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 $ε$ tends to zero. We derive an effective potential well, $σ^ε_{eff}(x)=-ε^2Λ_{eff}(x)$, such that $t^{q_ε}(k)-t^{σ^ε_{eff}}(k)$ is uniformly small on $\mathbb{R}$ and small in any bounded subset of a suitable complex strip. Within such a bounded subset, the scaled 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 zero. It follows that the Schrödinger operator $H_{q_ε}=-\partial_x^2+q_ε(x)$ has an $L^2$ bound state with negative energy situated at a distance $O(ε^4)$ from the edge of the continuous spectrum. Finally, we use this detailed information to prove a local energy time-decay estimate of the time-dependent Schrödinger equation.