## Nonuniqueness in inverse acoustic scattering on the line.(English)Zbl 0806.34014

Summary: The generalized one-dimensional Schrödinger equation $$d^ 2 \varphi/dx^ 2 + k^ 2 H(x)^ 2 \varphi = P (x) \varphi$$ is considered. The nonuniqueness is studied in the recovery of the function $$P(x)$$ when the scattering matrix, $$H(x)$$, and the bound state energies and norming constants are known. It is shown that when the reflection coefficient is unity at zero energy, there is a one-parameter family of functions $$P(x)$$ corresponding to the same scattering data. An explicitly solved example is provided. The construction of $$H(x)$$ from the scattering data is also discussed when $$H(x)$$ is piecewise continuous, and two explicitly solved examples are given with $$H(x)$$ containing a jump discontinuity.

### MSC:

 34A55 Inverse problems involving ordinary differential equations 81U40 Inverse scattering problems in quantum theory 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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### References:

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