Dynamical Systems Seminar

The spectrum of one-frequency quasi-periodic Schrödinger operators are widely expected to be Cantor. However, it turns out that `Cantor or non-Cantor', and the mechanism leads to Cantor depend strongly on the regularity of the potentials. After reviewing some recent works, I will first present a joint work with Artur Avila and David Damanik where we constructed a class of \(C^0\) potentials with non-Cantor spectrum, which are the first examples of this kind. Then I will show a recent joint work with Yiqian Wang where we obtained for a class of \(C^2\) potentials and for any fixed Diophantine frequency, the spectrum is Cantor. This is the first rigorous result of this type for quasi-periodic potentials beyond the \(C^0\) and the real analytic categories.