Symptotic behavior of solutions of the inhomogeneous Schrödinger equation on noncompact Riemannian manifolds

Abstract

The paper studies the behavior of bounded solutions of the inhomogeneous Schrödinger equation on non-compact Riemannian manifolds under a variation of the right side of the equation. Various problems for homogeneous elliptic equations, in particular the Laplace-Beltrami equation and the stationary Schrödinger equation, have been considered by a number of Russian and foreign authors since the second half of the 20th century. In the first part of this paper, an approach to the formulation of boundary value problems based on the introduction of classes of equivalent functions will be developed. The relationship between the solvability of boundary value problems on an arbitrary non-compact Riemannian manifold with variation of inhomogeneity is also established. In the second part of the work, based on the results of the first part, properties of solutions of the inhomogeneous Schrödinger equation on quasi-model manifolds are investigated, and exact conditions for unique solvability of the Dirichlet problem and some other boundary value problems on these manifolds are found.