The Direct Scattering Problem

Abstract

We wish to solve the Schrodinger equation

$$(\Delta + K^2 )\psi \; = \;V\psi ,$$

(1.1)

where $\Delta$ is the Laplacian in IR³, with boundary conditions appropriate to scattering. The function V, IR³ ↦ IR, is not assumed to have any particular symmetry properties but it is assumed to decrease to zero at infinity in a manner to be specified later. The solution is to describe a plane wave sent in the direction of the unit vector θ toward the “scattering center,” together with an outgoing spherical wave that describes the response of this center.