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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Jacobi matrices with absolutely continuous spectrum
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by Jan Janas and Serguei Naboko PDF
Proc. Amer. Math. Soc. 127 (1999), 791-800 Request permission

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
  • 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
  • © Copyright 1999 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 127 (1999), 791-800
  • MSC (1991): Primary 47B37; Secondary 47B39
  • DOI: https://doi.org/10.1090/S0002-9939-99-04586-4
  • MathSciNet review: 1469415