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.


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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