A new application of Schrödinger-type identity to singular boundary value problem for the Schrödinger equation

Abstract

Abstract In this paper, we present a modified Schrödinger-type identity related to the Schrödinger-type boundary value problem with mixed boundary conditions and spatial heterogeneities. This identity can be regarded as an L1 $L^{1}$-version of Fisher–Riesz’s theorem and has a broad range of applications. Using it and fixed point theory in L1 $L^{1}$-metric spaces, we prove that there exists a unique solution for the singular boundary value problem with mixed boundary conditions and spatial heterogeneities. We finally provide two examples, which show the effectiveness of the Schrödinger-type identity method.

Keywords

• Boundary value problem
• Schrödinger-type identity
• Uniqueness