One-dimensional free-electron density of states ($g_1(E)=L\sqrt{2m}/(\pi \hslash \sqrt{E})$)

For a length $L$ and parabolic free-electron dispersion relation $E={\hslash }^2{k}^2/(2m)$, the displayed total density of states includes both signs of $k$ and both spin states. Per spin it is half as large. Its $E^{-1/2}$ dependence follows by transforming the constant signed-$k$ density $L/(2\pi )$ to energy and including both momenta. The continuum singularity at the band bottom is integrable; a finite specimen still has discrete levels.