## 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.

### MSC:

 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 81U40 Inverse scattering problems in quantum theory
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### References:

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