The paper contains an extended version of the author’s lectures in the Edinburgh school on Spectral Theory and Geometry (1998). The main object of this paper is a Schrödinger operator \(H = -\Delta + V(x)\) on a non-compact Riemannian manifold \(M\). The author discusses two basic questions of the spectral theory for such operators: conditions of 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, he gives a shorter and a more transparent proof of a remarkable result given by Oleinik (1993-1997), 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 Schrödinger operator with a locally bounded potential in terms of the completeness of the dynamics for a related classical system. For proving this theorem the author explicitly uses the Lipschitz analysis on the Riemannian manifold and also additional geometrization arguments which include the 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, he considers the case when the potential \(V\) is semi-bounded below and the manifold \(M\) has bounded geometry. The author provides a necessary and sufficient condition for the spectrum of \(H\) to be discrete in terms of \(V\). This result is due to V. A. Kondrat’ev and M. Shubin and it extends the famous result of A. M. Molchanov for the case where \(M=\mathbb{R}^n\) . The author follows Molchanov’s scheme of the proof but simplifies and clarifies some moments, at the same time generalizing it to manifolds of bounded geometry.
For the entire collection see [Zbl 0924.00037].