Supercritical Ten Martini Problem

The supercritical Ten Martini problem asks whether the operator spectrum of a one-frequency quasiperiodic Schrödinger operator is Cantor in the region of positive Lyapunov exponent (Ge et al. 2026). The operator acts on square-summable sequences by

where is a nonconstant real analytic function, is irrational, and . Its operator spectrum is independent of . For the transfer matrix , put

Ge et al. (2026) report that , where , is a relatively perfect set with empty interior whenever the intersection is nonempty. It is closed relative to , so it need not be a compact Cantor set in the whole real line. In particular, the entire operator spectrum is a Cantor set if throughout it. For every such , this occurs after multiplication by a sufficiently large coupling constant (Sorets and Spencer 1991). The almost Mathieu operator is the special case with , whose Cantor-spectrum result is the Ten Martini theorem (Avila and Jitomirskaya 2009).