Self-adjointness of certain second order differential operators in Riemannian manifolds

Two theorems of Devinatz on self-adjointness of Schroedinger-type operators in Euclidean space are generalized to complete, non-compact, Riemannian manifolds. The sectional curvatures of the manifold are assumed to be positive and bounded above. The primary hypotheses place growth restrictions on the leading coefficients of the operator's local representations within certain annular regions of the manifold. The potential function is required to be bounded below and locally square integrable. Much of the proof is local, following the Euclidean case in its use of Kato's distributional inequality and a maximum principle of Littman. However, two additional results are needed and derived: one is a global version of the maximum principle and the other provides a uniform lower bound on the local components of the metric valid on any normal coordinate ball of sufficiently small radius. In the major result, the potential is assumed to be bounded below only locally. The possible decay of the potential to minus infinity is tied to a term involving the symbol of the operator and a distance function d(x,x0), where x0 is some fixed point of the manifold. The decay restrictions, furthermore, need only be imposed within annular regions. The central part of the proof is a truncation argument that involves a cutoff version of the operator defined on a compact manifold. In addition, global properties ofthe distance function are established, and the symbol of the operator is used to define a sesquilinear form that facilitates integration by parts. In the appendix a general self-adjointness result in Euclidean space is presented. The potential is assumed to be locally integrable and its decay to minus infinity is restricted within a sequence of pseudo-annular regions