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Bohr-Sommerfeld conditions for several commuting Hamiltonians

Summary: Bohr-Sommerfeld conditions for several commuting
3rd April 2003
Colette ANN 
Laboratoire de Mathematiques Jean Leray, UMR 6629
Universite de NANTES
Faculte des Sciences et des Techniques
BP 92208 44322 Nantes-Cedex 03, France
The goal of this paper is to nd the quantization conditions of Bohr-Sommerfeld
of several quantum Hamiltonians Q 1 (h); :::; Q k (h) acting on R n , depending on a
small parameter h, and which commute with each other. That is we determine,
around a regular energy level E 0 2 R k the principal term of the asymptotics in h of
eigenvalues  j (h); 1  j  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 ow of the joint principal
symbol q 0 is periodic with constant periods on the one energy level q 1
0 (E 0 ), we prove
that the part of the joint spectrum lying in a small neighborhood of E 0 is localized


Source: Anné, Colette - Laboratoire de Mathématiques Jean Leray,


Collections: Mathematics