The BCS Critical Temperature for Potentials with Negative Scattering Length

Abstract

We prove that the critical temperature for the BCS gap equation is given by $${T_c = \mu \left( \frac 8\pi {\rm e}^{\gamma -2} + o(1) \right) {\rm e}^{\pi/(2\sqrt \mu a)}}$$ in the low density limit μ→ 0, with γ denoting Euler’s constant. The formula holds for a suitable class of interaction potentials with negative scattering length a in the absence of bound states.