Solutions of the Dirichlet-Sch problem and asymptotic properties of solutions for the Schrödinger equation

Let a(X,y) be a nonnegative radical potential in a cylinder. In this paper, we study the solutions of the Dirichlet-Sch problem associated with a stationary Schrödinger operator. Existence of solutions for the Schrödinger equation and asymptotic properties as y→+∞ and y→−∞ are also considered under suitable conditions.