## Comportement semi-classique du spectre conjoint d’opérateurs pseudodifférentiels qui commutent. (Semiclassical behavior of the joint spectrum of commuting pseudodifferential operators).(French)Zbl 0665.35080

We consider $$\nu$$ pseudodifferential operators, $$Q_ 1(h),...,Q_{\nu}(h)$$, acting in $${\mathbb{R}}^ n$$, 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 $$\underline{\lim}_{| x| \to +\infty}V(x)>E$$. Under suitable conditions, we establish a functional calculus to define $$f(Q_ 1(h),...,Q_{\nu}(h))$$ as a pseudodifferential operator of the same type, when f belongs to $$C^{\infty}_ 0({\mathbb{R}}^{\nu})$$. We use it to study the semiclassical behaviour of the joint spectrum of $$Q_ 1(h),...,Q_{\nu}(h)$$, lying in a compact of $${\mathbb{R}}^{\nu}$$ 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 $${\mathbb{R}}^ n$$.

### MSC:

 35S05 Pseudodifferential operators as generalizations of partial differential operators 35P05 General topics in linear spectral theory for PDEs 35J10 Schrödinger operator, Schrödinger equation