Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

Finite and infinite gap Jacobi matrices

Christiansen, Jacob Stordal LU (2013) Fifth International Conference on Operator Theory Analysis and Mathematical Physics (OTAMP 2010) 227. p.43-55
Abstract
The present paper reviews the theory of bounded Jacobi matrices whose essential spectrum is a finite gap set, and it explains how the theory can be extended to also cover a large number of infinite gap sets. Two of the central results are generalizations of Denisov–Rakhmanov’s theorem and Szegő’s theorem, including asymptotics of the associated orthogonal polynomials. When the essential spectrum is an interval, the natural limiting object J0 has constant Jacobi parameters. As soon as gaps occur, ℓ say, the complexity increases and the role of J0 is taken over by an ℓ -dimensional isospectral torus of periodic or almost periodic Jacobi matrices.
Please use this url to cite or link to this publication:
author
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
keywords
Orthogonal polynomials, Szegő’s theorem, Isospectral torus, Parreau–Widom sets
host publication
Operator Theory Advances and Applications (Operator Methods in Mathematical Physics, Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2010, Bedlewo, Poland)
editor
Janas, Jan ; Kurasov, Pavel ; Laptev, Ari and Naboko, Sergey
volume
227
pages
13 pages
publisher
Birkhäuser Verlag
conference name
Fifth International Conference on Operator Theory Analysis and Mathematical Physics (OTAMP 2010)
conference location
Bedlewo, Poland
conference dates
2010-08-05 - 2010-08-12
external identifiers
  • scopus:84922287774
ISSN
2296-4878
0255-0156
ISBN
978-3-0348-0531-5
978-3-0348-0530-8 (print)
DOI
10.1007/978-3-0348-0531-5_2
language
English
LU publication?
no
id
0780b4bc-1eb8-49e1-8e38-fafa8c4c0ac1 (old id 3448252)
date added to LUP
2016-04-01 09:47:37
date last changed
2024-01-06 00:00:03
@inproceedings{0780b4bc-1eb8-49e1-8e38-fafa8c4c0ac1,
  abstract     = {{The present paper reviews the theory of bounded Jacobi matrices whose essential spectrum is a finite gap set, and it explains how the theory can be extended to also cover a large number of infinite gap sets. Two of the central results are generalizations of Denisov–Rakhmanov’s theorem and Szegő’s theorem, including asymptotics of the associated orthogonal polynomials. When the essential spectrum is an interval, the natural limiting object J0 has constant Jacobi parameters. As soon as gaps occur, ℓ say, the complexity increases and the role of J0 is taken over by an ℓ -dimensional isospectral torus of periodic or almost periodic Jacobi matrices.}},
  author       = {{Christiansen, Jacob Stordal}},
  booktitle    = {{Operator Theory Advances and Applications (Operator Methods in Mathematical Physics, Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2010, Bedlewo, Poland)}},
  editor       = {{Janas, Jan and Kurasov, Pavel and Laptev, Ari and Naboko, Sergey}},
  isbn         = {{978-3-0348-0531-5}},
  issn         = {{2296-4878}},
  keywords     = {{Orthogonal polynomials; Szegő’s theorem; Isospectral torus; Parreau–Widom sets}},
  language     = {{eng}},
  pages        = {{43--55}},
  publisher    = {{Birkhäuser Verlag}},
  title        = {{Finite and infinite gap Jacobi matrices}},
  url          = {{http://dx.doi.org/10.1007/978-3-0348-0531-5_2}},
  doi          = {{10.1007/978-3-0348-0531-5_2}},
  volume       = {{227}},
  year         = {{2013}},
}