The paper contains a rigorous proof of the statement formulated and supported by elegant heuristic arguments in the work by G. Andre and S. Aubry [Ann. Isr. Phys. Soc. 3, 133(1980)]. Theorem 1. Let \(u_ n\) be the solution of the equation \((1)\quad u_{n+1}+u_{n-1}+2g \cos (2\pi n\theta +\phi)u_ n=\lambda u_ n,\) \(u_{-1}=\cos \alpha\), \(u_ 0=\sin \alpha\), \(\alpha\in [0,\pi),\) \(g\geq 0\). Then the Lyapunov exponent \(\lim_{| n| \to \infty}| n|^{- 1}\sup_{\alpha \in [0,\pi)}\ln (u^ 2_ n+u^ 2_{n+1})=\gamma\) exists for almost every \(\phi\in [0,2\pi)\) and almost all \(\lambda\in {\mathbb{R}}\). This limit is strictly positive for irrational \(\theta\) and \(g>1\). The same results were proved by J. Avron and B. Simon [Bull. Am. Math. Soc., New Ser. 6, 81-85 (1982; Zbl 0491.47014); Duke Math. J. 50, 369-391 (1983; Zbl 0544.35030)] and stronger results were obtained by W. Craig and B. Simon [ibid. 50, 551-560 (1983; Zbl 0518.35027)] and M. Hermann [Ecole Polytechnique Preprint (1982)]. In particular using the Hermann approach it can be shown that the analogous statement is true if the sequence 2g cos(2\(\pi\) \(\theta\) n\(+\phi)\) is replaced by the sequence \(\sum_{k\leq | m|}a_ k\exp (2\pi i\) \(P_{\ell}(n))\), \(a^*_ k=a_{-k}\) and \(P_{\ell}(x)\) is the polynomial of degree \(\ell\) which has at least one irrational coefficient.
The important corollary of these results is the absence of absolutely continuous spectrum for the selfadjoint operator defined in \(\ell_ 2({\mathbb{Z}})\) by r.h.s. of Eq. (1).