The BCS energy gap at low density

Abstract

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.

The BCS Critical Temperature at High Density

“…for fixed κ > 0, which was done in a similar way for the low-density and weak-coupling limit of the critical temperature and the energy gap (see [9,10,12,18]). Indeed, using Lemma 1 from [9] one can easily see that…”

“…Since , the multiplication with is a bounded operator from to , and hence is a bounded operator from to . A further important ingredient in our proofs is to study the asymptotic behavior of for fixed , which was done in a similar way for the low-density and weak-coupling limit of the critical temperature and the energy gap (see [ 9 , 10 , 12 , 18 ]). Indeed, using Lemma 1 from [ 9 ] one can easily see that as long as .…”

“…BCS theory has previously been studied in the weak-coupling limit [ 6 , 9 ] and low-density limit [ 10 , 18 ]. In the weak-coupling limit one considers a potential for a fixed potential V for small coupling constant .…”

The BCS Critical Temperature at High Density

“…for fixed κ > 0, which was done in a similar way for the low-density and weak-coupling limit of the critical temperature and the energy gap (see [10,11,13,19]). Indeed, using Lemma 1 from [10] one can easily see that…”

“…BCS theory has previously been studied in the weak-coupling limit [6,10] and lowdensity limit [11,19]. In the weak-coupling limit one considers a potential λV for a fixed potential V for small coupling constant λ → 0.…”