Global rigidity for ultra-differentiable quasiperiodic cocycles and its spectral applications

TL;DR

This paper extends the understanding of quasiperiodic Schr"odinger operators by proving spectral and dynamical properties for ultra-differentiable potentials, relaxing the analyticity requirement.

Contribution

It demonstrates that key spectral and dynamical results hold for ultra-differentiable potentials, broadening the class of potentials where these properties are valid.

Findings

Quasiperiodic Schr"odinger cocycles are either rotations reducible or have positive Lyapunov exponent.

Spectral conjectures like Schr"odinger and Last's intersection spectrum are verified for ultra-differentiable potentials.

Results previously known for analytic potentials are extended to a broader class.

Abstract

For quasiperiodic Schr\"odinger operators with one-frequency analytic potentials, from dynamical systems side, it has been proved that the corresponding quasiperiodic Schr\"odinger cocycle is either rotations reducible or has positive Lyapunov exponent for all irrational frequency and almost every energy. From spectral theory side, the "Schr\"odinger conjecture" and the "Last's intersection spectrum conjecture" have been verified. The proofs of above results crucially depend on the analyticity of the potentials. People are curious about if the analyticity is essential for those problems, see open problems by Fayad-Krikorian and Jitomirskaya-Mar. In this paper, we prove the above mentioned results for ultra-differentiable potentials.