## Integral equation methods for the inverse problem with discontinuous wave speed.(English)Zbl 0897.34018

The inverse scattering problem for a one-dimensional generalised Schrödinger equation $\frac{d^{2}}{dx^{2}}\psi (k,x)+k^{2}H(x)^{2}\psi (x,k) =Q(x)\psi (x,k),\quad x\in {\mathbb{R}}$ is considered where $$H(x)$$ is strictly positive and piecewise continuous. One of the main results is an algorithm to obtain the discontinuities of $$H(x)$$ and $$\frac{d}{dx}H(x)/H(x)$$ from the asymptotics at large $$k$$ of the reflection coefficient. Also, Marchenko’s integral equation method is extended to the case at hand. Some explicit examples are worked out.

### MSC:

 34A55 Inverse problems involving ordinary differential equations 34L25 Scattering theory, inverse scattering involving ordinary differential operators 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 47A40 Scattering theory of linear operators 81U40 Inverse scattering problems in quantum theory

### Keywords:

Schrödinger equation; scattering data; inverse problems
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### References:

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