1 Nearest - neighbor distribution for singular billiards

The exact computation of the nearest-neighbor spacing distribution P (s) is performed for a rectangular billiard with point-like scat-terer inside for periodic and Dirichlet boundary conditions and it is demonstrated that when s → ∞ this function decreases exponentially. Together with the results of Ref. [13] 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 n-th nearest-neighbor spacing distribution P n (s) and its asymptotics is performed as well for any boundary conditions.