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On a class of anharmonic oscillators. (English. French summary) Zbl 1473.35376

Summary: In this work we study a class of anharmonic oscillators within the framework of the Weyl-Hörmander calculus. A prototype is an operator on \(\mathbb{R}^n\) of the form \((-\Delta)^\ell+|x|^{2k}\) for \(k,\ell\) integers \(\geq 1\). We obtain spectral properties in terms of Schatten-von Neumann classes for their negative powers and derive from them estimates on the rate of growth for the eigenvalues of the anharmonic oscillator \((-\Delta)^\ell+|x|^{2k}\). In particular we give a simple proof for the main term of the spectral asymptotics of these operators. We also study some examples of anharmonic oscillators arising from the analysis on Lie groups.

MSC:

35P20 Asymptotic distributions of eigenvalues in context of PDEs
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35J30 Higher-order elliptic equations
35Q41 Time-dependent Schrödinger equations and Dirac equations
47G30 Pseudodifferential operators
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