  We present a semiclassical theory for the excitation spectrum of a ballistic
quantum dot weakly coupled to a superconductor, for the generic situation that
the classical motion gives rise to a phase space containing islands of
regularity in a chaotic sea. The density of low-energy excitations is
determined by quantum energy scales that are related in a simple way to the
morphology of the mixed phase space. An exact quantum mechanical computation
for the annular billiard shows good agreement with the semiclassical
predictions, in particular for the reduction of the excitation gap when the
coupling to the regular regions is maximal.
