Differential operators on graphs and photonic crystals

P. Kuchment, L. Kunyansky

Research output: Contribution to journalArticle

29 Scopus citations

Abstract

Studying classical wave propagation in periodic high contrast photonic and acoustic media naturally leads to the following spectral problem: -Δu = λεu, where ε(x) (the dielectric constant) is a periodic function that assumes a large value ε near a periodic graph Σ in ℝ2 and is equal to 1 otherwise. High contrast regimes lead to appearence of pseudo-differential operators of the Dirichlet-to-Neumann type on graphs. The paper contains a technique of approximating these pseudo-differential spectral problems by much simpler differential ones that can sometimes be resolved analytically. Numerical experiments show amazing agreement between the spectra of the pseudo-differential and differential problems.

Original languageEnglish (US)
Pages (from-to)263-290
Number of pages28
JournalAdvances in Computational Mathematics
Volume16
Issue number2-3
DOIs
StatePublished - Dec 1 2002

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Keywords

  • Differential operators on graphs
  • Dirichlet-to-Neumann map
  • Photonic bandgap
  • Photonic crystal
  • Pseudo-differential operators on graphs
  • Spectrum

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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