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Finite and infinite gap Jacobi matrices

På svenska

Author

Editor

  • Jan Janas
  • Pavel Kurasov
  • Ari Laptev
  • Sergey Naboko

Summary, in English

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.

Publishing year

2013

Language

English

Pages

43-55

Publication/Series

Operator Theory Advances and Applications (Operator Methods in Mathematical Physics, Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2010, Bedlewo, Poland)

Volume

227

Document type

Conference paper

Publisher

Birkhäuser Verlag

Topic

  • Mathematics

Keywords

  • Orthogonal polynomials
  • Szegő’s theorem
  • Isospectral torus
  • Parreau–Widom sets

Conference name

Fifth International Conference on Operator Theory Analysis and Mathematical Physics (OTAMP 2010)

Conference date

2010-08-05 - 2010-08-12

Conference place

Bedlewo, Poland

Status

Published

ISBN/ISSN/Other

  • ISSN: 2296-4878
  • ISSN: 0255-0156
  • ISBN: 978-3-0348-0531-5
  • ISBN: 978-3-0348-0530-8 (print)