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Inverse wave scattering with discontinuous wave speed. (English) Zbl 0843.34080
Summary: The inverse scattering problem on the line is studied for the generalized Schrödinger equation $(d^2 \psi/dx^2) + k^2 H(x)^2 \psi = Q(x) \psi$, where $H(x)$ is a positive, piecewise continuous function with positive limits $H_\pm$ as $x \to \pm \infty$. This equation, in the frequency domain, describes the wave propagation in a nonhomogeneous medium, where $Q(x)$ is the restoring force and $1/H (x)$ is the variable wave speed changing abruptly at various interfaces. A related Riemann-Hilbert problem is formulated, and the associated singular integral equation is obtained and proved to be uniquely solvable. The solution of this integral equation leads to the recovery of $H(x)$ in terms of the scattering data consisting of $Q(x)$, a reflection coefficient, either of $H_\pm$, and the bound state energies and norming constants. Some explicitly solved examples are provided.

34L40Particular ordinary differential operators
81U40Inverse scattering problems (quantum theory)
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