# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1705.03718