the formn(R)=R1/2θ(R), where θ(R) is an almost periodic function of the Besicovitch classB1, and the Fourier amplitudes and the Fourier frequencies of θ(R) can be expressed via lengths of closed geodesics onQ and other simple geometric characteristics of these geodesics. We prove then that if the surfaceQ is generic then the limit distribution of θ(R) has a densityp(t), which is an entire function oft possessing an asymptotics on a real line, logp(t)≈−C±t4 ast→±∞. An explicit expression for the Fourier transform ofp(t) via Fourier amplitudes of θ(R) is also given. We obtain the analogue of the Guillemin-Duistermaat trace formula for the Liouville surfaces and discuss its accuracy.


Neural Network Fourier Transform Explicit Expression Entire Function Periodic Function 
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  1. 1.
    Berry, M.V., Mount, K.E.: Semiclassical approximations in wave mechanics. Rep. Progr. Phys.35, 315–397 (1972)Google Scholar
  2. 2.
    Berry, M.V., Tabor, M.: Closed orbits and the regular bound spectrum. Proc. R. Soc. Lond.A349, 101–123 (1976)Google Scholar
  3. 3.
    Bleher, P.M.: Quasiclassical expansion and the problem of quantum chaos. Lect. Notes in Math1469, 60–89 (1991)Google Scholar
  4. 4.
    Bleher, P.M.: On the distribution of the number of lattice points inside a family of convex ovals. Duke Math. J.67, 461–481 (1993)Google Scholar
  5. 5.
    Bleher, P.M.: Distribution of the error in the Weyl asymptotics for the Laplace operator on a two-dimensional torus and related lattice problems. Duke Math. J.70, 655–682 (1993)Google Scholar
  6. 6.
    Bleher, P.M.: Distribution of energy levels of a quantum free particle on a surface of revolution. Duke Math. J.74, 1–49 (1994)Google Scholar
  7. 6a.
    Bleher, P.M.: Semiclassical quantization rules near separatrices. Commun. Math. Phys.165, 621–640 (1994)Google Scholar
  8. 7.
    Bleher, P.M., Cheng, Zh., Dyson, F.J., Lebowitz, J.L.: Distribution of the error term for the number of lattice points inside a shifted circle. Commun. Math. Phys.154, 433–469 (1993)Google Scholar
  9. 8.
    Colin de Verdiére, Y.: Spectre du laplacien et longueurs des geodesiques periodiques. Comp. Math.27, 159–184 (1973)Google Scholar
  10. 9.
    Duistermaat, J.J., Guillemin, V.: Spectrum of elliptic operators and periodic bicharacteristics. Invent. Math.29, 39–79 (1975)Google Scholar
  11. 10.
    Guillemin, V.: Lectures on spectral theory of elliptic operators. Duke Math. J.44, 485–517 (1977)Google Scholar
  12. 11.
    Gutzwiller, G.: chaos in classical and quantum mechanics. New York: Springer-Verlag, 1990Google Scholar
  13. 12.
    Heath-Brown, D.R.: The distribution and the moments of the error term in the Dirichlet divisor problem. Acta Arithm.60, 389–415 (1992)Google Scholar
  14. 13.
    Hörmander, L.: Fourier integral operators I. Acta Math.127, 79–183 (1971)Google Scholar
  15. 14.
    Hörmander, L.: The spectral function of an elliptic operator. Acta Math.121, 193–218 (1968)Google Scholar
  16. 15.
    Kosygin, D.V., Minasov, A.A., Sinai, Ya.G.: Statistical properties of the Laplace-Beltrami operator on Liouville surfaces. Uspekhi Mat. Nauk48, no. 4, 3–130 (1993)Google Scholar
  17. 16.
    Lax, P., Phillips, R.: Scattering theory for automorphic functions. Princeton, NJ: Princeton University Press, 1976Google Scholar
  18. 17.
    Levitan, B.M., Zhikov, V.V.: Almost periodic functions and differential equations. Cambridge: Cambridge Univ. Press, 1968Google Scholar
  19. 18.
    McKean, H.B.: Selberg's trace formula as applied to a compact Riemann surface. Comm. Pure Appl. Math.25, 225–246 (1972)Google Scholar
  20. 19.
    Rubinstein, M., Sarnak, P.: Chebyshev bias (to appear)Google Scholar
  21. 20.
    Sarnak, P.: Arithmetic quantum chaos. Schur Lectures, Tel Aviv, 1992Google Scholar
  22. 21.
    Sinai, Ya.G.: Mathematical problems in the theory of quantum chaos. Lect. Notes in Math.1469, 41–59 (1991)Google Scholar
  23. 22.
    Nguen Tien Zung, Polyakova, L.S., Selivanova, E.N.: Topological classification of integrable geodesic flows with additional quadratic or linear in momenta integral on two-dimensional oriented riemannian manifolds. Funct. Analysis and Appl.27, 42–56 (1993)Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Pavel M. Bleher
    • 1
  • Denis V. Kosygin
    • 2
  • Yakov G. Sinai
    • 2
  1. 1.Department of Mathematical SciencesIndiana University-Purdue University at IndianapolisIndianapolisUSA
  2. 2.Department of MathematicsPrinceton UniversityPrincetonUSA

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