Arithmetical chaos

Abstract

Free motion on constant negative curvature surfaces with finite area gives rise to some of the best models for studying the quantum behavior of classically chaotic systems. Quite surprisingly, the results of numerical computations of quantum spectra for many such systems show a clear deviation from the predictions of random matrix theory. We show that such an anomaly is a property of a peculiar subclass of constant negative curvature models, namely the ones generated by the so-called arithmetic groups. A comprehensive review of these systems is presented. It is shown that arithmetical properties inherent in these models lead to an exponential degeneracy of the lengths of periodic orbits. This, using semiclassical formulas for the correlation functions, implies that the energy-level statistics are closer to the Poisson distribution typical of integrable systems than to any standard random matrix distribution typical of chaotic systems. A characteristic property of arithmetic systems is the existence of an infinite set of commuting operators of purely arithmetical origin. These pseudosymmetries allow one to build an exact Selberg-type trace formula giving not only the energy levels, but also the wavefunctions in terms of the periodic orbits. This formula is derived in detail for a specific case, the modular billiard with Dirichlet boundary conditions, and its relevance is checked numerically. Some results of the investigation of non-arithmetic models are also discussed.