= BCS electronic density of states
{c}
{title2=$N_s(E)=g_F|E|/\sqrt{E^2-\Delta^2}$}
For a particle-hole-symmetric normal <density of states> $g_F$ and isotropic <superconducting energy gap> $\Delta>0$, the electron spectral density is zero for $|E|<\Delta$ and has the displayed form outside the gap. The <BCS coherence factors> weight the two roots $\xi=\pm\sqrt{E^2-\Delta^2}$, where $\xi$ is normal-state energy relative to the <Fermi energy>. At positive $E$, their electron weights $(1+\xi/E)/2$ sum to one. Multiplying their sum by the Jacobian $|d\xi/dE|=E/\sqrt{E^2-\Delta^2}$ proves the formula. This electron spectral density differs from the unweighted count of all positive-energy <quasiparticles>, which receives both roots. Its gap-edge divergence is integrable and approaches the normal electronic density at large $|E|$.
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