  We calculate the power spectrum of curvature perturbations when the inflaton
field is rolling over the top of a local maximum of a potential. We show that
the evolution of the field can be decomposed into a late-time attractor, which
is identified as the slow roll solution, plus a rapidly decaying non-slow roll
solution, corresponding to the field rolling ``up the hill'' to the maximum of
the potential. The exponentially decaying transient solution can map to an
observationally relevant range of scales because the universe is also expanding
exponentially. We consider the two branches separately and we find that they
are related through a simple transformation of the slow roll parameter $\eta$
and they predict identical power spectra. We generalize this approach to the
case where the inflaton field is described by both branches simultaneously and
find that the mode equation can be solved exactly at all times. Even though the
slow roll parameter $\eta$ is evolving rapidly during the transition from the
transient solution to the late-time attractor solution, the resultant power
spectrum is an exact power-law spectrum. Such solutions may be useful for
model-building on the string landscape.
