  Some fraction of the material ejected in a core collapse supernova explosion
may remain bound to the compact remnant, and eventually turn around and fall
back. We show that the late time (> days) power associated with the accretion
of this "fallback" material may significantly affect the optical light curve,
in some cases producing super-luminous or otherwise peculiar supernovae. We use
spherically symmetric hydrodynamical models to estimate the accretion rate at
late times for a range of progenitor masses and radii and explosion energies.
The accretion rate onto the proto-neutron star or black hole decreases as Mdot
~ t^-5/3 at late times, but its normalization can be significantly enhanced at
low explosion energies, in very massive stars, or if a strong reverse shock
wave forms at the helium/hydrogen interface in the progenitor. If the resulting
super-Eddington accretion drives an outflow which thermalizes in the outgoing
ejecta, the supernova debris will be re-energized at a time when photons can
diffuse out efficiently. The resulting light curves are different and more
diverse than previous fallback supernova models which ignored the input of
accretion power and produced short-lived, dim transients. The possible outcomes
when fallback accretion power is significant include super-luminous (> 10^44
ergs / s) Type II events of both short and long durations, as well as luminous
Type I events from compact stars that may have experienced significant mass
loss. Accretion power may unbind the remaining infalling material, causing a
sudden decrease in the brightness of some long duration Type II events. This
scenario may be relevant for explaining some of the recently discovered classes
of peculiar and rare supernovae.
