Sensitivity of the eigenfunctions and the level curvature distribution in quantum billiards

Abstract

In searching for the manifestations of sensitivity of the eigenfunctions in quantum billiards (with Dirichlet boundary conditions) with respect to the boundary data (the normal derivative) we have performed instead various numerical tests for the Robnik billiard (quadratic conformal map of the unit disk) for 600 shape parameter values, where we look at the sensitivity of the energy levels with respect to the shape parameter. We show the energy level flow diagrams for three stretches of fifty consecutive (odd) eigenstates each with index 1,000 to 2,000. In particular, we have calculated the (unfolded and normalized) level curvature distribution and found that it continuously changes from a delta distribution for the integrable case (circle) to a broad distribution in the classically ergodic regime. For some shape parameters the agreement with the GOE von Oppen formula is very good, whereas we have also cases where the deviation from GOE is significant and of physical origin. In the intermediate case of mixed classical dynamics we have a semiclassical formula in the spirit of the Berry-Robnik (1984) surmise. Here the agreement with theory is not good, partially due to the localization phenomena which are expected to disappear in the semiclassical limit. We stress that even for classically ergodic systems there is no global universality for the curvature distribution, not even in the semiclassical limit.