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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
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References:

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