On a class of anharmonic oscillators

Abstract

In this work we study a class of anharmonic oscillators within the framework of the Weyl-H\"ormander 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 $\geq 1$. The simplest case corresponds to Hamiltonians of the form $|\xi|^2+|x|^{2k}$. Here by associating a H\"ormander 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