Inverse wave scattering with discontinuous wave speed

The inverse scattering problem on the line is studied for the generalized Schrödinger equation (d 2ψ/dx 2)+k 2 H(x)2ψ=Q(x)ψ, where H(x) is a positive, piecewise continuous function with positive limits H ± as x → ±∞. 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 ±, and the bound state energies and norming constants. Some explicitly solved examples are provided.