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 $\mathrm{\Xi }$ (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 $\lambda \ll 1$ and under appropriate assumptions on $V(x)$, our bounds show that $T_c\sim A\exp (-B\mathrm{/}\lambda )$ and $\mathrm{\Xi }\sim C\exp (-B\mathrm{/}\lambda )$ for some explicit coefficients $A$, $B$ and $C$ depending on the interaction $V(x)$ and the chemical potential $\mu$. The ratio $A\mathrm{/}C$ 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 $V(x)$.