We show that the energy gap for the BCS gap equation is

$$\begin{aligned} \varXi = \mu \left( 8 {\mathrm{e}}^{-2} + o(1)\right) \exp \left( \frac{\pi }{2\sqrt{\mu } a}\right) \end{aligned}$$

in the low density limit \(\mu \rightarrow 0\). 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.

中文翻译：

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低密度时的BCS能隙

我们证明BCS间隙方程的能隙为

$$ \ begin {aligned} \ varXi = \ mu \ left（8 {\ mathrm {e}} ^ {-2} + o（1）\ right）\ exp \ left（\ frac {\ pi} {2 \ sqrt {\ mu} a} \ right）\ end {aligned} $$

在低密度限制\（\ mu \ rightarrow 0 \）中。连同Hainzl和Seiringer对临界温度的类似结果（Lett Math Phys 84：99–107，2008），这表明，在低密度极限下，能隙与临界温度之比是一个独立的通用常数的相互作用势V。结果适用于一类具有负散射长度a且无束缚态的电势。