On a class of anharmonic oscillators II. General case

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 Schrödinger operators and anharmonic oscillators with fractional potentials. By associating a Hörmander 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.

19 pages