Nearest-neighbor distribution for singular billiards.

Abstract

The exact computation of the nearest-neighbor spacing distribution P(s) is performed for a rectangular billiard with a pointlike scatterer inside for periodic and Dirichlet boundary conditions, and it is demonstrated that when s-->infinity this function decreases exponentially. Together with the results of Bogomolny, Gerland, and Schmit [Phys. Rev. E 63, 036206 (2001)], it proves that spectral statistics of such systems is of intermediate type characterized by level repulsion at small distances and exponential fall-off of the nearest-neighbor distribution at large distances. The calculation of the nth nearest-neighbor spacing distribution P(n)(s) and its asymptotics is performed as well for any boundary conditions.