ARITHMETICAL CHAOS

Free motion on constant negative curvature surfaces with finite area g ives rise to some of the best models for studying the quantum behavior of classically chaotic systems. Quite surprisingly, the results of nu merical computations of quantum spectra for many such systems show a c lear deviation from the predictions of random matrix theory. We show t hat such an anomaly is a property of a peculiar subclass of constant n egative curvature models, namely the ones generated by the so-called a rithmetic groups. A comprehensive review of these systems is presented . It is shown that arithmetical properties inherent in these models le ad to an exponential degeneracy of the lengths of periodic orbits. Thi s, using semiclassical formulas for the correlation functions, implies that the energy-level statistics are closer to the Poisson distributi on typical of integrable systems than to any standard random matrix di stribution typical of chaotic systems. A characteristic property of ar ithmetic systems is the existence of an infinite set of commuting oper ators of purely arithmetical origin. These pseudosymmetries allow one to build an exact Selberg-type trace formula giving not only the energ y levels, but also the wavefunctions in terms of the periodic orbits. This formula is derived in detail for a specific case, the modular bil liard with Dirichlet boundary conditions, and its relevance is checked numerically. Some results of the investigation of non-arithmetic mode ls are also discussed. (C) 1997 Elsevier Science B.V.