Localization of eigenfunctions at zero for certain almost periodic operators

Abstract

For every irrational number $\alpha$ satisfying the property $\lim_{n\to\infty}|\!\sin \pi\alpha n|^{-1/n}=1$ and for every number $\beta>1$, it is shown that the difference equation $$ \xi_{n+1}+\xi_{n-1} +2\beta\cos(2\pi\alpha n+\th)\xi_n=0, \quad n\in\mathbb{Z} $$ has a non-trivial solution $\{\xi_n\}$ satisfying $\mathop{\overline{\lim}}_{|n|\to\infty}|\xi_n|^{1/|n|}\le|\beta|^{-1}$ if and only if $\theta=2\pi\alpha n+2\pi k\pm{\pi/2}$ for some $n,k\in\mathbb{Z}$.