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Absolute continuity of the integrated density of states for the almost Mathieu operator with non-critical coupling

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We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is non-critical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase, settling the measure-theoretical case of Problem 6 of Barry Simon’s list of Schrödinger operator problems for the twenty-first century.

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Authors and Affiliations

  1. CNRS UMR 7599, Laboratoire de Probabilités et Modèles aléatoires, Université Pierre et Marie Curie, Boîte courrier 188, 75252, Paris Cedex 05, France

    Artur Avila

  2. Department of Mathematics, Rice University, Houston, TX, 77005, USA

    David Damanik

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  1. Artur Avila
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  2. David Damanik
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Correspondence to Artur Avila.

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Avila, A., Damanik, D. Absolute continuity of the integrated density of states for the almost Mathieu operator with non-critical coupling . Invent. math. 172, 439–453 (2008). https://doi.org/10.1007/s00222-007-0105-7

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  • DOI: https://doi.org/10.1007/s00222-007-0105-7

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