Analytic solution of the BCS gap equation inD 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 andT _c. We work in the weak coupling limit, but allow the interaction widthν≡ħω _m/E _F to lie in the interval (0, ∞) Here,ħω _m is the maximum energy of a force-mediating boson, andE _F denotes the Fermi energy. We obtain expressions forT _c and ΔC, the jump in the electronic specific heat acrossT=T _c, in the limitsν≪1 (the usual phonon pairing) andν>1 (non-phononic pairing). This enables us to see howT _c scales with the mediating boson cut off. Our results predict a larger jump in the specific heat for the caseν>1, compared toν≪1. We also briefly touch upon the role of a van Hove singularity in the density of states.