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Bulk universality and
clock spacing of zeros for ergodic Jacobi matrices with
absolutely continuous spectrum
Artur Avila, Yoram Last and Barry Simon |
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Vol. 3 (2010), No. 1, 81–108
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Abstract |
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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 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 is , where is the density of zeros and is the absolutely continuous weight of the spectral measure. |
Keywords
orthogonal polynomials, clock behavior, almost Mathieu
equation
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Mathematical Subject Classification 2000
Primary: 26C10, 42C05, 47B36
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Milestones
Received: 20 October 2009
Accepted: 19 November 2009
Published: 4 March 2010
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Authors |
| Artur Avila | |
| CNRS UMR 7599 Laboratoire de Probabilités et Modèles Aléatoires Université Pierre et Marie Curie Boîte Courrier 188 75252 Paris Cedex 05 France |
Instituto Nacional de Matemática
Pura e Aplicada Estrada Dona Castorina 110 22460-320 Rio de Janeiro, RJ Brazil |
| http://w3.impa.br/~avila/ | |
| Yoram Last | |
| Institute of Mathematics The Hebrew University 91904 Jerusalem Israel |
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| Barry Simon | |
| Department of Mathematics California Institute of Technology MC 253-37 Pasadena, CA 91125 United States |
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| http://www.math.caltech.edu/people/simon.html | |
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