Recovery of Discontinuities in a Nonhomogeneous Medium

The recovery of the coe cient H(x) in the one-dimensional generalized Schrodinger equation d dx2 + kH(x) = Q(x) ; where H(x) is a positive, piecewise continuous function with positive limits H as x! 1; is studied. This equation describes the wave propagation in a one-dimensional nonhomogeneous medium in which the wavespeed 1=H(x) changes abruptly at a nite number of points and a restoring force Q(x) is present. When there are no bound states, the uniqueness of H(x) in the inversion is established for a proper choice of scattering data. When the transmission coe cient vanishes at k = 0; it is shown that the scattering data consisting of Q(x) and a reduced re ection coe cient uniquely determine H(x); and neither H+ nor H need to be given as part of the scattering data. If the transmission coe cient does not vanish when k = 0; then one needs to include either H+ or H in the scattering data to obtain H(x) uniquely. A simple algorithm is described giving the travel times from x = 0 to any discontinuity of H(x) and the relative changes in the wavespeed in terms of the large k-asymptotics of a (reduced) re ection coe cient. It is also shown that H+ and the transmission coe cient alone do not determine the number of discontinuities of H(x); let alone the travel times between them. Some examples are given to illustrate the algorithm. PACS Numbers: 03.65.Nk, 03.80.+r, 03.40.Kf, 43.20.+g Mathematics Subject Classi cation (1991): 81U40, 73D50, 34A55