Singular Sturm-Liouville theory on manifolds

In this paper we investigate Schrodinger operators L = -Delta (g) + a(x) on a compact Riemannian manifold (M, g), where the potential function a(x) is assumed to be continuous, but not necessarily bounded, outside of some closed set Sigma subset of M of measure zero. Under certain geometric hypotheses on Sigma and growth conditions on a(x) as x --> Sigma, we prove that the Dirichlet extension of L is bounded from below with discrete spectrum; in many cases, a(x) is allowed to approach - infinity as x --> Sigma. We also consider conditions on Sigma and a(x) under which the Sturm-Liouville theory of L is "singular" in that no boundary conditions are needed to specify the eigenvalues and eigenfunctions of L; in particular, this occurs when the domain of L does not depend on boundary conditions, for example, when L is essentially selfadjoint or more generally "essentially Dirichlet" (a new property that we define). The behavior of L on weighted Sobolev spaces is also discussed. In most of the paper we assume that Sigma is a k-dimensional submanifold without boundary, but in the last few sections we generalize our results to stratified sets. (C) 2001 Academic Press.