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Finite Gap Jacobi Matrices, II. The Szegő Class

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Abstract

Let \(\frak{e}\subset\mathbb{R}\) be a finite union of disjoint closed intervals. We study measures whose essential support is \({\frak{e}}\) and whose discrete eigenvalues obey a 1/2-power condition. We show that a Szegő condition is equivalent to

$$\limsup\frac{a_1\cdots a_n}{\mathrm{cap}(\frak{e})^n}>0$$

(this includes prior results of Widom and Peherstorfer–Yuditskii). Using Remling’s extension of the Denisov–Rakhmanov theorem and an analysis of Jost functions, we provide a new proof of Szegő asymptotics, including L 2 asymptotics on the spectrum. We make heavy use of the covering map formalism of Sodin–Yuditskii as presented in our first paper in this series.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen, Denmark

    Jacob S. Christiansen

  2. Mathematics 253-37, California Institute of Technology, Pasadena, CA, 91125, USA

    Barry Simon & Maxim Zinchenko

Authors
  1. Jacob S. Christiansen
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  2. Barry Simon
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  3. Maxim Zinchenko
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Corresponding author

Correspondence to Barry Simon.

Additional information

Communicated by Vilmos Totik.

J.S. Christiansen was supported in part by a Steno Research Grant from FNU, the Danish Research Council. B. Simon was supported in part by NSF grant DMS-0652919.

Maxim Zinchenko was supported in part by NSF grant DMS-0965411.

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Christiansen, J.S., Simon, B. & Zinchenko, M. Finite Gap Jacobi Matrices, II. The Szegő Class. Constr Approx 33, 365–403 (2011). https://doi.org/10.1007/s00365-010-9094-7

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  • DOI: https://doi.org/10.1007/s00365-010-9094-7

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