Optical-Model and Mandelstam Representation

It is shown that it is always possible to find explicitly an optical potential which, when used in the eikonal approximation, fits a scattering amplitude satisfying the Mandelstam assumptions. The potential can be constructed by making a series of integral transformations on the momentum-transfer discontinuity of the amplitude. In the case in which the asymptotic behavior in energy of the amplitude is given by the Reggepole hypothesis, it is shown that, although the range of the potential (behavior $e^{-{\mu }_{0}r}$) is constant, the size of the interaction region increases with energy owing to the nonuniformity in energy of the asymptotic behavior of the potential as a function of distance.