Abstract: 
We derive upper and lower bounds on the critical temperature Tc and the energy gap Ξ (at zero temperature) for the BCS gap equation, describing spin 1 2 fermions interacting via a local twobody interaction potential λV(x). At weak coupling λ 1 and under appropriate assumptions on V(x), our bounds show that Tc ∼A exp(B/λ) and Ξ∼C exp(B/λ) for some explicit coefficients A, B, and C depending on the interaction V(x) and the chemical potential μ. The ratio A/C turns out to be a universal constant, independent of both V(x) and μ. Our analysis is valid for any μ; for small μ, or low density, our formulas reduce to wellknown expressions involving the scattering length of V(x).
