Asymptotic laws for a class of quasi-periodic Schr\"odinger cocycles at the lowest energy of the spectrum

TL;DR

This paper analyzes the behavior of quasi-periodic Schrödinger cocycles near the spectrum's lowest energy, revealing asymptotic linearity in Oseledets-directions and growth in their norms, confirming numerical predictions.

Contribution

It provides the first rigorous proof of the asymptotic laws for Oseledets-directions at the spectrum's bottom in quasi-periodic Schrödinger cocycles.

Findings

Distance between Oseledets-directions is asymptotically linear near lowest energy.

C^2-norm of Oseledets-directions grows like the inverse square root of the distance.

Results confirm previous numerical observations.

Abstract

Let $(\omega, A_E)$ be a quasi-periodic Schr\"odinger cocycle, where $\omega$ is a Diophantine irrational. The potential is assumed to be $C^2$ with a unique non-degenerate minimum, and the coupling constant is assumed to be large. We show that, as the energy approaches the lowest energy of the spectrum from below, the distance between the Oseledets-directions, in projective coordinates, is asymptotically linear. Moreover, we show that the $C^2$-norm of the Oseledets-directions, in projective coordinates, grows asymptotically (almost) like the inverse of the square root of the distance. Both of these results confirm numerical observations.