Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations

Wenmeng Geng; Kai Tao

doi:10.3934/dcdsb.2020216

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2021, Volume 26, Issue 6: 2977-2996. Doi: 10.3934/dcdsb.2020216

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Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations

Wenmeng Geng and
Kai Tao

College of Sciences, Hohai University, No.1 Xikang Road, Nanjing, Jiangsu, 210098, China

^* Corresponding author: Kai Tao
^* Corresponding author: Kai Tao

Received: September 30, 2019

Revised: February 29, 2020

Early access: July 2020

Published: June 2021

The second author was supported by the Fundamental Research Funds for the Central Universities(Grant B200202004) and China Postdoctoral Science Foundation (Grant 2019M650094)

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Abstract

In the study of the continuity of the Lyapunov exponent for the discrete quasi-periodic Schrödinger operators, there is a pioneering result by Wang-You [21] that the authors constructed examples whose Lyapunov exponent is discontinuous in the potential with the $ C^0 $ norm for non-analytic potentials. In this paper, we consider this operators for some Gevrey potential, which is an analytic function having a Gevrey small perturbation, with Diophantine frequency. We prove that in the large coupling regions, the Lyapunov exponent is positive and jointly continuous in all parameters, such as the energy, the frequency and the potential. Note that all analytic functions are also Gevrey ones. Therefore, we also obtain that all of the large analytic potentials are the non-perturbative weak Hölder continuous points of the Lyapunov exponent in the Gevrey topology with $ C^0 $ norm. It is the first result about the continuity in non-analytic potential with this norm and is complementary to Wang-You's result.

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Mathematics Subject Classification: Primary: 37C55, 37D25; Secondary: 30D60.

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