Comportement semi-classique du spectre conjoint d'opérateurs pseudodifférentiels qui commutent

Article type: Research Article

Authors: Charbonnel, Anne-Marie

Affiliations: Département de Mathématiques et d'Informatique, Faculté des Sciences et des Techniques, Université de Nantes, 44072 Nantes Cedex 03, France

Abstract: We consider ν pseudodifferential operators, Q1(h),…,Qν(h), acting in Rn, commuting together, depending on a small parameter h. For instance, one of them is the Schrödinger operator with a potential V(x) satisfying the condition $\underline{lim}_{|x|\rightarrow +\infty}V(x)>E$. Under suitable conditions, we establish a functional ca1culus to define f(Q1(h),…,Qν(h)) as a pseudodifferential operator of the same type, when f belongs to C∞0(Rν). We use it to study the semiclassical behaviour of the joint spectrum of Q1(h),…,Qν(h), lying in a compact of Rν where it is discrete. When these operators form a quantically integrable system, we give more precise estimations. Our results generalize those obtained by Helffer and Robert for one operator acting in Rn.

DOI: 10.3233/ASY-1988-1305

Journal: Asymptotic Analysis, vol. 1, no. 3, pp. 227-261, 1988