  We theoretically explore the crossover from three dimensions (3D) to two (2D)
in a strongly interacting atomic Fermi superfluid through confining the
transverse spatial dimension. Using the gaussian pair fluctuation theory, we
determine the zero-temperature equation of state and Landau critical velocity
as functions of the spatial extent of the transverse dimension and interaction
strength. In the presence of strong interactions, we map out a dimensional
crossover diagram from the location of maximum critical velocity, which
exhibits distinct dependence on the transverse dimension from 2D to quasi-2D,
and to 3D. We calculate the dynamic structure factor to characterize the
low-energy excitations of the system and propose that the intermediate quasi-2D
regime can be experimentally probed using Bragg spectroscopy.
