Uniqueness of reflectionless Jacobi matrices and the DenisovRakhmanov Theorem
Author:
Christian Remling
Journal:
Proc. Amer. Math. Soc. 139 (2011), 21752182
MSC (2010):
Primary 42C05, 47B36, 81Q10
Published electronically:
November 30, 2010
MathSciNet review:
2775395
Fulltext PDF
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Additional Information
Abstract: If a Jacobi matrix is reflectionless on and has a single equal to , then is the free Jacobi matrix , . 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 's close to , then one assumption in the DenisovRakhmanov Theorem can be dropped.
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 J.A. Cima, A.L. Matheson, and W.T. Ross, The Cauchy Transform, Mathematical Surveys and Monographs, 125, American Mathematical Society, Providence, RI, 2006. MR 2215991 (2006m:30003)
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 W. Craig, The trace formula for Schrödinger operators on the line, Comm. Math. Phys. 126 (1989), 379407. MR 1027503 (90m:47063)
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 D. Damanik, D. Hundertmark, R. Killip, and B. Simon, Variational estimates for discrete Schrödinger operators with potentials of indefinite sign, Comm. Math. Phys. 238 (2003), 545562. MR 1993385 (2004i:81068)
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 P. Deift and B. Simon, Almost periodic Schrödinger operators. III. The absolutely continuous spectrum in one dimension, Comm. Math. Phys. 90 (1983), 389411. MR 719297 (85i:34009b)
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Additional Information
Christian Remling
Affiliation:
Department of Mathematics, University of Oklahoma, Norman, Oklahoma 73019
Email:
cremling@math.ou.edu
DOI:
http://dx.doi.org/10.1090/S000299392010107475
Keywords:
Reflectionless Jacobi matrix,
DenisovRakhmanov 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.
