Abstract
The inverse-scattering theory at a fixed energy for the scattering of a particle by a potential in the Schr\"odinger equation formulated by Alam and Malik, which is based on the earlier work of Hooshyar and Razavy, is extended, in this paper, to the scattering of spinless particles at relativistic energies governed by the Klein-Gordon equation. The differential equation is replaced by a set of difference equations. This reduces the inverse-scattering problem to solving a continued fraction equation. The solution provides the values of the potential at a number of points which are equal to (one plus the number of partial waves). The theory is tested for three widely different complex potentials, one of which is relevant to pion-nucleus scattering. The points of the potentials determined from the inverse-scattering formalism are in accord with the actual ones in all three cases. Since the Klein-Gordon equation is effectively a Schr\"odinger equation with an energy-dependent potential, the method may, in the appropriate cases, be suitable for the latter case.
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