Distribution of Energy Levels in Quantum Systems with Integrable Classical Counterpart. Rigorous Results

  • P. M. Bleher
Conference paper


Let E 0E 1E 2 ≤... be the energy levels (eigenvalues) of the Schrödinger operator H = -1/2Δ + U(q) on a closed d-dimensional Riemannian manifold M d . Here
$$- \Delta = - \frac{1}{{\sqrt {g} }}\frac{\partial }{{\partial {q^{i}}}}(\sqrt {g} {g^{{ij}}}\frac{\partial }{{\partial {g^{i}}}})] $$
is the Laplace-Beltrami operator and to ensure the discreteness of the spectrum of H we assume, in the case of a non-compact M d , that limq→∞ U(q) = ∞. For simplicity we assume also that M d has no boundary. Otherwise it is neccessary to supply H with Dirichlet (or some other) boundary conditions.


Spectral Interval Quantum Chaos Schrodinger Operator Revolution Surface Smooth Periodic Function 
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  1. 1.
    Bleher, P.M.: Quasi-classical expansion and the problem of quantum chaos. Preprint CARR Rept. in Math.Phys., 1990, No 9/90 (to appear in Lecture Notes in Mathematics)Google Scholar
  2. 2.
    Bleher, P.M.: J.Statist.Phys. 61, 869–876 (1990)MathSciNetCrossRefADSGoogle Scholar
  3. 3.
    Bleher, P.M.: J.Statist.Phys. 63, 261–283 (1991)MathSciNetCrossRefADSGoogle Scholar
  4. 4.
    Berry, M.V., Tabor, M.: Proc.Roy.Soc.London Ser. A 356, 375–394 (1977)CrossRefMATHADSGoogle Scholar
  5. 5.
    Major, P.: Poisson law for the number of lattice points in a random strip with finite area. Preprint Mathematical Institute of the Hungarian Academy of Sciences, 1991Google Scholar
  6. 6.
    Pandey, A., Bohigas, O., Giannoni, M.J.: J.Phys. A: Math.Gen. 22, 4083–4088 (1989)CrossRefADSGoogle Scholar
  7. 7.
    Sinai, Ya.G.: Mathematical problems in the theory of quantum chaos. Distinguished Raimond and Beverly Sackler Lectures. Tel Aviv University, 1990 (to appear in Lecture Notes in Mathematics)Google Scholar
  8. 8.
    Sinai, Ya.G.: Poisson distribution in a geometrical problem. Advances in Soviet Mathematics, Publications of AMS (1991)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • P. M. Bleher
    • 1
  1. 1.Raymond and Beverly Sackler Faculty of Exact Sciences, School of Mathematical SciencesTel Aviv UniversityRamat AvivIsrael

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