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.


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