  We study the scattering properties of Schr\"{o}dinger operators with
potentials that have short-range decay along a collection of rays in $\bbR^d$.
This generalizes the classical setting of short-range scattering in which the
potential is assumed to decay along \emph{all} rays. For these operators, we
show that any state decomposes into an asymptotically free piece and a piece
which may interact with the potential for long times. We give a microlocal
characterization of the scattering states in terms of the dynamics and a
corresponding description of their complement. We also show that in certain
cases these characterizations can be purely spatial.
