Statistical properties of random banded matrices with strongly fluctuating diagonal elements

Random banded matrices (RBM’s) whose diagonal elements fluctuate more than the off-diagonal elements were introduced recently by Shepelyansky as a convenient means to model the coherent propagation of two interacting particles in a random potential. We treat the problem analytically by using a mapping onto the same supersymmetric nonlinear σ model that was used earlier when considering standard RBM ensemble, but with renormalized parameters. A Lorentzian form of the local density of states and a two-scale spatial structure of the eigenfunctions presented recently by Jacquod and Shepelyansky are reproduced by direct calculation of the distribution of eigenfunction components.