  We analytically calculate the local density of states for Cauchy random band
matrices with strongly fluctuating diagonal elements. The Breit-Wigner form for
ordinary band matrices is replaced by a Levy distribution of index $\mu=1/2$
and the characteristic energy scale $\alpha$ is strongly enhanced as compared
to the Breit-Wigner width. The unperturbed eigenstates decay according to the
non-exponential law $\propto e^{-\sqrt{\alpha t}}$. We analytically determine
the localization length by a new method to derive the supersymmetric non-linear
$\sigma$ model for this type of band matrices.
