On a class of anharmonic oscillators II. General case

TL;DR

This paper analyzes a broad class of anharmonic oscillators with Hamiltonians involving smooth functions, extending previous polynomial cases to include fractional operators and potentials, and studies their spectral properties and eigenvalue growth.

Contribution

It generalizes spectral analysis of anharmonic oscillators to include fractional and non-polynomial cases using Hörmander metrics.

Findings

Spectral properties characterized in Schatten-von Neumann classes.

Eigenvalue growth rates estimated for the operators.

Extension of previous polynomial case analysis to fractional and smooth functions.

Abstract

In this work we study a class of anharmonic oscillators on $\mathbb{R}^n$ corresponding to Hamiltonians of the form $A(D)+V(x)$, where $A(\xi)$ and $V(x)$ are $C^{\infty}$ functions enjoying some regularity conditions. Our class includes fractional relativistic Schr\"odinger operators and anharmonic oscillators with fractional potentials. By associating a H\"ormander metric 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 operators $A(D)+V(x)$. This extends the analysis in the first part of our work, where the case of polynomial $A$ and $V$ has been analysed.