## 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.## References

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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