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Jacobi matrices with absolutely continuous spectrum

Authors:	Jan Janas Serguei Naboko

Affiliation:	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 such that 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 , then has an absolutely continuous spectrum covering . Moreover, the asymptotics of the solution is also given.

Keywords:	Jacobi matrix absolutely continuous spectrum asymptotics behaviour

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