Schroedinger operators involving singular potentials and measure data

Abstract

We study the existence of solutions of the Dirichlet problem for the Schroedinger operator with measure data $$\{\begin{array}{rlrl}-\Delta u+Vu& =\mu & & \text{in }\Omega ,\\ u& =0& & \text{on }\partial \Omega .\end{array}$$ 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.

Fixed a display problem in arxiv's abstract. Original tex file unchanged

References in corpus (1)

• Inverse Problem for a Curved Quantum Guide

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