 Positive-energy quasiparticle density from a folded spectrum

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 $D_{+}(E)=2g_{F}E/\sqrt{E^2-{\Delta }^2}$. A single branch instead gives $g_{F}E/\sqrt{E^2-{\Delta }^2}$. The distinction persists when $\Delta \to 0$: 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.