  Using phase-space complexification, an Initial Value Representation (IVR) for
the semiclassical propagator in position space is obtained as a composition of
inverse Segal-Bargmann (S-B) transforms of the semiclassical coherent state
propagator. The result is shown to be free of caustic singularities and
identical to the Herman-Kluk (H-K) propagator, found ubiquitously in physical
and chemical applications. We contrast the theoretical aspects of this
particular IVR with the van Vleck-Gutzwiller (vV-G) propagator and one of its
IVRs, often employed in order to evade the non-linear "root-search" for
trajectories required by vV-G. We demonstrate that bypassing the root-search
comes at the price of serious numerical instability for all IVRs except the H-K
propagator. We back up our theoretical arguments with comprehensive numerical
calculations performed using the homogeneous Kerr system, about which we also
unveil some unexpected new phenomena, namely: (1) the observation of a clear
mark of half the Ehrenfest's time in semiclassical dynamics; and (2) the
accumulation of trajectories around caustics as a function of increasing time
(dubbed "caustic stickiness"). We expect these phenomena to be more general
than for the Kerr system alone.
