Trace class conditions for functions of Schr\"odinger operators

Abstract

We consider the difference $f(-\Delta +V)-f(-\Delta)$ of functions of Schr\"odinger operators in $L^2(\mathbb R^d)$ and provide conditions under which this difference is trace class. We are particularly interested in non-smooth functions $f$ and in $V$ belonging only to some $L^p$ space. This is motivated by applications in mathematical physics related to Lieb--Thirring inequalities. We show that in the particular case of Schr\"odinger operators the well-known sufficient conditions on $f$, based on a general operator theoretic result due to V. Peller, can be considerably relaxed. We prove similar theorems for $f(-\Delta +V)-f(-\Delta)-\frac{d}{d\alpha} f(-\Delta +\alpha V)|_{\alpha=0}$. Our key idea is the use of the limiting absorption principle.