Theoretical considerations for inverse scattering

Inversion theory is a guide for handling inverse problems rather than a set of inversion techniques. The usual ill-posed character of inverse problems has first to be dealt with by reassessing the concept of solution and by giving a detailed answer to the basic mathematical questions of the solutions existence, uniqueness, construction, stability, and approximation. But this is not sufficient. If the results are to be used by physicists, they should take into account the fact that no physical problem can be given a complete study separately from all others. In other words, investigating mathematical questions, and in particular describing the sets of solutions, must be done in such a way that physical motivations and physical considerations are always apparent. These remarks are illustrated by several examples taken from scattering theory.