Critical temperature and energy gap for the BCS equation

We derive upper and lower bounds on the critical temperature T-c and the energy gap Xi (at zero temperature) for the BCS gap equation, describing spin-1/2 fermions interacting via a local two-body interaction potential lambda V(x). At weak coupling lambda << 1 and under appropriate assumptions on V(x), our bounds show that T-c similar to A exp(-B/lambda) and Xi similar to C exp(-B/lambda) for some explicit coefficients A, B, and C depending on the interaction V(x) and the chemical potential mu. The ratio A/C turns out to be a universal constant, independent of both V(x) and mu. Our analysis is valid for any mu; for small mu, or low density, our formulas reduce to well-known expressions involving the scattering length of V(x).