Essential self-adjointness for magnetic Schrödinger operators on non-compact manifolds

We give a condition of essential self-adjointness for magnetic Schrödinger operators on non-compact Riemannian manifolds with a given positive smooth measure which is fixed independently of the metric. This condition is related to the classical completeness of a related classical hamiltonian without magnetic field. The main result generalizes the result by I. Oleinik [46, 47, 48], a shorter and more transparent proof of which was provided by the author in [59]. The main idea, as in [59], consists in 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 electric potential. We also prove a magnetic version of the A. Povzner theorem [49] on essential self-adjointness of semi-bounded Schrödinger operators, as well as its generalization to manifolds.