Random matrix theory in semiclassical quantum mechanics of chaotic systems

Abstract

The statistical properties of the spectrum of systems which have a chaotic classical limit have been found to be similar to those of random matrix ensembles. The author explains this correspondence, including the fact that long-ranged spectral statistics show deviations from the results of random matrix theory. The method depends on the ambiguity of quantisation of a given classical system; although the energy levels depend on the particular quantisation used, the spectral statistics are assumed to depend only on the classical motion. The ambiguity of quantisation can be represented by a small perturbation acting on the Hamiltonian. This perturbation executes a random walk, which causes its matrix elements to undergo a random walk. If the matrix elements are completely uncorrelated, the spectral statistics are those of random matrix theory. In the case of classically chaotic systems, the matrix elements are constrained by sum rules related to classical periodic orbits. This leads to deviations of the long-ranged spectral statistics from the predictions of random matrix theory.