Homogenized graphs with spectrally embedded defect states

Abstract

By homogenizing high-contrast periodic metric graphs, we create continuous media that admit bound states produced by local defects at energies embedded in the continuous spectrum. This is achieved by fitting two copies of a lattice graph simultaneously into space and coupling them by connecting edges. The homogenization of an ODE on a graph to a PDE in space is achieved by representing the associated operators in a common functional space through interpolation by Fourier analysis. The two coupled graphs result in an effective medium with internal degrees of freedom that allow a decomposition into two orthogonal spaces of hybrid states. Critical high contrast retains microscopic resonances in the effective medium and allows a spectral band of one hybrid state space to overlap a spectral gap of the other. This serves as the mechanism for creating bound states at frequencies embedded in the continuous spectrum of the effective medium. These bound states in the continuum persist for $\varepsilon$-scale metric graphs due to $O(\varepsilon)$ operator-norm estimates for reduced resolvents on the defect region.