Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations

Abstract

<p style='text-indent:20px;'>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 [<xref ref-type="bibr" rid="b21">21</xref>] that the authors constructed examples whose Lyapunov exponent is discontinuous in the potential with the <inline-formula><tex-math id="M1">\begin{document}$ C^0 $\end{document}</tex-math></inline-formula> norm for non-analytic potentials. In this paper, we consider this operators for some Gevrey potential, which is an analytic function having a Gevrey small perturbation, with Diophantine frequency. We prove that in the large coupling regions, the Lyapunov exponent is positive and jointly continuous in all parameters, such as the energy, the frequency and the potential. Note that all analytic functions are also Gevrey ones. Therefore, we also obtain that all of the large analytic potentials are the non-perturbative weak Hölder continuous points of the Lyapunov exponent in the Gevrey topology with <inline-formula><tex-math id="M2">\begin{document}$ C^0 $\end{document}</tex-math></inline-formula> norm. It is the first result about the continuity in non-analytic potential with this norm and is complementary to Wang-You's result.