Almost Mathieu Operator

The almost Mathieu operator is given by

$$ H^{\lambda,\alpha}_\omega u = u(n+1) + u(n-1) + 2 \lambda \cos(2\pi (\omega + n\alpha)) u(n) $$

It's important because it is one of the best-understood examples of an ergodic Schrödinger operator

10 Martini Problem

For all $\lambda \neq 0$, all irrational $a$, and all $\omega$, the spectrum of the almost Mathieu operator is a Cantor set.

Hofstader Butterfly

Hofstadter's butterfly is a graph of the spectral properties of non-interacting two-dimensional electrons in a perpendicular magnetic field in a lattice.