Essential self-adjointness of Schrödinger-type operators on manifolds

Abstract

We obtain several essential self-adjointness conditions for a Schroedinger\ntype operator D*D+V acting in sections of a vector bundle over a manifold M.\nHere V is a locally square-integrable bundle map. Our conditions are expressed\nin terms of completeness of certain metrics on M; these metrics are naturally\nassociated to the operator. We do not assume a priori that M is endowed with a\ncomplete Riemannian metric. This allows us to treat e.g. operators acting on\nbounded domains in the euclidean space.\n For the case when the principal symbol of the operator is scalar, we\nestablish more precise results. The proofs are based on an extension of the\nKato inequality which modifies and improves a result of Hess, Schrader and\nUhlenbrock.\n