Recovery of discontinuities in a non-homogeneous medium

The recovery of the coefficient H(x) in the one-dimensional generalized Schrodinger equation d(2) psi/dx(2) + k(2)H(x)(2) psi = Q(x)psi, where H(x) is a positive, piecewise continuous function with positive limits H-+/- as x --> +/-infinity, is studied. This equation describes the wave propagation in a one-dimensional non-homogeneous medium in which the wavespeed 1/H(x) changes abruptly at a finite 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 coefficient vanishes at k = 0, it is shown that the scattering data consisting of Q(x) and a reduced reflection coefficient uniquely determine H(x), and neither H-+ nor H-- need to be given as part of the scattering data If the transmission coefficient 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) reflection coefficient. It is also shown that H-+ and the transmission coefficient 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.