Trace Class Conditions for Functions of Schrödinger Operators

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