On the Inverse Problem for the Klein-Gordon s-Wave Equation

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Abstract
The s‐wave Klein‐Gordon equation of a particle in an electriclike, spherically symmetrical, local, and energy‐independent potential is studied with the aim of deriving exact relationships between the potential and the experimental data. We obtain the value of the potential and of all its derivatives at the origin as explicit functionals of the phase shift of the particle and the antiparticle and of their bound‐state parameters. Furthermore, we find a set of infinitely many nonlinear sum rules for the phase shift and the bound‐state parameters of both the particle and the antiparticle; these sum rules embody the requirement that the potential be electriclike. This restriction on the potential is also briefly discussed in the framework of the Gel'fand‐Levitan approach to the inverse problem.