Quantum-classical correspondence and the transition to chaos in coupled quartic oscillators

In this paper we investigate the semiclassical mechanics of a system of two quartic oscillators coupled by a quartic perturbation γ${\mathit{q}}_1^2$ ${\mathit{q}}_2^2$. Our focus is on the evolution of the quantum density of states from the integrable limit (γ=0) to the strongly coupled regime (γ=15.0). In the integrable limit, the Berry-Tabor analysis of the semiclassical density of states in terms of rational tori is appropriate. We extend this analysis to treat the contributions of resonant tori at the boundaries of physical action space. Computation of the power spectrum of the quantum density of states for a sequence of γ values reveals the evolution of the underlying classical periodic orbit structure. The influence of several resonant, symmetric isochronous, and tangent bifurcations on the density of states is identified. Localization of eigenstates in the vicinity of the shortest periodic orbits is also discussed.