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Bulk universality and clock spacing of zeros for ergodic Jacobi matrices with absolutely continuous spectrum

Avila, Artur and Last, Yoram and Simon, Barry (2010) Bulk universality and clock spacing of zeros for ergodic Jacobi matrices with absolutely continuous spectrum. Analysis & PDE, 3 (1). pp. 81-108. ISSN 1948-206X.

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By combining ideas of Lubinsky with some soft analysis, we prove that universality and clock behavior of zeros for orthogonal polynomials on the real line in the absolutely continuous spectral region is implied by convergence of (1/n)K_n(x, x) for the diagonal CD kernel and boundedness of the analog associated to second kind polynomials. We then show that these hypotheses are always valid for ergodic Jacobi matrices with absolutely continuous spectrum and prove that the limit of (1/n)K_n(x, x) is ρ_∞(x)/w(x) where ρ_∞ is the density of zeros and w is the absolutely continuous weight of the spectral measure.

Item Type:Article
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Simon, Barry0000-0003-2561-8539
Additional Information:© 2010 Mathematical Sciences Publishers. Received 20 Oct 2009. Accepted 19 Nov 2009. Published: 4 March 2010. Y. Last was supported in part by grant 1169/06 from the Israel Science Foundation; B. Simon by grant DMS-0652919 from the NSF; and both by grant 2006483 from the United States–Israel Binational Science Foundation (BSF), Jerusalem. A. Avila thanks M. Flach and T. Tombrello for the hospitality of Caltech. B. Simon would like to thank E. de Shalit for the hospitality of Hebrew University. This research was partially conducted during the period Avila served as a Clay Research Fellow. We would like to thank H. Furstenberg and B. Weiss for useful comments.
Funding AgencyGrant Number
Israel Science Foundation1169/06
Binational Science Foundation (BSF)2006483
Subject Keywords:orthogonal polynomials, clock behavior, almost Mathieu equation
Issue or Number:1
Classification Code:MSC2000: 26C10, 42C05, 47B36
Record Number:CaltechAUTHORS:20100928-144330506
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:20205
Deposited By: Tony Diaz
Deposited On:28 Sep 2010 22:22
Last Modified:03 Oct 2019 02:06

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