  By means of empirical fits to the differential cross section data on pp and
p(bar)p elastic scattering, above 10 GeV (center-of-mass energy), we determine
the eikonal in the momentum - transfer space (q^2- space). We make use of a
numerical method and a novel semi-analytical method, through which the
uncertainties from the fit parameters can be propagated up to the eikonal in
the $q^2$- space. A systematic study of the effect of the experimental
information at large values of the momentum transfer is developed and discussed
in detail. We present statistical evidence that the imaginary part of the
eikonal changes sign in the q^2- space and that the position of the zero
decreases as the energy increases; after the position of the zero, the eikonal
presents a minimum and then goes to zero through negative values. We discuss
the applicability of our results in the phenomenological context, outlining
some connections with nonperturbative QCD. A short review and a critical
discussion on the main results concerning "model-independent" analyses are also
presented.
