## Inverse scattering problem for the wave equation with discontinuous wavespeed.(English)Zbl 0843.34079

Bainov, D. (ed.) et al., Proceedings of the fifth international colloquium on differential equations, Plovdiv, Bulgaria, August 18-23, 1994. Utrecht: VSP. 3-21 (1995).
Summary: The inverse 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 $$k$$ is the wavenumber, $$1/H (x)$$ is the wavespeed, and $$Q(x)$$ is the restoring force per unit length. $$H (x)$$ is a positive, piecewise continuous function having limits $$H_\pm$$ as $$x \to \pm \infty$$, and $$Q(x)$$ satisfies certain integrability conditions. This equation describes wave propagation in a nonhomogeneous medium in which the wavespeed is allowed to change abruptly at certain interfaces. The inverse problem considered here consists in determining the function $$H(x)$$ from a suitable set of scattering data and for a given $$Q(x)$$. At the heart of the solution are a Riemann-Hilbert problem and a related singular integral equation. The solvability of the integral equation is discussed, and the solution method is illustrated by some explicitly solved examples.
For the entire collection see [Zbl 0835.00016].

### MSC:

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