Absence of embedded eigenvalues for Riemannian Laplacians

In this paper we study absence of embedded eigenvalues for Schrodinger operators on non-compact connected Riemannian manifolds. A principal example is given by a manifold with an end (possibly more than one) in which geodesic coordinates are naturally defined. In this case one of our geometric conditions is a positive lower bound of the second fundamental form of angular submanifolds at infinity inside the end. Another condition is an upper bound of the trace of this quantity, while a third one is a bound of the derivatives of part of the trace (some oscillatory behaviour of the trace is allowed). In addition to geometric bounds we need conditions on the potential, a regularity property of the domain of the Schrodinger operator and the unique continuation property. Examples include ends endowed with asymptotic Euclidean or hyperbolic metrics. (C) 2013 Elsevier Inc. All rights reserved.