Cantor spectrum for a class of $C^2$ quasiperiodic Schr\"odinger operators

TL;DR

This paper proves that for a class of smooth quasiperiodic Schrödinger operators with Diophantine frequencies, the spectrum is a Cantor set, using dynamical systems methods and analyzing hyperbolic cocycles.

Contribution

It introduces a dynamical systems approach to establish Cantor spectrum for $C^2$ quasiperiodic Schrödinger operators and applies it to $ ext{SL}(2,b R)$ cocycles.

Findings

Spectrum is Cantor for $C^2$ quasiperiodic potentials with Diophantine frequencies.

Uniform hyperbolic systems are open and dense in certain $ ext{SL}(2,b R)$ cocycle families.

Method relies on analysis of asymptotic stable and unstable directions.

Abstract

We show that for a class of $C^2$ quasiperiodic potentials and for any Diophantine frequency, the spectrum of the corresponding Schr\"odinger operators is Cantor. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to general $\mathrm{SL}(2,\mathbb R)$ cocycles, and obtain that uniform hyperbolic systems form a open and dense set in some one-parameter family.