Hofstadter's butterfly

Remarks

Check at $\alpha = 0$: the potential is the constant $c = 2\cos(2\pi\theta)$, the spectrum is $[c - 2, c + 2]$, and the union over $\theta$ is $[-4, 4]$, one band.

Check at $\alpha = 1/2$: the potential is $(-1)^n c$ with $c = 2\cos(2\pi\theta)$, of period $2$. Its two bands are given by $E^2 = c^2 + 4\cos^2 k$, $k \in [-\pi, \pi]$, so they are $[-\sqrt{c^2 + 4}, -\abs{c}]$ and $[\abs{c}, \sqrt{c^2 + 4}]$. Taking the union over $c \in [-2, 2]$ gives $[-2\sqrt2, 0] \cup [0, 2\sqrt2]$: two bands that touch at $E = 0$.

For $\alpha = p/q$ in lowest terms, $\sigma(\alpha)$ is a union of $q$ closed intervals, the two central ones touching when $q$ is even. For irrational $\alpha$, $\sigma(\alpha)$ does not depend on $\theta$; it is a Cantor set (the ten martini problem) of Lebesgue measure zero. The picture is symmetric under $\alpha \mapsto 1 - \alpha$ and $E \mapsto -E$, and its parts repeat on smaller scales, which makes it look like a fractal.