The BCS Energy Gap at Low Density

We show that the energy gap for the BCS gap equation is $\Xi = \mu \left( 8 e^{-2} + o(1)\right) \exp\left( \frac{\pi}{2\sqrt{\mu} a}\right)$ in the low density limit $\mu \to 0$. Together with the similar result for the critical temperature [arXiv:0803.3324] this shows that, in the low density limit, the ratio of the energy gap and critical temperature is a universal constant independent of the interaction potential $V$. The results hold for a class of potentials with negative scattering length $a$ and no bound states.