Scattering theory for CMV matrices: Uniqueness, helson–szegő and strong szegő theorems

L. Golinskii, A. Kheifets, F. Peherstorfer, P. Yuditskii

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11 Scopus citations

Abstract

We develop a scattering theory for CMV matrices, similar to the Faddeev–Marchenko theory. A necessary and sufficient condition is obtained for the uniqueness of the solution of the inverse scattering problem. We also obtain two sufficient conditions for uniqueness, which are connected with the Helson–Szegő and the strong Szegő theorems. The first condition is given in terms of the boundedness of a transformation operator associated with the CMV matrix. In the second case this operator has a determinant. In both cases we characterize Verblun-sky parameters of the CMV matrices, corresponding spectral measures and scattering functions.

Original languageEnglish
Pages (from-to)479-508
Number of pages30
JournalIntegral Equations and Operator Theory
Volume69
Issue number4
DOIs
StatePublished - Apr 2011
Externally publishedYes

Bibliographical note

Publisher Copyright:
© Springer Basel AG 2011.

Keywords

  • Arov regularity
  • Faddeev-Marchenko space
  • Gelfand-Levitan-Marchenko equation
  • Scattering function
  • Schur algorithm
  • Spectral measure
  • Transformation operator
  • γ-generating pair

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