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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 $\left({d}^{2}\psi /d{x}^{2}\right)+{k}^{2}H{\left(x\right)}^{2}\psi =Q\left(x\right)\psi$, where $H\left(x\right)$ is a positive, piecewise continuous function with positive limits ${H}_{±}$ as $x\to ±\infty$. This equation, in the frequency domain, describes the wave propagation in a nonhomogeneous medium, where $Q\left(x\right)$ is the restoring force and $1/H\left(x\right)$ 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 $H\left(x\right)$ in terms of the scattering data consisting of $Q\left(x\right)$, a reflection coefficient, either of ${H}_{±}$, and the bound state energies and norming constants. Some explicitly solved examples are provided.
##### MSC:
 34L40 Particular ordinary differential operators 81U40 Inverse scattering problems (quantum theory)