Essential self-adjointness of Schroedinger type operators on manifolds

Details Essential self-adjointness of Schroedinger type operators on manifolds Essential self-adjointness of Schroedinger type operators on manifolds

2002-01-24

arxiv.org/abs/math/0201231v3

Mathematics

Spectral Theory

52 pages, Minor corrections are made; To appear in Russian Math. Surveys

Scientific paper

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

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