Cantor Spectrum for Schr\"odinger Operators with Potentials arising from Generalized Skew-shifts

TL;DR

This paper proves that for Schr"odinger operators with potentials from generalized skew-shifts, the spectrum is generically a Cantor set due to the density of uniform hyperbolicity in the associated cocycles.

Contribution

It establishes the density of uniform hyperbolicity for cocycles over generalized skew-shifts, leading to the generic Cantor spectrum for related Schr"odinger operators.

Findings

Uniform hyperbolicity is dense among cocycles over generalized skew-shifts.

Generic potentials lead to Cantor spectrum in Schr"odinger operators.

Obstructions to hyperbolicity vanish for Schr"odinger cocycles.

Abstract

We consider continuous $SL(2,\mathbb{R})$-cocycles over a strictly ergodic homeomorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an $SO(2,\mathbb{R})$-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruction to uniform hyperbolicity, then it can be $C^0$-perturbed to become uniformly hyperbolic. For cocycles arising from Schr\"odinger operators, the obstruction vanishes and we conclude that uniform hyperbolicity is dense, which implies that for a generic continuous potential, the spectrum of the corresponding Schr\"odinger operator is a Cantor set.