  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.
