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Jacobi matrices with absolutely continuous spectrum

Authors:	Jan Janas Serguei Naboko

Institution:	Institute of Mathematics, Polish Academy of Sciences, Cracow Branch, Sw. Tomasza 30, 31-027 Krakow, Poland ; Department of Mathematical Physics, Institute for Physics, St. Petersburg University, Ulianovskaia 1, 198904, St. Petergoff, Russia

Abstract:	Let be a Jacobi matrix defined in as , where 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 has an absolutely continuous spectrum covering . Moreover, the asymptotics of the solution $Ju = \lambda u, \, \lambda \in \mathbb{R}$ is also given.

Keywords:	Jacobi matrix absolutely continuous spectrum asymptotics behaviour

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