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

Let M be a general complete Riemannian manifold and consider a Schrödinger operator − + V on L2(M). We prove Cwikel-Lieb-Rozenblum as well as Lieb-Thirring type estimates for − + 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 Schrödinger operators with complex-valued potentials