Models of intermediate spectral statistics

Abstract

Based on numerical results it is conjectured that the spectral statistics of certain pseudointegrable billiards have a special form similar to that of the Anderson model at the transition point. A simple theoretical model where such statistics can be obtained analytically is briefly discussed. A few models with similar behavior are considered. In particular, we analytically found the eigenvalue statistics of a Poisson-distributed matrix perturbed by a rank one matrix, which is a good model for spectral statistics of a singular billiard.