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, $q_{\epsilon }=q(x,x/\epsilon )$, 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, ${\sigma }_{eff}^{\epsilon }(x)=-{\epsilon }^2{\Lambda }_{eff}(x)$, such that $t^{q_{\epsilon }}(k)-t^{{\sigma }_{eff}^{\epsilon }}(k)$ 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 Schrödinger 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 Schrödinger equation.

to appear in Communications on Pure and Applied Mathematics

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