Localization on Quantum Graphs with Random Edge Lengths

2008/10/31 by Frédéric Klopp, Konstantin Pankrashkin · 10 citations
Mathematics · Physics and Astronomy · #Combinatorics #Discrete mathematics #Graph #Independent and identically distributed random variables #Lattice (music) #Mathematics #Physics #Quantum #Quantum chaos and dynamical systems #Quantum graph #Quantum mechanics #Random Matrices and Applications #Random graph #Random variable #Spectral Theory in Mathematical Physics #math-ph #math.MP #msc:35R60 #msc:47B80 #msc:60H25 #msc:81Q10

paper · pdf · doi:10.1007/s11005-009-0293-8

published in Letters in Mathematical Physics 87(1-2), 99-114 (Springer Science+Business Media)

arxiv created 2009/01/14 · openalex publication_date 2009/01/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

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 \ZZ^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 at the bottom of the spectrum almost surely. We also study the case of other spectral edges.

Citations

• Equilateral quantum graphs and boundary triples
• Disorder and interactions in Aharonov-Bohm cages
• Continuity of the Integrated Density of States on Random Length Metric Graphs
• Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder
• A random necklace model
• SPECTRA OF SELF-ADJOINT EXTENSIONS AND APPLICATIONS TO SOLVABLE SCHRÖDINGER OPERATORS
• Eigenfunction expansion for Schrodinger operators on metric graphs
• Sch'nol's Theorem For Strongly Local Forms
• Anderson Localization for radial tree-like random quantum graphs