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Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems

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Abstract

In this paper, we prove that a quasi-periodic linear differential equation in sl(2,ℝ) with two frequencies (α,1) is almost reducible provided that the coefficients are analytic and close to a constant. In the case that α is Diophantine we get the non-perturbative reducibility. We also obtain the reducibility and the rotations reducibility for an arbitrary irrational α under some assumption on the rotation number and give some applications for Schrödinger operators. Our proof is a generalized KAM type iteration adapted to all irrational α.

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Author notes

  1. Xuanji Hou

    Present address: School of Mathematics and Statistics, Huazhong Normal University, Wuhan, 430079, P.R. China

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  1. Department of Mathematics, Nanjing University, Nanjing, 210093, P.R. China

    Xuanji Hou & Jiangong You

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  1. Xuanji Hou
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  2. Jiangong You
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Correspondence to Jiangong You.

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Hou, X., You, J. Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems. Invent. math. 190, 209–260 (2012). https://doi.org/10.1007/s00222-012-0379-2

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