  We study a generalized resolvent for the generator of a Markovian semigroup.
The Markovian generator appears in a linear Boltzmann equation modeling a
one-dimensional test particle in a periodic potential and colliding elastically
with particles from an ideal background gas. We obtain bounds for the
state-modulated resolvent which are relevant in the regime where the mass ratio
between the test particle and a particle from the gas is large. These bounds
relate to the typical amount of time that the particle spends in different
regions of phase space before arriving to a region around the origin.
