Affiliations: Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152‐6429, USA | Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, I‐40126 Bologna, Italy
Abstract: 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.
