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Letters in Mathematical Physics

, Volume 87, Issue 1–2, pp 99–114 | Cite as

Localization on Quantum Graphs with Random Edge Lengths

  • Frédéric Klopp
  • Konstantin Pankrashkin
Article

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.

Mathematics Subject Classification (2000)

81Q10 35R60 47B80 60H25 

Keywords

Quantum graph Random operator Random metric Anderson localization 

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Copyright information

© Springer 2009

Authors and Affiliations

  1. 1.Laboratoire Analyse, Géométrie et Applications, CNRS UMR 7539, Institut GaliléeUniversité Paris-NordVilletaneuseFrance
  2. 2.Institut Universitaire de FranceParisFrance
  3. 3.Institut für MathematikHumboldt-UniversitätBerlinGermany
  4. 4.Laboratoire de mathématiques, CNRS UMR 8628Unversité Paris-SudOrsayFrance

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