Signatures of Quantum Chaos

Abstract

Ten years ago, the statistical properties of quantum levels of chaotic systems and integrable systems were described by random matrix theory (RMT) and all was well; v., Boghigas, Giannoni and Schmit (1986) for the Sinai billiard and the stadium billiard, Berry and Tabor (1977) for integrable systems; also note, McDonald and Kaufmann (1988), Berry (1981, 1985), Berry and Robnick (1986), Seligman, Verbarshoot and Zirnbauer (1985) and Berry and Mondragon (1987). Subsequently, problems arose with this description and it was determined that quantum chaotic systems should be characterized as belonging to two classes, arithmetical or generic; (v. Schmit (1991)). The arithmetical class exhibits neither level repulsion nor spectral rigidity with spectrum which is asymptotically Poisson in nature, although the billiard is a strongly chaotic system. Here arithmetical chaotic systems were noted to have exponentially growing multiplicities of lengths of periodic orbits. Schmit also observed that the spectra of the billiard on the modular domain, i.e., the noncompact triangle with angles (π/2, π/3, 0), has spectral statistics similar to Poisson.