Essential self-adjointness of Schrödinger-type operators on manifolds

Abstract

Several conditions are obtained ensuring the essential self-adjointness of a Schrödinger-type operator , where is a first-order elliptic differential operator acting on the space of sections of a Hermitian vector bundle over a manifold with positive smooth measure and is a Hermitian bundle endomorphism. These conditions are expressed in terms of completeness of certain metrics on naturally associated with . The results generalize theorems of Titchmarsh, Sears, Rofe-Beketov, Oleinik, Shubin, and Lesch. It is not assumed a priori that is endowed with a complete Riemannian metric. This enables one to treat, for instance, operators acting on bounded domains in with Lebesgue measure. Singular potentials are also admitted. In particular, a new self-adjointness condition is obtained for a Schrödinger operator on whose potential has a Coulomb-type singularity and can tend to at infinity. For the special case in which the principal symbol of is scalar, more precise results are established for operators with singular potentials. The proofs of these facts are based on a refined Kato-type inequality modifying and improving a result of Hess, Schrader, and Uhlenbrock.