Title: Inclusion of pairing fluctuations in the differential equation for the gap parameter for superfluid fermions in the presence of nontrivial spatial constraints

Abstract: Most theoretical treatments of inhomogeneous superconductivity/fermionic superfluidity have been based on the Bogoliubov-deGennes equations (or, else, on their various simplified forms), which implement a standard mean-field decoupling in the presence of spatial inhomogeneities. This approach is reliable even at finite temperature for weak inter-particle attraction, when the Cooper pair size is much larger than the average inter-particle distance (corresponding to the BCS limit of the BCS-BEC crossover). However, it looses accuracy for increasing attraction when the Cooper pair size becomes comparable or even smaller than the average inter-particle distance (corresponding to the BEC limit of the BCS-BEC crossover), in particular when finite-temperature effects are considered. In these cases, inclusion of pairing fluctuations beyond mean field is required, a task that turns out to be especially difficult in the presence of inhomogeneities. Here, we implement the inclusion of pairing fluctuations directly on a coarse-graining version of the Bogoliubov-deGennes equations, which makes it simpler and faster to obtain a solution over the whole sector of the temperature-coupling phase diagram of the BCS-BEC crossover in the broken-symmetry phase. We apply this method in the presence of a super-current flow, such that problems related to the Josephson effect throughout the BCS-BEC crossover can be addressed under a variety of circumstances. This is relevant in the view of recent experimental data with ultra-cold Fermi atoms, to which the results of the present approach favorably compare.