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

Abstract

We consider ν pseudodifferential operators, Q₁(h),…,Q_ν(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),…,Q_ν(h)) as a pseudodifferential operator of the same type, when f belongs to C^∞₀(R^ν). We use it to study the semiclassical behaviour of the joint spectrum of Q₁(h),…,Q_ν(h), lying in a compact of R^ν 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.