Nonuniqueness in the inverse scattering problem

The inverse scattering problem consists of determining the functional form of a scattering potential given the scattering matrix A (k0s, k0s0) for all scattering directions s and one or more values of the wave vector k0s0. In this paper it is shown that within the framework of the first Born approximation the inverse scattering problem as defined above does not possess a unique solution. It is also shown that within the framework of exact (potential) scattering theory the problem does not admit a unique solution given only the scattering matrix for a single fixed value of the wave vector k0s0 as data. The final section in the paper considers scattering experiments using incident fields other than plane waves and where knowledge of the scattered field at all points exterior to the scattering volume is available as data. It is found that, within the framework of exact scattering theory, the data generated by any single such experiment is not sufficient to uniquely specify the scattering potential while, within the framework of the first Born approximation, the data generated by any finite number of such experiments is not sufficient to uniquely specify the potential.