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Scattering and inverse scattering for a second-order differential equation. (English) Zbl 0773.34011

Summary: The scattering and three inverse scattering problems for \((d/dx)[a(x)(d\psi/dx)]+k^ 2h(x)\psi=Q(x)\psi\) on the real axis are considered herein. In the first inverse scattering problem, \(Q(x)\) is recovered when \(a(x)\), \(h(x)\), and the scattering data are given. In the second inverse problem \(h(x)\) is recovered when \(a(x)\), \(Q(x)\), and the scattering data are known. In the third inverse problem, in case \(Q(x)=0\), \(a(x)\) is recovered when \(h(x)\) and the scattering data are known. The inversion is illustrated with examples.

MSC:

34A55 Inverse problems involving ordinary differential equations
34L25 Scattering theory, inverse scattering involving ordinary differential operators
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References:

[1] Faddeev L. D., Am. Math. Soc. Transl. 2 pp 139– (1964)
[2] Faddeev L. D., Tr. Mat. Inst. Stekl. 73 pp 314– (1964)
[3] DOI: 10.1063/1.524447 · Zbl 0446.34029
[4] DOI: 10.1063/1.529650 · Zbl 0760.35032
[5] DOI: 10.1063/1.529714 · Zbl 0756.34083
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