BOUNDARY-VALUE PROBLEMS FOR THE INHOMOGENEOUS SCHR ̈ODINGER EQUATION WITH VARIATIONS OF ITS POTENTIAL ON NON-COMPACT RIEMANNIAN MANIFOLDS

Abstract

We study solutions of the inhomogeneous Schr ̈odinger equation Δ𝑢 − 𝑐(𝑥)𝑢 = 𝑔(𝑥), where 𝑐(𝑥), 𝑔(𝑥) are Holder functions, with variations of its potential 𝑐(𝑥) >= 0 on a noncompact Riemannian manifold 𝑀 . Our technique essentially relies on an approach from the papers by E. A. Mazepa and S. A. Korol’kov connected with introduction of equivalency classes of functions. It made it possible to formulate boundary-value problems on 𝑀 independently from a natural geometric compactification. In the present work, we obtain conditions under which the solvability of boundary-value problems of the inhomogeneous Schr ̈odinger equation is preserved for some variations of the coefficient 𝑐(𝑥) >= 0 on 𝑀 .

Keywords

• inhomogeneous schr ̈odinger equation
• variations of coefficients
• boundary-value problems
• riemannian manifold