Scattering and Localization Properties of Highly Oscillatory Potentials

TL;DR

This paper studies how highly oscillatory, microstructured potentials affect scattering, localization, and decay in the 1D Schrödinger equation, revealing effective potentials and bound states not captured by homogenization.

Contribution

It derives an effective potential approximation for microstructured potentials, identifies bound states near zero energy, and analyzes their impact on scattering and time-decay properties.

Findings

Effective potential well captures low-energy scattering behavior.

Existence of bound states near zero energy for small oscillation scales.

Universal form of scaled transmission coefficient depending on a computable parameter.

Abstract

We investigate scattering, localization and dispersive time-decay properties for the one-dimensional Schr\"odinger equation with a rapidly oscillating and spatially localized potential, $q_\epsilon=q(x,x/\epsilon)$, 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, $t^{q_\epsilon}(k)$, as $\epsilon$ tends to zero. We derive an effective potential well, $\sigma^\epsilon_{eff}(x)=-\epsilon^2\Lambda_{eff}(x)$, such that $t^{q_\epsilon}(k)-t^{\sigma^\epsilon_{eff}}(k)$ 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 $\epsilon$, 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 $\epsilon^2$ from zero. It follows that the Schr\"odinger operator $H_{q_\epsilon}=-\partial_x^2+q_\epsilon(x)$ has an $L^2$ bound state with negative energy situated at a distance $O(\epsilon^4)$ 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 Schr\"odinger equation.