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Localization on Quantum Graphs with Random Edge Lengths

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Abstract

The spectral properties of the Laplacian on a class of quantum graphs with random metric structure are studied. Namely, we consider quantum graphs spanned by the simple \({\mathbb Z^d}\) -lattice with δ-type boundary conditions at the vertices, and we assume that the edge lengths are randomly independently identically distributed. Under the assumption that the coupling constant at the vertices does not vanish, we show that the operator exhibits the Anderson localization near the spectral edges situated outside a certain forbidden set.

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Author notes

  1. Konstantin Pankrashkin

    Present address: Laboratoire de mathématiques, CNRS UMR 8628, Unversité Paris-Sud, Bâtiment 425, 91405, Orsay, France

Authors and Affiliations

  1. Laboratoire Analyse, Géométrie et Applications, CNRS UMR 7539, Institut Galilée, Université Paris-Nord, 99 av. Jean-Baptiste Clément, 93430, Villetaneuse, France

    Frédéric Klopp & Konstantin Pankrashkin

  2. Institut Universitaire de France, 103 bd. Saint-Michel, 75005, Paris, France

    Frédéric Klopp

  3. Institut für Mathematik, Humboldt-Universität, Rudower Chaussee 25, 12489, Berlin, Germany

    Konstantin Pankrashkin

Authors
  1. Frédéric Klopp
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  2. Konstantin Pankrashkin
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Correspondence to Konstantin Pankrashkin.

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Klopp, F., Pankrashkin, K. Localization on Quantum Graphs with Random Edge Lengths. Lett Math Phys 87, 99–114 (2009). https://doi.org/10.1007/s11005-009-0293-8

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  • DOI: https://doi.org/10.1007/s11005-009-0293-8

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