Bounds on Momentum Dependence of Phase Shifts and Magnitude of Coupling Constants

The analytic structure of partial-wave amplitudes is used to derive a dispersion relation for the scattering phase shift. This dispersion relation is used to obtain a lower bound on the momentum derivative of the phase shift. The bound depends on an integral over the unphysical (left-hand) cut in the momentum squared plane and can be expressed in terms of the number of zeros of the real part of the $S$ matrix along the unphysical cut. Stronger bounds are also presented involving the position of these zeros and the locations and widths of resonances and virtual states. The same approach is used to obtain limits on the magnitude of coupling constants.