A time-dependent approach to the inverse backscattering problem

We describe a time-dependent approach to the inverse backscattering problem within the framework of the Lax-Phillips theory of scattering,. We apply t his approach to the inverse problem, arising in quantum mechanics, of deter mining a potential from the backscattering amplitude. As a consequence, we give an alternative proof of the fact that the backscattering map is global ly Fredholm. A corollary is that knowledge of the backscattering amplitude determines a potential which is close, in an appropriate topology, to an op en and dense set of potentials. We also give an outline of a procedure on h ow to recover the singularities of a conormal potential from the singularit ies of backscattering data. Another inverse problem that we discuss, arisin g in ultrasound, is to determine the index of refraction (sound speed) of a medium from backscattering information, i.e. by measuring the echoes.