  The dynamics of molecular motors which occasionally detach from a
heterogeneous track like DNA or RNA is considered. Motivated by recent single
molecule experiments, we study a simple model for a motor moving along a
disordered track using chemical energy while an external force opposes its
motion. The motors also have finite processivity, i.e., they can leave the
track with a position dependent rate. We show that the response of the system
to disorder in the hopping off rate depends on the value of the external force.
For most values of the external force, strong disorder causes the motors which
survive for long times on the track to be localized at preferred positions.
However, near the stall force, localization occurs for any amount of disorder.
Existence of localized states near the top of the band implies a stretched
exponential contribution to the decay of the survival probability. To obtain
these results, we study the complex eigenvalue spectrum of the time evolution
operator. A similar spectral analysis also provides a very efficient method for
studying the dynamics of motors with infinite processivity.
