Remarks on the Cwikel–Lieb–Rozenblum and Lieb–Thirring Estimates for Schrödinger Operators on Riemannian Manifolds

Abstract

Let M be a general complete Riemannian manifold and consider a Schrodinger operator −Δ+V on L2(M). We prove Cwikel–Lieb–Rozenblum as well as Lieb–Thirring type estimates for −Δ+V. These estimates are given in terms of the potential and the heat kernel of the Laplacian on the manifold. Some of our results hold also for Schrodinger operators with complex-valued potentials.