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Scattering and inverse scattering in one-dimensional nonhomogeneous media. (English) Zbl 0760.35032

Summary: The wave propagation in a one-dimensional nonhomogeneous medium is considered, where the wave speed and the restoring force depend on location. In the frequency domain this is equivalent to the Schrödinger equation \(d^ 2\psi/dx^ 2+k^ 2\psi=k^ 2P(x)\psi+Q(x)\psi\) with an added potential proportional to energy. The scattering and bound-state solutions of this equation are studied and the properties of the scattering matrix are obtained; the inverse scattering problem of recovering the restoring force when the wave speed and the scattering data are known are also solved.

MSC:

35P25 Scattering theory for PDEs
35Q40 PDEs in connection with quantum mechanics
81U40 Inverse scattering problems in quantum theory
35R30 Inverse problems for PDEs
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