Arithmetical chaos and violation of universality in energy level statistics

Abstract

A class of strongly chaotic systems revealing a strange arithmetical structure is discussed whose quantal energy levels exhibit level attraction rather than repulsion. As an example, the nearest-neighbor level spacings for Artin's billiard have been computed in a large energy range. It is shown that the observed violation of universality has its root in the existence of an infinite number of Hermitian operators (Hecke operators) which commute with the Hamiltonian and generate nongeneric correlations in the eigenfunctions.