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The BCS anomalous average is the pair expectation $⟨b_k⟩=⟨c_{-k\downarrow }{c}_{k\uparrow }⟩$. In the BCS ground state it is $u_k{v}_k={\Delta }_k/(2E_k)$ for real gap at zero temperature. At temperature $T$ it becomes ${\Delta }_k\tanh (E_k/(2k_{B}T))/(2E_k)$. Substitution into the pairing-field definition gives the BCS gap equation.

In each factor of the BCS ground state, the BCS anomalous average is $⟨b_k⟩=u_k{v}_k={\Delta }_k/(2E_k)$. Self-consistency therefore gives the zero-temperature BCS gap equationPut ${\Omega }_D=\hslash {\omega }_D$. The specified constant attractive interaction makes ${\Delta }_k$ independent of $k$ inside the energy shell and zero outside it:The printed ${\delta }_{k,0}$ is incompatible with this interaction: $k$ labels relative pair momentum, not the total momentum of a Cooper pair. Every pair here has zero total momentum, while many relative momenta contribute.

For the nonzero solution, the constant-shell BCS gap equation becomes $1=(V/L^3){\sum }_{|{\xi }_k|<{\Omega }_D}(2\sqrt{{\xi }_k^2+{\Delta }^2}{)}^{-1}$. Let ${\nu }_s$ denote the approximately constant single-spin density of states per unit volume at the Fermi level. ThenIn weak coupling, $\Delta \simeq 2\hslash {\omega }_D{e}^{-1/({\nu }_{s}V)}$. The displayed answer in the question uses $\nu ={\nu }_s$. If “total electronic density of states” includes both spin species, ${\nu }_{\mathrm{tot}}=2{\nu }_s$, the argument instead reads $2/({\nu }_{\mathrm{tot}}V)$. If the density counts the whole box rather than unit volume, divide it by $L^3$ before using this formula.

Put $\xi =v_F(k)\ell$ and use the same ${\nu }_F$. With a symmetric cutoff $|\xi |<\Lambda$, the BCS gap equation becomesThe frequency integral is $1/[2\sqrt{{\xi }^2+{\Delta }^2}]$, soThusThe gap and the one-loop strong-coupling scale have the same nonperturbative exponential dependence; their order-one prefactors depend on the cutoff convention and microscopic completion. The renormalization-group divergence is therefore the normal-state signal of the paired, gapped phase found from the self-consistent gap equation.

The single-spin density of states counts momentum states for one spin species. For degenerate up and down species, the spin-summed density of states is twice this value. Specify separately whether the density is per unit volume or extensive. In the BCS gap equation, the momentum sum labels opposite-spin pairs once and therefore uses the single-spin density per volume when the interaction has normalization $V/\mathcal{V}$.