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Abstract We consider one-dimensional quasi-periodic Schrödinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates, which lead to refined Hölder continuity of the Lyapunov exponents and the integrated density of states, in both small Lyapunov exponent and large coupling regimes. Our results cover all the Diophantine frequencies and some Liouville frequencies.
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Effective multi-scale approach to the Schrödinger cocycle over a skew-shift base
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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- PAR ID:
- 10105652
- Date Published:
- Journal Name:
- Ergodic Theory and Dynamical Systems
- ISSN:
- 0143-3857
- Page Range / eLocation ID:
- 1 to 66
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/etds.2019.19
