Explicit expressions of the potential and its derivatives at the origin in terms of the scattering data

The quantum mechanical theory of scattering of a particle by a spherically symmetrical potential is presented. As in the inverse scattering problem, the input of the calculation is the scattering and bound-state data, and the output is data on the potential. The results discussed are explicit expressions for the values of the potential and its derivatives at the origin in terms of the scattering and boundstate data. Various methods to obtain these results are outlined. The presentation is aimed at introducing these various approaches. The simplest scattering problem (nonrelativistic S-wave scattering on a holomorphic potential without bound states) is used as the basis for discussion, and technicalities are omitted whenever possible without loss of clarity. A complete compilation is given of the results obtained to date in this field, including the treatment of higher partial waves and the Klein-Gordon and Dirac equations.