A note on the level spacings distribution of the Hamiltonians in the transition region between integrability and chaos

Abstract

It is well known that there is no strict universality of the spectral fluctuations of quantum Hamiltonians whose classical counterparts undergo the transition from integrability to complete chaos. The author discusses the level spacings distribution P(S), and explains why the semiclassical formulae of Berry and Robnik cannot be correct for small S. There is no global universality of P(S) for nearly integrable systems, but the approach to the integrability as the perturbation parameter in goes to zero can be universal. This is reflected in the fact that the slope of dP/dS at S=0 for small in is universally inversely proportional to in . The author gives two models in terms of two-dimensional random matrices, one of them being based on maximum entropy considerations. The author also points out the connection to the statistics of zeros of random functions, and discusses the numerical evidence.