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

In: Mathematical Physics X

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  • P. M. Bleher

    (Tel Aviv University, Raymond and Beverly Sackler Faculty of Exact Sciences, School of Mathematical Sciences)

Abstract

Let E 0 ≤ E 1 ≤ E 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 (1) $$- \Delta = - \frac{1}{{\sqrt {g} }}\frac{\partial }{{\partial {q^{i}}}}(\sqrt {g} {g^{{ij}}}\frac{\partial }{{\partial {g^{i}}}})] $$ 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.

Suggested Citation

  • P. M. Bleher, 1992. "Distribution of Energy Levels in Quantum Systems with Integrable Classical Counterpart. Rigorous Results," Springer Books, in: Konrad Schmüdgen (ed.), Mathematical Physics X, pages 298-302, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-77303-7_25
    DOI: 10.1007/978-3-642-77303-7_25
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