Inverse scattering in one-dimensional nonconservative media
The authors give a careful analysis of the direct and inverse scattering problem for the (generalized) Schrödinger equation \[ \psi''+k^2 \psi = (\pm i k P +Q) \psi \] on the real line. Here, as usual, the prime denotes the derivative with respect to the spatial variable \(x\), \(k\) is the wavenumber, and the real-valued functions \(P(x)\) and \(Q(x)\) denote the energy absorption/generation and the restoring force density, respectively. The scattering data of the above equation with the \(+\) sign are suitably defined and are complemented with those of the \(-\) sign equation in order to recover \(P(x)\) and \(Q(x)\). The inverse scattering procedure developed by the authors makes use of a pair of uncoupled Marchenko integral equations. Conditions for the uniqueness of solutions to such equations are presented. The paper concludes with some illustrative examples.
scientific article; zbMATH DE number 1140087