Almost Mathieu Operator

The almost Mathieu operator acts on a doubly infinite sequence by

It is a quasiperiodic discrete Schrödinger operator. The parameter is the coupling, is the frequency, and is the phase. For irrational and nonzero , its operator spectrum is a Cantor set (Avila and Jitomirskaya 2009). The critical case is closely related to the Harper equation and generates Hofstadter's butterfly as varies.