We investigate scattering, localization, and dispersive time decay properties for the one‐dimensional Schrödinger equation with a rapidly oscillating and spatially localized potential q ε = q ( x , x / ε ), where q ( x , y ) is periodic and mean zero with respect to y. Such potentials model a microstructured medium. Homogenization theory fails to capture the correct low‐energy (k small) behavior of scattering quantities, e.g., the transmission coefficient t q ε ( k ) as ∊ tends to zero. We derive an effective potential well σeffε ( x ) = − ε 2 Λ eff ( x ) such that t q ε ( k ) − t σeffε ( k ) is small, uniformly for k ∈ R as well as in any bounded subset of a suitable complex strip. Within such a bounded subset, the scaled limit of the transmission coefficient has a universal form, depending on a single parameter, which is computable from the effective potential. A consequence is that if ϵ, the scale of oscillation of the microstructure potential, is sufficiently small, then there is a pole of the transmission coefficient (and hence of the resolvent) in the upper half‐plane on the imaginary axis at a distance of order ε 2 from 0. It follows that the Schrödinger operator H q ε = − ∂ x 2 + q ε ( x ) has an L 2 bound state with negative energy situated a distance frakturO ( ε 4 ) from the edge of the continuous spectrum. Finally, we use this detailed information to prove the local energy time decay estimate: | ( 1 + | · | ) − 3 e − i t H q e P c ψ 0 | L ∞ ≤ * C t − 1 / 2 true( 1 + e 4   true( ∫ l ∫ R Λ eff ) 2 t ) − 1 | ( 1 + | · | 3 ) ψ 0 | L 1 , where P c denotes the projection onto the continuous spectral part of H q ε. © 2013 Wiley Periodicals, Inc.

We present an analytical theory of topologically protected photonic states for the two-dimensional Maxwell equations for a class of continuous periodic dielectric structures, modulated by a domain wall. We further numerically confirm the applicability of this theory for three-dimensional structures.

International audienceWaves in extended periodic structures are well-known to spatially disperse and decay in amplitude as time advances. This dispersion is associated with the continuous spectrum of the underlying differential operator and the absence of discrete eigenvalues. The introduction of localized perturbations, leads to defect modes, states in which energy remains trapped and spatially localized. In this paper we present results on the bifurcation of such defect modes, associated with the emergence of discrete eigenvalues from the continuous spectrum