The Berry-Tabor conjecture,

Abstract. One of the central observations of quantum chaology is that statistical properties of quantum spectra exhibit surprisingly universal features, which seem to mirror the chaotic or regular dynamical properties of the underlying classical limit. I will report on recent studies of simple regular systems, where some of the observed phenomena can be established rigorously. The results discussed are intimately related to the distribution of values of quadratic forms, and in particular to a quantitative version of the Oppenheim conjecture. Quantum Chaos One of the main objectives of quantum chaology is to identify characteristic properties of quantum systems which, in the semiclassical limit, reflect the regular or chaotic features of the underlying classical dynamics. Take for example the geodesic flow on the unit tangent bundle of a compact two-dimensional Riemannian surface M . The corresponding quantum system is described by the stationary Schrödinger equatio