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Inverse scattering in one-dimensional nonconservative media. (English) Zbl 0916.34070

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.

MSC:

34L25 Scattering theory, inverse scattering involving ordinary differential operators
81U40 Inverse scattering problems in quantum theory
74J25 Inverse problems for waves in solid mechanics
34A55 Inverse problems involving ordinary differential equations
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