We investigate scattering, localization, and dispersive time decay properties for the one-dimensional Schrödinger equation with a rapidly oscillating and spatially localized potential , where 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 as ∊ tends to zero. We derive an effective potential well such that is small, uniformly for 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 from . It follows that the Schrödinger operator has an bound state with negative energy situated a distance from the edge of the continuous spectrum. Finally, we use this detailed information to prove the local energy time decay estimate:
where denotes the projection onto the continuous spectral part of . © 2013 Wiley Periodicals, Inc.