Dependence of the phase of the elastic scattering amplitude on momentum transfer

We discuss the extent to which the often assumed independence of the phase of the elastic scattering amplitude from momentum transfer in the domain of only small $t$ constrains the dependence of the phase on $t$ in general. Based on analyticity, we prove that if the scattering amplitude phase is independent of the transferred momentum in the domain of its small values in strong couplings, then this remains the case in the whole physical domain. Moreover, if such independence holds in any domain of physical energies including values infinitely close to the first inelastic threshold from below, then the whole scattering amplitude vanishes. We also discuss the relation between the dependence of the phase on $t$ and the size of the interaction domain.