On a class of anharmonic oscillators : II. General case

In this work we study a class of anharmonic oscillators on Rn corresponding to Hamiltonians of the form A(D) + V(x), where A(xi) and V(x) are C infinity functions enjoying some reg-ularity conditions. Our class includes fractional relativistic Schrodinger operators and anharmonic oscillators with frac-tional potentials. By associating a Hormander metric we ob-tain spectral properties in terms of Schatten-von Neumann classes for their negative powers and derive from them esti-mates on the rate of growth for the eigenvalues of the opera-tors A(D) + V(x). This extends the analysis in the first part [1], where the case of polynomial A and V has been analysed.(c) 2022 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).