Recovery of discontinuities in a non-homogeneous medium

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Abstract
The recovery of the coefficient H(x) in the one-dimensional generalized Schrödinger equation , where H(x) is a positive, piecewise continuous function with positive limits as , 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 nor 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 or 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 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.

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