= BCS gap equation
{c}
{title2=$\Delta_k=-\sum_{k^\prime}V_{kk^\prime}\Delta_{k^\prime}\tanh(E_{k^\prime}/(2k_BT))/(2E_{k^\prime})$}
{wiki}
The <BCS gap equation> makes the <superconducting gap function> self-consistent with the <BCS anomalous average> that creates it. For a translation-invariant spin-singlet <mean-field approximation>, define $\xi_k=\epsilon_k-\mu$ and $E_k=\sqrt{\xi_k^2+|\Delta_k|^2}$. Then
$$
\Delta_k=-\sum_{k'}V_{kk'}\frac{\Delta_{k'}}{2E_{k'}}
\tanh\!\left(\frac{E_{k'}}{2k_BT}\right).
$$
The interaction $V_{kk'}$ and its normalization/cutoff are part of the model. A nonzero solution describes pairing; $\Delta_k=0$ is always a solution, so finding any solution alone does not establish which state minimizes the <free energy>.
For constant attraction $-V$ in a symmetric shell $|\xi|<\Lambda$ with constant single-spin <density of states> $\nu_s$, the zero-<temperature> equation reduces to
$$
1=\nu_sV\int_0^\Lambda\frac{d\xi}{\sqrt{\xi^2+\Delta^2}},\qquad
\Delta=\frac{\Lambda}{\sinh[1/(\nu_sV)]}.
$$
At <weak coupling>, this gives $\Delta\simeq2\Lambda e^{-1/(\nu_sV)}$. The exponential explains how a small attraction produces an extremely small pairing scale, matching the <Cooper instability>. The finite-<temperature> equation can be linearized as the gap tends to zero to determine the mean-field transition. <Theory of Superconductivity by John Bardeen and others>[Sections II–III, variational paired state and finite-temperature self-consistency; notation and shell reduction stated here]{ref}
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