Characterization of the shape of a scattering obstacle using the spectral data of the far field operator

This paper is concerned with the inverse obstacle scattering problem for time harmonic plane waves. We derive a factorization of the far field operator F in the form GSG* and prove that the ranges of root F and G coincide. Then we give an explicit characterization of the scattering obstacle which uses only the spectral data of the far field operator F. This result is used to prove a convergence result for a recent numerical method proposed by Colton, Kirsch, Monk, Piana and Potthast. We illustrate this method by some numerical examples.