Schroedinger operators involving singular potentials and measure data

TL;DR

This paper characterizes the measures for which the Dirichlet problem involving the Schrödinger operator with measure data admits solutions, depending on the potential's integrability and capacity conditions.

Contribution

It provides a complete characterization of measure data allowing solutions for all potentials in Lebesgue spaces, using capacity theory, and extends results to potentials in Hardy spaces.

Findings

Characterization of measures for solution existence based on capacity.

Solution existence for potentials in Lebesgue and Hardy spaces.

Extension of classical results to singular potentials and measure data.

Abstract

We study the existence of solutions of the Dirichlet problem for the Schroedinger operator with measure data $$ \left\{ \begin{alignedat}{2} -\Delta u + Vu & = \mu && \quad \text{in } \Omega,\\ u & = 0 && \quad \text{on } \partial \Omega. \end{alignedat} \right. $$ We characterize the finite measures $\mu$ for which this problem has a solution for every nonnegative potential $V$ in the Lebesgue space $L^p(\Omega)$ with $1 \le p \le N/2$. The full answer can be expressed in terms of the $W^{2,p}$ capacity for $p > 1$, and the $W^{1,2}$ (or Newtonian) capacity for $p = 1$. We then prove the existence of a solution of the problem above when $V$ belongs to the real Hardy space $H^1(\Omega)$ and $\mu$ is diffuse with respect to the $W^{2,1}$ capacity.