Scattering and localization properties of highly oscillatory potentials

Abstract

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 {small element of} 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{lunate}, 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 O ( 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 ( 1 + e 4 ( ∫ 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.