  In this article we have investigated the effect of weak random disorder in
the BCS-BEC crossover region. The disorder is included in the mean field
formalism through NSR theory of superconducting fluctuations. A self consistent
numerical solution of the coupled equations involving the superfluid gap
parameter and density as a function of the disorder strength, albeit unaffected
in the BCS phase, yields a depleted order parameter in the BEC regime and an
interesting nonmonotonic behavior of the condensate fraction in the vicinity of
the unitary region, and a gradual depletion thereafter, as the pairing
interaction is continuously tuned across the BCS-BEC crossover. The unitary
regime thus demonstrates a robust paradigm of superfluidity even when the
disorder is introduced. To support the above feature and shed light on a
lingering controversial issue, we have computed the behavior of the sound mode
across the crossover that distinctly reveals a suppression of the sound
velocity. We also find the Landau critical velocity that shows similar
nonmonotonicity as that of the condensate fraction data, thereby supporting a
stable superfluid scenario in the unitary limit.
