  The analytic properties of the eikonal and U-matrix unitarization schemes are
examined. It is shown that the basic properties of these schemes are identical.
Both can fill the full circle of unitarity, and both can lead to standard and
non-standard asymptotic relations for the ratio of the elastic cross section to
the total cross section. The relation between the phases of the unitarized
amplitudes in each scheme is examined, and it is shown that demanding
equivalence of the two schemes leads to a bound on the phase in the U-matrix
scheme.
