= Solution
Put $\xi=\epsilon-\epsilon_F$ and $\Delta=|\Delta|>0$. The <BCS quasiparticle spectrum> has $E=\sqrt{\xi^2+\Delta^2}$, so on either inverse branch
$$
\xi_\pm=\pm\sqrt{E^2-\Delta^2},\qquad
\left|\frac{d\xi_\pm}{dE}\right|=\frac E{\sqrt{E^2-\Delta^2}}.
$$
Conservation of states on one branch gives $D_{\rm branch}(E)dE=g(\epsilon_F)|d\xi|$. Consequently the single-branch expression requested in the source is
$$
\boxed{D_{\rm branch}(E)=\frac{g(\epsilon_F)E}{\sqrt{E^2-\Delta^2}},\qquad E>\Delta.}
$$
There are no such states for $0<E<\Delta$ in this ideal fully gapped model. Near the edge the <density of states> diverges as $(E-\Delta)^{-1/2}$, an integrable singularity; at large $E/\Delta$ it approaches its normal-state value.
\b[Counting qualification.] The <energy> map in the source is two-to-one. If $g(\epsilon)$ literally counts all normal-state levels both above and below $\epsilon_F$, the unweighted <positive-energy quasiparticle density from a folded spectrum> must sum both preimages, and is instead
$$
\boxed{D_{\rm qp}(E)=\sum_{\pm}g(\epsilon_F+\xi_\pm)
\left|\frac{d\xi_\pm}{dE}\right|
=\frac{2g(\epsilon_F)E}{\sqrt{E^2-\Delta^2}}.}
$$
Already at $\Delta=0$, folding $\xi$ into $|\xi|$ doubles the positive-axis density. Thus the printed formula is not the unweighted count of all states under the literal mapping without an additional normalization convention.
There is also a standard physical quantity with exactly the printed normalization: the <BCS electronic density of states>, measured for example by <single-electron superconducting tunneling>. At positive $E$, the electron spectral weights of the two roots are the <BCS coherence factors> $u_\pm^2=(1+\xi_\pm/E)/2$. They sum to one, so the weighted sum is $g(\epsilon_F)E/\sqrt{E^2-\Delta^2}$. This supplies a valid electronic-DOS interpretation, while keeping it distinct from counting every positive-<energy> excitation. Spin degeneracy must be included consistently in $g$ and the transformed density; it does not remove the branch-counting issue.
Back to article page
