Bohr-Sommerfeld conditions for several commuting Hamiltonians

Abstract

The goal of this paper is to find the quantization conditions of Bohr-Sommerfeld of several quantum Hamiltonians Q₁(h), . . . , Q_k(h) acting on â„ⁿ, depending on a small paremeter h, and which commute to each other. That is we determine, around a regular energy level E₀ â‹² â„^k the principal term of the asymptotic in h of eigenvalues Î»_j(h), 1 â‰¤ j â‰¤ k of the operators Q_j(h) that are associated to a common eigenfunction. Thus we localize the so-called joint spectrum of the operators.

Under the assumption that the classical Hamiltonian flow of the joint principal symbol q0 is periodic with constant periods on the one energy level q₀^-1 (E₀), we prove that the part of the joint spectrum lying in a small neighbourhood of E₀ is localized near a lattice of size h determined in terms of actions and Maslov indices. The multiplicity of the spectrum is also determined.