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Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder

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Abstract

We consider the Laplacian on a rooted metric tree graph with branching number K≥2 and random edge lengths given by independent and identically distributed bounded variables. Our main result is the stability of the absolutely continuous spectrum for weak disorder. A useful tool in the discussion is a function which expresses a directional transmission amplitude to infinity and forms a generalization of the Weyl-Titchmarsh function to trees. The proof of the main result rests on upper bounds on the range of fluctuations of this quantity in the limit of weak disorder.

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Authors and Affiliations

  1. Departments of Mathematics and Physics, Princeton University, Princeton, NJ, 08544, USA

    Michael Aizenman & Simone Warzel

  2. Department of Mathematics, University of California at Davis, Davis, CA, 95616, USA

    Robert Sims

Authors
  1. Michael Aizenman
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  2. Robert Sims
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  3. Simone Warzel
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Communicated by B. Simon

Copyright rests with the authors. Faithful reproduction of the article for non-commercial purpose is permitted.

Simone Warzel: On leave from: Institut für Theoretische Physik, Universität Erlangen-Nürnberg, Germany

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Aizenman, M., Sims, R. & Warzel, S. Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder. Commun. Math. Phys. 264, 371–389 (2006). https://doi.org/10.1007/s00220-005-1468-5

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  • DOI: https://doi.org/10.1007/s00220-005-1468-5

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