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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Jacobi matrices with absolutely
continuous spectrum


Authors: Jan Janas and Serguei Naboko
Journal: Proc. Amer. Math. Soc. 127 (1999), 791-800
MSC (1991): Primary 47B37; Secondary 47B39
MathSciNet review: 1469415
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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.


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Additional Information

Jan Janas
Affiliation: Institute of Mathematics, Polish Academy of Sciences, Cracow Branch, Sw. Tomasza 30, 31-027 Krakow, Poland
Email: najanas@cyf-kr.edu.pl

Serguei Naboko
Affiliation: Department of Mathematical Physics, Institute for Physics, St. Petersburg University, Ulianovskaia 1, 198904, St. Petergoff, Russia
Email: naboko@snoopy.phys.spbu.ru

DOI: http://dx.doi.org/10.1090/S0002-9939-99-04586-4
PII: S 0002-9939(99)04586-4
Keywords: Jacobi matrix, absolutely continuous spectrum, asymptotics behaviour
Received by editor(s): June 25, 1997
Additional Notes: The research of the first author was supported by grant PB 2 PO3A 002 13 of the Komitet Badań Naukowych, Warsaw.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1999 American Mathematical Society