Reliability of small matrices for large spectra with nonuniversal fluctuations

Abstract

We study the change of quantum spectra under two types of perturbations. One of them corresponds to the breaking of classical integrability and amounts to a crossover from level clustering to level repulsion. The second type of perturbation breaks time-reversal invariance; under conditions of classical chaos the degree of level repulsion then grows from linear to quadratic. To characterize the spectral changes we propose, for each type of transition, a distribution of nearest-neighbor spacings. Both of these ‘‘generalized Wigner surmises’’ are rigorous for suitable ensembles of 2×2 matrices but prove reliable for dynamical systems with many levels.