ON THE INVERSE SCATTERING PROBLEM FOR THE HELMHOLTZ-EQUATION IN ONE DIMENSION

A scheme is presented for the solution of inverse scattering problems for the one-dimensional Helmholtz equation. The scheme is based on a combination of the standard Riccati equation for the impedance function with a new trace formula for the derivative of the potential, and can be viewed as a frequency-domain version of the layer-stripping approach. The pricipal advantage of our proceedure is that if the scatterer to be reconstructed has m greater-than-or-equal-to 1 continuous derivatives, the accuracy of the reconstruction is proportional to 1/a(m), where a is the highest frequency for which scattering data are available. Thus a smooth scatterer is reconstructed very accurately from a limited amount of available data. The scheme has an asymptotic cost O(n2), where n is the number of features to be recovered (as do classical layer-stripping algorithms), and is stable with respect to perturbations of the scattering data. The performance of the algorithm is illustrated by several examples.