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 μ=1/2
and the characteristic energy scale α is strongly enhanced as compared
to the Breit-Wigner width. The unperturbed eigenstates decay according to the
non-exponential law ∝e−αt. We analytically determine
the localization length by a new method to derive the supersymmetric non-linear
σ model for this type of band matrices.Comment: 4 pages, 1 figur