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

  • P. M. Bleher
Conference paper

Abstract

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
$$- \Delta = - \frac{1}{{\sqrt {g} }}\frac{\partial }{{\partial {q^{i}}}}(\sqrt {g} {g^{{ij}}}\frac{\partial }{{\partial {g^{i}}}})] $$
(1)
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.

Keywords

Manifold limE 

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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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