Trace Class Conditions for Functions of Schrödinger Operators

We consider the difference f(−Δ+V)−f(−Δ) of functions of Schrödinger operators in L^2(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 Schrödinger 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(−Δ+V)−f(−Δ)−^d/_(dα)f(−Δ+αV)|α=0 . Our key idea is the use of the limiting absorption principle.