Schatten class conditions for functions of Schrödinger operators

Abstract

We consider the difference $f(H_1)-f(H_0)$, where $H_0=-\Delta$ and $H_1=-\Delta +V$ are the free and the perturbed Schrödinger operators in $L^2({\mathbb{R}}^d)$, and $V$ is a real-valued short range potential. We give a sharp sufficient condition for this difference to belong to a given Schatten class ${\mathbf{S}}_p$, depending on the rate of decay of the potential and on the smoothness of $f$ (stated in terms of the membership in a Besov class). In particular, for $p>1$ we allow for some unbounded functions $f$.