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

of the wave speed Tuncay Aktosun Department of Mathematics, Southern Methodist University. Dallas, Texas 75275 Martin Klaus Department of Mathematics, Virginia Polytechnic Institute and State University, Blackburg, Virginia 24061 Cornelis van der Mee Department of Physics and Astronomy, Free University, Amsterdam, The Netherlands (Received 17 September 1991; accepted for publication 2 1 November 199 1) The inverse scattering problem for the 1-D Schriidinger equation d2$/dx2 + k21c, = mJ441, + Q(x) is studied. This equation is equivalent to the 1-D wave equation with speed l/ +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.