Bohr-Sommerfeld conditions for several commuting Hamiltonians
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Colette Anné
colette.anne@math.univ-nantes.fr
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Anne-Marie Charbonnel
colette.anne@math.univ-nantes.fr
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Abstract
The goal of this paper is to find the quantization conditions of Bohr-Sommerfeld of several quantum Hamiltonians Q1(h), . . . , Qk(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„k the principal term of the asymptotic in h of eigenvalues λj(h), 1 ≤ j ≤ k of the operators Qj(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.
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