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Effective multi-scale approach to the Schrödinger cocycle over a skew-shift base

  • R. HAN (a1), M. LEMM (a2) and W. SCHLAG (a3)

Abstract

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.

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Effective multi-scale approach to the Schrödinger cocycle over a skew-shift base

  • R. HAN (a1), M. LEMM (a2) and W. SCHLAG (a3)

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