Energy-level statistics of the Hadamard-Gutzwiller ensemble

Abstract

The statistical properties of the quantal energy levels of the Hadamard-Gutzwiller ensemble - whose classical members belong to the class of systems with hard chaos - are investigated. Based on a sample of 4500 energy levels, it is shown that the short-range statistics as nearest-neighbour spacing distributions are governed by the GOE predictions of random-matrix theory, which was first surmised by Wigner and by Landau and Smorodinsky for nuclear level statistics. This result strengthens the hypothesis that quantum systems with chaotic classical counterpart display level repulsion as predicted by random-matrix theory. However, the level statistics describing correlations over greater level distances deviate from the GOE predictions, which is explained as a simple consequrnce of the fact that the spectral rigidity Δ 3 ( L) introduced by Dyson and Mehta saturates non-universally at a finite value Δ ∞ for L → Δ in complete agreement with the semiclassical theory developed by Berry.