= Solution
For parabolic bands, the <density of states> near a subband edge scales as \b[$g_{1D}\propto(E-E_n)^{-1/2}$], \b[$g_{2D}\propto\Theta(E-E_n)$], and \b[$g_{3D}\propto\sqrt{E-E_c}$]. Thus a quasi-1D wire has inverse-square-root singularities at successive transverse subband thresholds, a quasi-2D well has a staircase, and a bulk 3D band rises smoothly from zero. Each expression vanishes below its threshold.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2019/paper-2-density-of-states.png]
{title=Parabolic-band density of states in one, two and three unconfined dimensions}
{source=paper-2-density-of-states.py}
These sketches assume extended noninteracting states with simple quadratic dispersion. Finite <temperature>, disorder and finite lifetime broaden measured threshold features; the ideal singularity is integrable and does not represent infinitely many states in a finite <energy> interval.
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