Scattering and localization properties of highly oscillatory potentials
Résumé
We investigate scattering, localization and dispersive time-decay properties for the one-dimensional Schrödinger equation with a rapidly oscillating and spatially localized potential, $q_\eps=q(x,x/\eps)$, 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, {\it e.g.} the transmission coefficient, $t^{q_\eps}(k)$, as $\eps$ tends to zero. We derive an {\it effective potential well}, $\seff(x)=-\eps^2\Lambda_{\rm eff}(x)$, such that $t^{q_\eps}(k)-t^\seff(k)$ 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 $\eps$, 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 $\eps^2$ from zero. It follows that the Schrödinger operator $H_{q_\eps}=-\partial_x^2+q_\eps(x)$ has an $L^2$ bound state with negative energy situated a distance $\mathcal{O}(\eps^4)$ from the edge of the continuous spectrum. Finally, we use this detailed information to prove the local energy time-decay estimate $\big|e^{-i t H_{q_\eps}} P_c \psi_0\big|_{\mathcal{L}^\infty_3}\ \le\ C\ t^{-1/2}\ \left(1+\eps^4\ \left(\int_\mathbb{R}\Lambda_{\rm eff}\right)^2\ t\ \right)^{-1}\ \big|\psi_0\big|_{\mathcal{L}^1_3}$, where $P_c$ denotes the projection onto the continuous spectral part of $H_{q_\eps}$.
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