Scattering and localization properties of highly oscillatory potentials

We investigate scattering, localization and dispersive time-decay properties for the onedimensional Schrödinger equation with a rapidly oscillating and spatially localized potential, q = q(x, x/ ), 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 lowenergy (k small) behavior of scattering quantities, e.g. the transmission coefficient, t (k), as tends to zero. We derive an effective potential well, σ eff(x) = − Λeff(x), such that t (k) − t eff(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 , 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 2 from zero. It follows that the Schrödinger operator Hq = −∂ x + q (x) has an L bound state with negative energy situated a distance O( ) from the edge of the continuous spectrum. Finally, we use this detailed information to prove the local energy time-decay estimate: ∣∣(1 + | · |)−3e−itHq Pcψ0∣∣L∞ ≤ C t−1/2 (1 + 4 ( ∫R Λeff)2 t )−1∣∣(1 + | · |3)ψ0∣∣L1 , where Pc denotes the projection onto the continuous spectral part of Hq .