Exponentially decaying eigenvectors for certain almost periodic operators

Abstract

For every point $\chi$ in the spectrum of the operator $(h(\delta)\xi)=\xi_{n+1}+\xi_{n-1}+\beta(\delta e^{2\pi\alpha ni}+\delta^{-1}e^{-2\pi\alpha ni})\xi_n$ on $\ell^2(\mathbb{Z})$ there exists a complex number x of modulus one such that the equation $\xi_{n+1}+\xi_{n-1}+\beta(x\delta e^{2\pi\alpha ni}+\bar{x}\delta^{-1}e^{-2\pi\alpha ni})\xi_n=\chi\xi_n$ has a non-trivial solution satisfying the condition $\limsup_{|n|\to\infty}|\xi_n|^{1/|n|}\le\delta^{-1}\beta^{-1}$ provided that $\beta,\delta>1$ and $\alpha$ satisfies the diophantine condition $\lim_{n\to\infty}|{\rm sin}\,\pi\alpha n|^{-{1/n}}=1$. The parameters $x\delta$ and $\chi$ are in the range of analytic functions which are defined on a Riemann surface covering the resolvent set of the operator h(1).