On a class of anharmonic oscillators II. General case

Abstract

In this work we study a class of anharmonic oscillators on Rn corresponding to Hamiltonians of the form A(D)+V(x), where A(ξ) and V(x) are C∞ 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 [1], where the case of polynomial A and V has been analysed.