Set of Crossing-Symmetric Dynamical Equations and Infinitely Rising Regge Trajectories

An equation based on unitarity and analyticity is written down. It is a relativistic generalization of a Schr\"odinger equation with a local energy-dependent potential of the Yukawa type, in contrast to the usual Mandelstam equations, which correspond to a nonlocal energy-independent potential. A prescription is given for our potential, which makes the amplitude exactly crossing symmetric. It is argued that a first guess at the potential might simply be to take the low-energy crossed-channel resonances of the Veneziano model. The cutoff on the number of resonances is then an undetermined parameter, which, however, merely serves to fix the energy scale in the limit of a vanishingly small pion mass. A determinantal approximation to our scheme (which differs from the determinantal approximation to the usual N/D equations) is then found to lead to Regge trajectories which rise indefinitely. The $I=1$ output resonances agree approximately with the input ones, average duality is found to be satisfied, and the P-wave scattering length comes out close to the Weinberg value. These results do not change much, at least at low engeries, even if a strong dose of inelasticity is introduced.