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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
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