Analytic solution of the BCS gap equation in D dimensions (D=1, 2, 3), at finite temperatures

The Bardeen-Cooper-Schrieffer (BCS) gap equation is solved analytically in one, two and three dimensions, for temperatures close to zero and T-c. We work in the weak coupling limit, but allow the interaction width nu = (h) over bar omega(m)/E(F) to lie in the interval (0, infinity) Here, (h) over bar omega(m) is the maximum energy of a force-mediating boson, and E(F) denotes the Fermi energy. We obtain expressions for T-c and Delta C, the jump in the electronic specific heat across T = T-c, in the limits nu much less than 1 (the usual phonon pairing) and nu > 1 (non-phononic pairing). This enables us to see how T-c scales with the mediating boson cut off. Our results predict a larger jump in the specific heat for the case nu > 1, compared to nu much less than 1. We also briefly touch upon the role of a van Hove singularity in the density of states.