Absolutely continuous Jacobi operators
Author:
Steen Pedersen
Journal:
Proc. Amer. Math. Soc. 130 (2002), 23692376
MSC (2000):
Primary 33C45, 39A70; Secondary 47A10, 47B39
Published electronically:
February 4, 2002
MathSciNet review:
1897462
Fulltext PDF Free Access
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References 
Similar Articles 
Additional Information
Abstract: We show (among other results) that a symmetric Jacobi matrix whose diagonal is the zero sequence and whose superdiagonal satisfies , and has purely absolutely continuous spectrum when considered as a selfadjoint operator on .
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 [CFKS87]
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 [Cla96]
 S.L. Clark, A spectral analysis for selfadjoint operators generated by a class of secondorder difference equations, J. Math. Anal. Appl. 197 (1996), 267285. MR 96m:47061
 [dMS98]
 A. Boutet de Monvel and J. Sahbani, On the spectral properties of the spinboson Hamiltonians, Lett. Math. Phys. 44 (1998), 2333. MR 2000m:81047
 [dMS99]
 A. Boutet de Monvel and J. Sahbani, On the spectral properties of discrete Schrödinger operators: the multidimensional case, Rev. Math. Phys. 11 (1999), 10611078. MR 2000j:47064
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 J. Dombrowski and S. Pedersen, Orthogonal polynomials, spectral measures and absolute continuity, J. Comput. Appl. Math. 65 (1995), 115124. MR 97b:47027
 [DP97]
 J. Dombrowski and S. Pedersen, Spectral measures and Jacobi matrices related to Laguerre type systems of orthogonal polynomials, Constr. Approx. 13 (1997), 421433. MR 98e:47041
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 [HL78]
 D.B. Hinton and R.T. Lewis, Spectral analysis of secondorder difference equations, J. Math. Anal. Appl. 63 (1978), 421438. MR 58:29512
 [JL99]
 S. Jitomirskaya and Y. Last, Powerlaw subordinacy and singular spectra I. Halfline operators, Acta Math. 183 (1999), 171189. MR 2001a:47033
 [JM00]
 J. Janas and M. Moszynski, The alternative approaches to the absolute continuity of Jacobi matrices with monotonic weights, Integral Equations Operator Theory, to appear.
 [JN99a]
 J. Janas and S. Naboko, Jacobi matrices with powerlike weights  grouping in blocks approach, J. Funct. Anal. 166 (1999), 218243. MR 2000k:47032
 [JN99b]
 J. Janas and S. Naboko, Multitreshold spectral phase transition examples in a class of unbounded Jacobi matrices, Recent advances in operator theory (Groningen, 1998), 267285. Oper. Theory Adv. Appl., 124, Birkhäuser, Basel, 2001.
 [KL00]
 A. Kiselev and Y. Last, Solutions, spectrum and dynamics for Schrödinger operators on infinite domains, Duke Math. J. 102 (2000), 125150. CMP 2000:09
 [LS99]
 Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of onedimensional Schrödinger operators, Invent. Math. 135 (1999), 329367. MR 2000f:47060
 [Mou81]
 E. Mourre, Absence of singular continuous spectrum for certain selfadjoint operators, Commun. Math. Phys. 78 (1981), 391408. MR 82c:47030
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 C.R. Putnam, Commutation Properties of Hilbert Space Operators and Related Topics, Ergebnisse der Math., vol. 36, SpringerVerlag, Berlin, 1967. MR 36:707
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 G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Latices, Mathematical Surveys and Monograhps, vol. 72, Amer. Math. Soc., Providence, RI, 2000. MR 2001b:39019
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Additional Information
Steen Pedersen
Affiliation:
Department of Mathematics, Wright State University, Dayton, Ohio 45435
Email:
steen@math.wright.edu
DOI:
http://dx.doi.org/10.1090/S0002993902063396
PII:
S 00029939(02)063396
Keywords:
Orthogonal polynomials,
weighted shift,
absolute continuity,
Jacobi matrix
Received by editor(s):
September 1, 2000
Received by editor(s) in revised form:
March 21, 2001
Published electronically:
February 4, 2002
Communicated by:
David R. Larson
Article copyright:
© Copyright 2002
American Mathematical Society
