Schroedinger operators involving singular potentials and measure data

We study the existence of solutions of the Dirichlet problem for the Schroedinger operator with measure data $$\{\begin{array}{rlrl}{\displaystyle -\mathrm{\Delta }u+Vu}& {\displaystyle =\mu }& {\displaystyle }& {\displaystyle \text{in }\mathrm{\Omega },}\\ {\displaystyle u}& {\displaystyle =0}& {\displaystyle }& {\displaystyle \text{on }\mathrm{\partial }\mathrm{\Omega }\mathrm{.}}\end{array}$$ We characterize the finite measures $\mu$ for which this problem has a solution for every nonnegative potential $V$ in the Lebesgue space $L^p(\mathrm{\Omega })$ with $1\le p\le N\mathrm{/}2$. 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 $H^1(\mathrm{\Omega })$ and $\mu$ is diffuse with respect to the $W^{2,1}$ capacity.