Jacobi matrices with absolutely continuous spectrum

Abstract:

Let $J$ be a Jacobi matrix defined in $l^2$ as $Re W$, where $W$ is a unilateral weighted shift with nonzero weights $\lambda _k$ such that $\lim _k \lambda _k = 1.$ Define the seqences: $\varepsilon _k:= \frac {\lambda _{k-1}}{\lambda _k} -1,$ $\delta _k:= \frac {\lambda _k -1}{\lambda _k}, \eta _k:= 2 \delta _k + \varepsilon _k.$ If $\varepsilon _k = O(k^{-\alpha }) , \eta _k = O(k^{-\gamma }), \frac {2}{3}< \alpha \leq \gamma , \alpha + \gamma > 3/2$ and $\gamma > 3/4$, then $J$ has an absolutely continuous spectrum covering $(-2,2)$. Moreover, the asymptotics of the solution $Ju = \lambda u, \lambda \in \mathbb {R}$ is also given.