 Solution

For parabolic bands, the density of states near a subband edge scales as $g_{1D}\propto (E-E_n{)}^{-1/2}$, $g_{2D}\propto \Theta (E-E_n)$, and $g_{3D}\propto \sqrt{E-E_c}$. 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.

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.