Differential operators on graphs and photonic crystals
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Studying classical wave propagation in periodic high contrast photonic and acoustic media naturally leads to the following spectral problem: -u = u, where (x) (the dielectric constant) is a periodic function that assumes a large value near a periodic graph in 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.