Potential Scattering as Opposed to Scattering Associated with Independent Particles in the $S$-Matrix Theory of Strong Interactions

A definition of a relativistic generalized potential is given, suitable at arbitrary energies for a pair of particles whose elastic scattering amplitude satisfies the Mandelstam representation. It is shown that the generalized potential plays a role in the dynamics analogous to that of the ordinary nonrelativistic potential in a Schrödinger equation and determines the scattering to the same extent. Below the threshold for inelastic processes the generalized potential is real and its energy dependence in the elastic region is expected for certain particle combinations (such as the nucleon-nucleon) to be weak. In such cases one may uniquely define, for use in the Schrödinger equation, an energy-independent ordinary potential that coincides with the potential of Charap and Fubini. In general, when the potential is complex and energy-dependent the dynamical problem involves iteration of an integral equation deduced by Mandelstam. The generalized potential may be decomposed according to range and it is shown that keeping only the long- and medium-range parts, corresponding to transfer of one or two particles, is almost equivalent to the "strip approximation." Finally, a general definition is given of "pure potential scattering" as opposed to scattering associated with "independent" particles, either stable or unstable, and a variety of experimental situations are discussed with respect to this distinction, which is shown to be susceptible to experimental test.