Values of the Potential and Its Derivatives at the Origin in Terms of the S-Wave Phase Shift and Bound-State Parameters

Abstract

The framework of this paper is the quantum‐mechanical theory of nonrelativistic scattering of a particle by a spherically symmetrical, local, and energy‐independent potential. The principle results are explicit expressions for the values at the origin of the potential and of its derivatives in terms of the S‐wave scattering phase shift and of the parameters of the S‐wave bound states (if any). Previous work in this direction is reviewed, and different routes to the establishing of these results are discussed. Some remarks concerning the characterization of S‐wave physically equivalent potentials (i.e., potentials having the same S‐wave phase shift and S‐wave binding energies) are presented. The properties of the S‐wave phase shift and bound state parameters which are necessary and sufficient to ensure that the corresponding potential be an even function of r are ascertained. The analysis is restricted to potentials represented by holomorphic functions of r.