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

TL;DR

This paper constructs explicit solutions for the Dirichlet problem of the radial Schrödinger equation in a unit ball with complex potentials, introducing a generalized Poisson kernel and analyzing solution conditions.

Contribution

It provides an explicit orthogonal set of solutions, series expansion for boundary data, and develops a generalized Poisson kernel for complex boundary measures.

Findings

Explicit solution construction for the Dirichlet problem

Series representation using formal spherical polynomials

Conditions for solvability and uniqueness

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.