  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.
