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Inverse scattering in 1-D nonhomogeneous media and recovery of the wave speed. (English) Zbl 0756.34083

Summary: The inverse scattering problem for the 1-D Schrödinger equation \(d^ 2\psi/dx^ 2+k^ 2\psi=k^ 2P(x)\psi+Q(x)\psi\) is studied. This equation is equivalent to the 1-D wave equation with speed \(1/\sqrt{1- P(x)}\) in a nonhomogeneous medium where \(Q(x)\) acts as a restoring force. When \(Q(x)\) is integrable with a finite first moment, \(P(x)<1\) and bounded below and satisfies two integrability conditions, \(P(x)\) is recovered uniquely when the scattering data and \(Q(x)\) are known. Some explicitly solved examples are provided.

MSC:

34L25 Scattering theory, inverse scattering involving ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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References:

[1] Faddeev L. D., Am. Math. Soc. Transl. 2 pp 139– (1964)
[2] Faddeev L. D., Trudy Mat. Inst. Steklova 73 pp 314– (1964)
[3] DOI: 10.1002/cpa.3160320202 · Zbl 0388.34005
[4] DOI: 10.1063/1.524447 · Zbl 0446.34029
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