  The time-dependent transmission coefficient for the Kramers problem exhibits
different behaviors in different parameter regimes. In the high friction regime
it decays monotonically ("non-adiabatic"), and in the low friction regime it
decays in an oscillatory fashion ("energy-diffusion-limited"). The generalized
Kramers problem with an exponential memory friction exhibits an additional
oscillatory behavior in the high friction regime ("caging"). In this paper we
consider an oscillatory memory kernel, which can be associated with a model in
which the reaction coordinate is linearly coupled to a nonreactive coordinate,
which is in turn coupled to a heat bath. We recover the non-adiabatic and
energy-diffusion-limited behaviors of the transmission coefficient in
appropriate parameter regimes, and find that caging is not observed with an
oscillatory memory kernel. Most interestingly, we identify a new regime in
which the time-dependent transmission coefficient decays via a series of rather
sharp steps followed by plateaus ("stair-like"). We explain this regime and its
dependence on the various parameters of the system.
