Generalized optical theorem

We present the generalized optical theorem and its applications with special emphasis on the roles of bound states. First, we prove the theorem which gives a necessary and sufficient condition for a function ' k ' | T |k'of two variables k' and k to be physically acceptable as a half-on-shell T-matrix, i.e., to have an underlying Hermitian potential V . Secondly, using the theorem, we construct a scattering theory starting from a physically acceptable half-on-shell T-matrix'k' | T |k' , which in turn introduces a very useful classification scheme of Hermitian potentials. In the end, as an application of our theory, we present the most general solution of the inverse scattering problem with numerical examples.