Local Level Velocity Variances and the Fidelity in Integrable Systems

In a chaotic system, a wave packet's Wigner transform will ergodically explore the entire energy surface on an extremely short time scale. However, for integrable systems the Wigner transform will remain on constant action surfaces and can access only a fraction of the available phase space. Thus, the local properties of integrable systems can be more important than the global properties. An example occurs when a system is perturbed and the energy levels are redistributed. Using first order quantum perturbation theory, this redistribution is largely responsible for changes in the evolution of the wave packet. More specifically, it is the local variance of the level velocities, which are defined by the changes in the eigenenergies due to the perturbation, that defines the rate of decoherence between the same initial wave packet evolved through the unperturbed and perturbed systems. We derive a semiclassical expression for these local variances and show an application of the variances to the fidelity. The results are demonstrated in the rectangle billiard which is fully integrable.