  We consider Hermitian random matrices of the form $H = W + \lambda V$, where
$W$ is a Wigner matrix and $V$ a diagonal random matrix independent of $W$. We
assume subexponential decay for the matrix entries of $W$ and we choose
$\lambda \sim 1$ so that the eigenvalues of $W$ and $\lambda V$ are of the same
order in the bulk of the spectrum. In this paper, we prove for a large class of
diagonal matrices $V$ that the local deformed semicircle law holds for $H$,
which is an analogous result to the local semicircle law for Wigner matrices.
We also prove complete delocalization of eigenvectors and other results about
the positions of eigenvalues.
