Floquet solutions for the Schrödinger equation with fast-oscillating quasi-periodic potentials

Abstract

<p style='text-indent:20px;'>In this paper, we study the one-dimensional stationary Schrödinger equation with quasi-periodic potential <inline-formula><tex-math id="M1">\begin{document}$ u(\omega t) $\end{document}</tex-math></inline-formula>. We show that if the frequency vector <inline-formula><tex-math id="M2">\begin{document}$ \omega $\end{document}</tex-math></inline-formula> is sufficient large, the Schrödinger equation admits two linear independent Floquet solutions for a set of positive measure of energy <inline-formula><tex-math id="M3">\begin{document}$ E $\end{document}</tex-math></inline-formula>. In contrast with previous results, the conditions of small potential <inline-formula><tex-math id="M4">\begin{document}$ u $\end{document}</tex-math></inline-formula> or large energy <inline-formula><tex-math id="M5">\begin{document}$ E $\end{document}</tex-math></inline-formula> are no longer needed.