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Differential Operators on Graphs and Photonic Crystals

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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 R 2 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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Wichita State University, Wichita, KS

    P. Kuchment

  2. Math. Dept., Texas A&M University, College Station, TX, 77543, USA

    P. Kuchment

  3. ACM, Caltech, Pasadena, CA, USA

    L. Kunyansky

  4. Department of Mathematics, University of Arizona, Tucson, AZ, USA

    L. Kunyansky

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Kuchment, P., Kunyansky, L. Differential Operators on Graphs and Photonic Crystals. Advances in Computational Mathematics 16, 263–290 (2002). https://doi.org/10.1023/A:1014481629504

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