“…Analyzing finite range interactions is of relevance since, while indeed, the BEC-BCS crossover has been identified in several systems, mostly in ultracold atomic clouds [23,24,[57][58][59][60], in some others the effective interaction is fixed and the crossover is tuned by means of the density [23,24,28,38,61] and, while the strength of the interaction is still characterized by the s-wave scattering length, it has been observed that next leading-order contributions, such as the effective range, are important for the density-induced BEC-BCS crossover, even at the mean-field level [28,38,62]. At any rate, the main requirement is the use of a finite-range interaction, a feature of the present study, and which has been known to pose a very difficult problem for solving the gap equation [62].…”