The Inverse Problem

Abstract

We now turn to the inverse problem, in which it is assumed that the scattering amplitude is given as a function, IR × S² × S² ↦ ℂ, and the aim is to infer the underlying potential in the Schrodinger equation (1.1). It should be noted that from a physical point of view the experimental data are, at best, given by the differential scattering cross section ∣A∣² rather than the complex scattering amplitude itself. There is, therefore, a first step in the inverse problem that will not be discussed here: to infer the scattering amplitude from a knowledge of its modulus. For this step we refer to the literature (see, for example, [Ne82c, Section 20.2]). For our purposes the data will be considered to be the scattering amplitude A(k, θ, θ’).