Method of Variations of Potential of Quasi-Periodic Schrödinger Equations

We study the one-dimensional discrete quasi-periodic Schrödinger equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$-{\varphi}(n + 1) -{\varphi}(n - 1) + {\lambda}V (x + n\omega){\varphi}(n) = E{\varphi}(n),\quad n \in {\mathbb{Z}}$$ \end{document}. We introduce the notion of variations of potential and use it to define typical potential. We show that for typical C3 potential V, if the coupling constant λ is large, then for most frequencies ω, the Lyapunov exponent is positive for all energies E.