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BCS-BEC crossover in a trapped Fermi super-fluid using a density-functional equation

We derive a generalized time-dependent Galilean-invariant density-functional (DF) equation appropriate to study the stationary and non-stationary properties of a trapped Fermi super-fluid in the Bardeen-Cooper-Schrieffer (BCS) to Bose-Einstein condensation (BCS) crossover. This equation is equivalent to a quantum hydrodynamical equation for a Fermi super-fluid. The bulk chemical potential of this equation has the proper (model-independent) dependence on the Fermi-Fermi scattering length in the BCS and BEC limits. We apply this DF equation to the study of stationary density profile and size of a cigar-shaped Fermi super-fluid of $^6$Li atoms and the results are in good agreement with the experiment of Bartenstein {\it et al.} in the BCS-BEC crossover. We also apply the DF equation to the study of axial and radial breathing oscillation and our results for these frequencies are in good agreement with experiments in the BCS-BEC crossover.