 
Summary: Perturbation of several commuting
hpseudodi#erential operators
Colette ANN ’
E & AnneMarie CHARBONNEL
June 15, 2006
UMR 6629 de Math’ematiques, BP 92208
Universit’e de NANTES
Facult’e des Sciences et des Techniques
44322 NantesCedex 03, France
Abstract
We consider k pseudodi#erential operators Q 1 (h), . . . , Q k (h), acting on R n com
muting together, and depending on a small parameter h. Under the assumption
that the classical Hamiltonian flow of the joint principal symbol q 0 is periodic with
constant period on one given energy level q 1 0 (E 0 ), in [1] we have shown that the
joint spectrum of these operators lying in a hdepending neighborhood I(h) of E 0
is localized near a lattice. This paper followed the now classical method initiated
by Hel#er and Robert in [11], and in [4] for several operators, which is based on the
FIO theory. We investigate in this article the method proposed by Colin de Verdi‘ere
to Dozias [8] for one operator acting on R n and obtain a theorem of perturbation.
