Generalized optical theorem

Abstract

Abstract 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 $\langle {\boldsymbol {k}}^{\prime } | T | {\boldsymbol {k}} \rangle$ of two variables ${\boldsymbol {k}}^{\prime }$ and ${\boldsymbol {k}}$ 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 $\langle {\boldsymbol {k}}^{\prime } | T | {\boldsymbol {k}} \rangle$, 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.