Abstract
Avila’s Almost Reducibility Conjecture (ARC) is a powerful statement linking purely analytic and dynamical properties of analytic one-frequency \(SL(2,{\mathbb{R}})\) cocycles. It is also a fundamental tool in the study of spectral theory of analytic one-frequency Schrödinger operators, with many striking consequences, allowing to give a detailed characterization of the subcritical region. Here we give a proof, completely different from Avila’s, for the important case of Schrödinger cocycles with trigonometric polynomial potentials and non-exponentially approximated frequencies, allowing, in particular, to obtain all the desired spectral consequences in this case.
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Notes
Let F be a bounded analytic (possibly matrix valued) function defined on {θ||ℑθ|<h}, ∥F∥h=sup|ℑθ|<h∥F(θ)∥. \(C^{\omega}_{h}({\mathbb{T}},*)\) denotes the set of all these ∗-valued functions (∗ will usually denote \({\mathbb{R}}\), \(sl(2,{\mathbb{R}})\) or \(SL(2,{\mathbb{R}})\)). Denote by \(C^{\omega}({\mathbb{T}},{\mathbb{R}})\) the union \(\cup _{h>0}C_{h}^{\omega}({\mathbb{T}},{\mathbb{R}})\).
Note that ωd(E)>0 if E∈ΣV,α.
we omit E in the following formula for simplicity.
ΣV,α is the spectrum of HV,α,θ.
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Acknowledgements
The first version of this paper was finished in Oct, 2019 when I was a Visiting Assistant Professor at UCI and I would like to thank Professor Svetlana Jitomirskaya for many helpful discussions and kind support. We told Professor Artur Avila our different proof of the ARC in Dec 2019 and I would like to thank him for explaining to me his proof and the histories of the ARC. I would also like to thank Professor Jiangong You for many helpful discussions. I am also grateful to the organizers of QMath15, Sep.2022, for giving me the opportunity to present some details of my approach. L. Ge was partially supported by NSFC grant (12371185) and the Fundemental Research Funds for the Central Universities (the start-up fund), Peking University.
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Ge, L. On the Almost Reducibility Conjecture. Geom. Funct. Anal. 34, 32–59 (2024). https://doi.org/10.1007/s00039-024-00671-0
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DOI: https://doi.org/10.1007/s00039-024-00671-0