  The three-disk system, which for many years has served as a paradigm for the
usefulness of cycle expansion methods, represents an extremely hard problem to
semiclassical quantization when the disks are moved closer and closer together,
since (1) pruning of orbits sets in, rendering the symbolic code incomplete,
and (2) the number of orbits necessary to obtain accurate semiclassical
eigenvalues proliferates exponentially. In this note we show that an
alternative method, viz. harmonic inversion, which does not rely on the
existence of complete symbolic dynamics or other specific properties of
systems, provides a key to solving the problem of semiclassical quantization of
systems with strong pruning. For the closed three-disk system we demonstrate
how harmonic inversion, augmented by a signal cross-correlation technique,
allows one to semiclassically calculate the energies up to the 28th excited
state with high accuracy.
