Spectral theory of the Schrodinger operators on non-compact manifolds: qualitative results

摘要

The paper contains an expanded version of my lectures in the Edinburgh school on Spectral Theory and Geometry (Spring 1998). I tried to make the exposition as self-contained as possible. The main object of this paper is a Schrodinger operator H = -Δ + V(x) on a noncompact Riemannian manifold M. We discuss two basic questions of the spectral theory for such operators: conditions of the essential self-adjointness (or quantum completeness), and conditions for the discreteness of the spectrum in terms of the potential V. In the first part of the paper we provide a shorter and a more transparent proof of a remarkable result by I. Oleinik [81, 82, 83], which implies practically all previously known results about essential self-adjointness in absence of local singularities of the potential. This result gives a sufficient condition of the essential self-adjointness of a Schrodinger operator with a locally bounded potential in terms of the completeness of the dynamics for a related classical system. The simplification of the proof given by I. Oleinik is achieved by an explicit use of the Lipschitz analysis on the Riemannian manifold and also by additional geometrization arguments which include a use of a metric which is conformal to the original one with a factor depending on the minorant of the potential. In the second part of the paper we consider the case when the potential V is semibounded below and the manifold M has bounded geometry. We provide a necessary and sufficient condition for the spectrum of H to be discrete in terms of V. It is formulated by use of the harmonic (Newtonian) capacity in geodesic coordinates on M. This result is due to V.A. Kondrat'ev and M. Shubin and it extends the famous result of A.M. Molchanov [78] where the case M = R~n was considered. We follow Molchanov's scheme of the proof but simplify and clarify some moments of this proof, at the same time generalizing it to manifolds of bounded geometry. Somewhat shorter versions of the two parts of this paper can also be found in the papers [100], [58].