The BCS energy gap at low density

We show that the energy gap for the BCS gap equation is $$\begin{aligned} \varXi = \mu \left( 8 {\mathrm{e}}^{-2} + o(1)\right) \exp \left( \frac{\pi }{2\sqrt{\mu } a}\right) \end{aligned}$$ Ξ = μ 8 e - 2 + o ( 1 ) exp π 2 μ a in the low density limit $$\mu \rightarrow 0$$ μ → 0 . Together with the similar result for the critical temperature by Hainzl and Seiringer (Lett Math Phys 84: 99–107, 2008), 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.