The stability of one-dimensional inverse scattering

The stability of one-dimensional inverse scattering is discussed. The analysis is performed analytically using perturbation theory for reflection coefficients that are rational functions of the wavenumber and numerically for realistic reflection coefficients. It is shown that the same instability features occur for real reflections as for rational reflection coefficients. There are three types of error considered which are common in the application of the data. Explicitly examined are an erroneously chosen DC level, an amplitude error and a time error. It is shown that the physical reason for the instabilities of the one-dimensional inverse problem is the lack of sufficient energy penetration in the potential that is examined.