Manifestation of classical chaos in the statistics of quantum energy levels

We investigate the quantum energy-level sequence of two coupled quartic oscillators. The classical system is a scaling one and has been shown elsewhere to undergo a transition from regular to irregular motion for increasing coupling strength. The distribution of spacings between adjacent energy levels P(S) and the Δ₃ statistic indicate a corresponding transition from an uncorrelated spectrum where successive levels arrive randomly to a spectrum described by the Gaussian orthogonal ensemble of random matrices. For P(S) we use Berry and Robnik's semiclassical formulas which depend on the volumes of regions in phase space filled with chaotic trajectories and the extension of this ansatz to the Δ₃ statistic. We investigate systematic deviations from this picture due to quantum effects and discuss their universality. We find that P(S) is mainly sensitive to the total irregular fraction of phase space, whereas Δ₃ strongly reflects its partitioning into several parts. For vanishing coupling strength, where the system becomes integrable, we find extremely slow convergence to the semiclassical limit.