## 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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