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Uniqueness of reflectionless Jacobi matrices and the Denisov-Rakhmanov Theorem


Author: Christian Remling
Journal: Proc. Amer. Math. Soc. 139 (2011), 2175-2182
MSC (2010): Primary 42C05, 47B36, 81Q10
DOI: https://doi.org/10.1090/S0002-9939-2010-10747-5
Published electronically: November 30, 2010
MathSciNet review: 2775395
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Abstract: If a Jacobi matrix $ J$ is reflectionless on $ (-2,2)$ and has a single $ a_{n_0}$ equal to $ 1$, then $ J$ is the free Jacobi matrix $ a_n\equiv 1$, $ b_n\equiv 0$. The paper discusses this result and its generalization to arbitrary sets and presents several applications, including the following: if a Jacobi matrix has some portion of its $ a_n$'s close to $ 1$, then one assumption in the Denisov-Rakhmanov Theorem can be dropped.


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Additional Information

Christian Remling
Affiliation: Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email: cremling@math.ou.edu

DOI: https://doi.org/10.1090/S0002-9939-2010-10747-5
Keywords: Reflectionless Jacobi matrix, Denisov-Rakhmanov Theorem
Received by editor(s): June 14, 2010
Published electronically: November 30, 2010
Additional Notes: The author’s work was supported by NSF grant DMS 0758594
Communicated by: Walter Van Assche
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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