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].


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