  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.
