Spectral properties of high contrast band-gap materials and operators on graphs

The theory of classical waves in periodic high contrast photonic and acoust ic media leads to the spectral problem -Delta u = lambda epsilon u where the dielectric constant epsilon(x) is a periodic function which assum es a large value a near a periodic graph Sigma in R-2 and is equal to 1 oth erwise. Existence and locations of spectral gaps are of primary interest. T he high contrast asymptotics naturally leads to pseudodifferential operator s of the Dirichlet-to-Neumann type on graphs and on more general structures . Spectra of these operators are studied numerically and analytically. New spectral effects are discovered, among them the "almost discreteness" of th e spectrum for a disconnected graph and the existence of "almost localized" waves in some connected purely periodic structures.