SENSITIVITY OF THE EIGENFUNCTIONS AND THE LEVEL CURVATURE DISTRIBUTION IN QUANTUM BILLIARDS

In searching for the manifestations of the sensitivity of the eigenfun ctions in quantum billiards (with Dirichlet boundary conditions) with respect to the boundary data (the normal derivative), we have performe d instead various numerical tests for the Robnik billiard (a quadratic conformal map of the unit disk) for 600 shape parameter values, where we have looked at the sensitivity of the energy levels with respect t o the shape parameter. We show the energy level flow diagrams for thre e stretches of fifty consecutive (odd) eigenstates, each with index 10 00 to 2000. In particular, we have calculated the (unfolded and normal ized) level curvature distribution and found that it continuously chan ges from a delta distribution for the integrable case (circle) to a br oad distribution in the classically ergodic regime. For some shape par ameters the agreement with the GOE von Oppen formula is very good, whe reas we have also cases where the deviation from GOE is significant an d of physical origin. In the intermediate case of mixed classical dyna mics 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 s ystems there is no global universality for the curvature distribution, not even in the semiclassical limit.