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Localized eigenvectors on metric graphs

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  • Kravitz, H.
  • Brio, M.
  • Caputo, J.-G.

Abstract

We analyze the eigenvectors of the generalized Laplacian for two metric graphs occurring in practical applications. In accordance with random network theory, localization of an eigenvector is rare and the network should be tuned to observe exactly localized eigenvectors. We derive the resonance conditions to obtain localized eigenvectors for various geometric configurations and their combinations to form more complicated resonant structures. These localized eigenvectors suggest new indicators based on the energy density; in contrast to standard criteria, ours provide the number of active edges. We also suggest practical ways to design resonating systems based on metric graphs. Finally, numerical simulations of the time-dependent wave equation on the metric graph show that localized eigenvectors can be excited by a broadband initial condition, even with leaky boundary conditions.

Suggested Citation

  • Kravitz, H. & Brio, M. & Caputo, J.-G., 2023. "Localized eigenvectors on metric graphs," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 214(C), pages 352-372.
    • Handle: RePEc:eee:matcom:v:214:y:2023:i:c:p:352-372
    DOI: 10.1016/j.matcom.2023.07.011
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    References listed on IDEAS

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    1. Michele Gaio & Dhruv Saxena & Jacopo Bertolotti & Dario Pisignano & Andrea Camposeo & Riccardo Sapienza, 2019. "A nanophotonic laser on a graph," Nature Communications, Nature, vol. 10(1), pages 1-7, December.
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