 
Summary: BohrSommerfeld conditions for several commuting
Hamiltonians
3rd April 2003
Colette ANN
E & AnneMarie CHARBONNEL
Laboratoire de Mathematiques Jean Leray, UMR 6629
Universite de NANTES
Faculte des Sciences et des Techniques
BP 92208 44322 NantesCedex 03, France
Abstract
The goal of this paper is to nd the quantization conditions of BohrSommerfeld
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 socalled 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
