Schrödinger-type identity to the existence and uniqueness of a solution to the stationary Schrödinger equation

In this article we study applications of the Schrödinger-type identity for obtaining transmutations via the fixed point index for nonlinear integral equations. It is possible to derive a wide range of transmutation operators by this method. Classical Riesz transforms are involved in the Schrödinger-type identity method as basic blocks, among them are Fourier, sine and cosine-Fourier, Hankel, Mellin, Laplace, and some generalized transforms. In this paper, we present a modified Schrödinger-type identity for solutions of a class of linear Schrödinger equations with mixed boundary conditions. The techniques used in our proofs are quite different, and most remarkably some of the proofs become simpler and more straightforward. As an application, we obtain the existence and uniqueness of a solution to the stationary Schrödinger equation in the sense of the Weyl law, which advances the recent results obtained in several articles even in a more general setting.