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

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

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

((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 lim_q→∞ 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.