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Person# Rui Han

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Rui Han, Marius Christopher Lemm

We study the one-dimensional discrete Schrödinger operator with the skew-shift potential $2\lambda\cos\left(2\pi \left(\binom{j}{2} \omega+jy+x\right)\right)$. This potential is long conjectured to behave like a random one, i.e., it is expected to produce Anderson localization for arbitrarily small coupling constants $\lambda>0$. In this paper, we introduce a novel perturbative approach for studying the zero-energy Lyapunov exponent $L(\lambda)$ at small $\lambda$. Our main results establish that, to second order in perturbation theory, a natural upper bound on $L(\lambda)$ is fully consistent with $L(\lambda)$ being positive and satisfying the usual Figotin-Pastur type asymptotics $L(\lambda)\sim C\lambda^2$ as $\lambda\to 0$. The analogous quantity behaves completely differently in the Almost-Mathieu model, whose zero-energy Lyapunov exponent vanishes for $\lambda

2018Rui Han, Marius Christopher Lemm

We prove a conditional theorem on the positivity of the Lyapunov exponent for a Schrödinger cocycle over a skew shift base with a cosine potential and the golden ratio as frequency. For coupling below 1, which is the threshold for Herman's subharmonicity trick, we formulate three conditions on the Lyapunov exponent in a finite but large volume and on the associated large deviation estimates at that scale. Our main results demonstrate that these finite-size conditions imply the positivity of the infinite volume Lyapunov exponent. This paper shows that it is possible to make the techniques developed for the study of Schrödinger operators with deterministic potentials, based on large deviation estimates and the avalanche principle, effective.

2018