Classical and quantum completeness for the Schrödinger operators on non-compact manifolds

We provide a shorter and more transparent proof of a result by I. Oleinik [27, 28, 29]. It gives a sufficient condition of the essential self-adjointness of a Schrödinger operator on a non-compact Riemannian manifold 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.