Scattering and Localization Properties of Highly Oscillatory Potentials

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Abstract

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

display math

where math formula denotes the projection onto the continuous spectral part of math formula. © 2013 Wiley Periodicals, Inc.

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