Boundary Value Problems for the Inhomogeneous Schrödinger Equation with Variations of Its Potential on Quasi-Model Riemannian Manifolds

In this paper, solutions of the inhomogeneous Schrodinger equation ¨ are studied Lu = Δu − c(x)u = h(x), (9) where c(x), h(x) — locally continuous Holder functions, with variations of its ¨ potential c(x) ≥ 0 on quasi-model Riemannian manifolds. It is clear that if h(x) ≡ ≡ 0, then the equation (9) is a stationary Schrodinger equation. Therefore, we ¨ assume that h(x) ̸= 0. In this paper, we obtain the exact conditions under which the solvability of boundary value problems of the inhomogeneous Schrodinger ¨ equation is preserved with some variations of the coefficient c(x) ≥ 0 on M. At present one of the directions of the development of this topic is the question of the solvability on a non-compact manifold of the Dirichlet problem on the restoring the solution of the stationary Schrodinger equation from boundary ¨ data on "infinity" and other boundary value problems. It can be noted that the formulation of the Dirichlet problem on non-compact manifolds may be problematic. One of the possible approaches to solving this problem is based on the introduction of classes of equivalent functions on a non-compact manifold (see [2; 9]). In some cases, geometric compactification of a manifold allows us to do this in the same way as when setting the classical Dirichlet problem in bounded domains Rn (see [3; 6; 7; 11; 15; 16]). The proof of the main results of the work is based on classical statements of the theory of partial differential equations: the maximum principle, the comparison and uniqueness theorems for solutions of linear elliptic differential equations. This work we study the existence of solutions and the solvability of boundary value problems of the equation (9) on non-compact Riemannian manifolds of the following form. Let M be a complete Riemannian manifold without the partial, represented as a union M = B ∪ D, where B — some compact, and the subset D is isometric to the product [r0, +∞) × S1 × S2... × Sk (where r0 > 0, Si — compact Riemannian manifolds without partial) and has the metric Theorem 1. [10] Let the Riemannian manifold M and the right-hand side of the equation (9) be such that Ik < ∞ (Kh< ∞, if c(x) ≡ 0), I1 = ... = Is = = ∞, Li < ∞ for everyone i = s + 1, ..., K, 0 ≤ s ≤ k. Then for any continuous on S1 × S2 × ... × Sk function Φ(0s+1, ..., 0k) there is a unique bounded solution u(x) on M to the equation (9) such that on D holds. Theorem 2. Let the Riemannian manifold M and the coefficients of the equation (10) be such that Ih < ∞, I1 = ... = Is = ∞, Ii < ∞ for everyone i = s+1, ..., k, 0 ≤ s ≤ k. Then for any continuous on S1×S2×...×Sk function Ф(0s+1, ..., 0k) there is a unique bounded solution u(x) on M to the equation (10) such that on D holds.