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Resonances in domains of size \(h\). (Résonances dans des domaines de taille \(h\).) (French) Zbl 1034.35084

The author considers a self-adjoint unbounded operator \(P\) which tends to \((-h^2\Delta)\) at infinity and gives an estimation for the number of the resonances of \(P\) in domains of size \(h\) around a regular value \(E_0\). Here \(h\) stands for any number in the open interval \((0,1)\). The main result, which is stated as a theorem, shows that the number in question is of \(O(h^{1-n})\). In the proof, the definition of the resonances through the analytical dilatations method plays an important role.

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35S05 Pseudodifferential operators as generalizations of partial differential operators
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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