Differential Operators on Graphs and Photonic Crystals

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