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On Agmon Metrics and Exponential Localization for Quantum Graphs

Research output: Contribution to journalArticle

  • M. Harrell II Evans
  • Anna V. Maltsev
Original languageEnglish
Pages (from-to)429–448
Number of pages20
JournalCommunications in Mathematical Physics
Issue number2
Early online date28 Mar 2018
DateAccepted/In press - 22 May 2016
DateE-pub ahead of print - 28 Mar 2018
DatePublished (current) - Apr 2018


We investigate the rate of decrease at infinity of eigenfunctions of quantum graphs by using Agmon’s method to prove L2 and (Formula presented.) 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.

    Research areas

  • math-ph, math.MP, math.SP

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