Inverse scattering in 1-D nonhomogeneous media and recovery of the wave speed

The inverse scattering problem for the 1-D Schrodinger equation d2-psi/dx2 + k2-psi = k2P(x)psi + Q(x)psi is studied. This equation is equivalent to the 1-D wave equation with speed 1/ square-root 1 - P(x) in a nonhomogeneous medium where Q(x) acts as a restoring force. When Q(x) is integrable with a finite first moment, P(x) < 1 and bounded below and satisfies two integrability conditions, P(x) is recovered uniquely when the scattering data and Q(x) are known. Some explicitly solved examples are provided.