= Positive-energy quasiparticle density from a folded spectrum
{title2=$D_+(E)=\sum_{\xi:E(\xi)=E}g(\xi)|d\xi/dE|$}
A many-to-one change of energy variable requires summing every inverse branch. For $E(\xi)=\sqrt{\xi^2+\Delta^2}$ and constant normal <density of states> $g_F$ on both sides of zero, there are two roots for $E>\Delta$, giving $D_+(E)=2g_FE/\sqrt{E^2-\Delta^2}$. A single branch instead gives $g_FE/\sqrt{E^2-\Delta^2}$. The distinction persists when $\Delta\to0$: folding $\xi$ into $|\xi|$ doubles a two-sided constant density on the positive axis. Spin degeneracy is a separate normalization choice; it must be applied consistently to both normal and transformed densities. The <BCS electronic density of states> has the single-factor expression because it is spectrally weighted rather than an unweighted branch count.
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