Mean-field transition temperature of strongly disordered superconductors

I discuss the mean-field theory of superconductivity in a strongly disorder ed system of fermions with short-range attraction. It is argued that in thi s limit the effective theory at low energies is equivalent to the disordere d Bose-Hubbard model, and I consider both the infinite-range and the "neare st-neighbor" hopping of bosons between the localized states. In the infinit e-range case the mean-field theory is exact, and the superconducting gap is uniform in space, while in the latter case, the gap becomes highly non-uni form in space, but, surprisingly, is finite everywhere at T < T-MF. I find that the mean-field transition temperature T-MF > 0 always, and argue that the superconductor-insulator transition at T = 0 in models with net attract ion between fermions is in the universality class of "dirty-bosons".