Low-rank perturbations and the spectral statistics of pseudointegrable billiards.

We present an efficient method to solve Schrödinger's equation for perturbations of low rank. The method is ideally suited for systems with short range interactions or quantum billiards. It involves a secular equation of low dimension, which directly returns the level counting function. For illustration, we calculate the number variance for two pseudointegrable quantum billiards: the barrier billiard and a right triangle billiard. In this way, we obtain precise estimates for the level compressibility in the semiclassical (high energy) limit. In both cases, our results confirm recent theoretical predictions, based on periodic orbit summation, disregarding diffractive orbits.