Scattering theory tells how solutions of one abstract Schrödinger equation (of the form
${\rm i}({\rm d}u/{\rm d}t)=Hu$
with
$H=H^*)$
are asymptotic to solutions of another (in principle simpler) abstract Schrödinger equation. We extend this theory to inhomogeneous problems of the form
${\rm i}({\rm d}u)/({\rm d}t)=Hu+h(t)$
, with special emphasis on factored equations of the form
$\prod_{j=1}^N (({\rm d}/{\rm d}t)-{\rm i}A_j)u(t) = h(t),$
where
$A_1,\ldots, A_N$
are commuting selfadjoint operators. As a special case, corresponding to
$N=4$
and two‐space scattering, we conclude that every solution
$u(\cdot, t)$
of the inhomogeneous elastic wave equation in the exterior of a bounded star shaped obstacle is of the form
$u=v+w+z,$
where
$v(\cdot, t)$
solves the free (homogeneous) elastic wave equation with no obstacle,
$w(\cdot,t)$
is determined by the (rather general) inhomogeneity, and
$z(\cdot,t)={\rm o}(1)$
as
$t\to \pm \infty.$
Some of the results are presented in a more general Banach space context.
