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On Agmon Metrics and Exponential Localization for Quantum Graphs

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Abstract

We investigate the rate of decrease at infinity of eigenfunctions of quantum graphs by using Agmon’s method to prove L2 and \({L^\infty}\) bounds on the product of an eigenfunction with the exponential of a certain metric. A generic result applicable to all graphs is that the exponential rate of decay is controlled by an adaptation of the standard estimates for a line, which are of classical Liouville–Green (WKB) form. Examples reveal that this estimate can be the best possible, but that a more rapid rate of decay is typical when the graph has additional structure. In order to understand this fact, we present two alternative estimates under more restrictive assumptions on the graph structure that pertain to a more rapid decay. One of these depends on how the eigenfunction is distributed along a particular chosen path, while the other applies to an average of the eigenfunction over edges at a given distance from the root point.

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Author notes

  1. Anna V. Maltsev

    Present address: School of Mathematical Sciences, Queen Mary University of London, London, E1 4NS, UK

Authors and Affiliations

  1. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Evans M. Harrell II

  2. Department of Mathematics, University of Bristol, Bristol, BS81SD, UK

    Anna V. Maltsev

Authors
  1. Evans M. Harrell II
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  2. Anna V. Maltsev
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Correspondence to Evans M. Harrell II.

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Communicated by P. Deift

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Harrell, E.M., Maltsev, A.V. On Agmon Metrics and Exponential Localization for Quantum Graphs. Commun. Math. Phys. 359, 429–448 (2018). https://doi.org/10.1007/s00220-018-3124-x

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  • DOI: https://doi.org/10.1007/s00220-018-3124-x

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