Generalized Poisson kernel and solution of the Dirichlet problem for the radial Schr\"odinger equation

Abstract

We present an explicit construction of the solution to the Dirichlet boundary value problem for the radial Schr\"odinger equation in the unit ball, with a complex-valued potential $V$ satisfying the condition $\int_0^1r|V(r)|dr<\infty$. The solution is based on the construction of an explicit orthogonal set of solutions for the radial equation. In the case of a Dirichlet problem with boundary data in $W^{\frac{1}{2},2}(\mathbb{S}^{d-1})$, the solution is expressed as a series expansion in terms of the so-called formal spherical polynomials. We establish conditions for the solvability and uniqueness of the Dirichlet problem. Based on this series representation, we introduce the concept of generalized Poisson kernel, develop its main properties, and investigate the conditions under which the Dirichlet problem, with a boundary condition being a complex Radon measure on $\mathbb{S}^{d-1}$, admits a solution in the sense of a distributional boundary values.