Spectral statistics of scale invariant systems

Abstract

We give both numerical and analytical results for the spectral statistics of two dimensional systems with a homegeneous polynomial as potential. When the hamiltonian is time reversal invariant we find a gradual transition from the Poisson ensemble to the GOE going from the classically integrable to the classically chaotic case. When the time reversal symmetry is broken we find the statistics of the GUE. Most features observed in the numerical calculations, with inclusion of the ‘kink’ in the Δ₃ statistic for integrable integrable systems and the asymptotic logarithmic behaviour for chaotic systems, are described by the semiclassical limit of the fluctuating part of the level density. It is shown that the transition to the chaotic regime can be described in the semiclassical limit. Finally, we construct a random matrix model that describes well the short range behaviour of all statistics studied.