On a class of anharmonic oscillators

Abstract

In this work we study a class of anharmonic oscillators within the framework of the Weyl-Hörmander calculus. The anharmonic oscillators arise from several applications in mathematical physics as natural extensions of the harmonic oscillator. A prototype is an operator on ${\mathbb{R}}^n$ of the form $(-\Delta {)}^{\ell }+|x{|}^{2k}$ for $k,\ell$ integers $\ge 1$. The simplest case corresponds to Hamiltonians of the form $|\xi {|}^2+|x{|}^{2k}$. Here by associating a Hörmander metric $g$ to a given anharmonic oscillator we investigate several properties of the anharmonic oscillators. We obtain spectral properties in terms of Schatten-von Neumann classes for their negative powers. We also study some examples of anharmonic oscillators arising from the analysis on Lie groups

arXiv commentThis is an updated version with some corrections