  For the description of nonsequential double ionization (NSDI) of rare-gas
atoms by a strong linearly polarized laser field, the quantum-mechanical
$S$-matrix diagram that incorporates rescattering impact ionization is
evaluated in the strong-field approximation. We employ a uniform approximation,
which is a generalization of the standard saddle-point approximation. We
systematically analyze the manifestations of the electron-electron interaction
in the momentum distributions of the ejected electrons: for the interaction, by
which the returning electron frees the bound electron, we adopt either a
(three-body) contact interaction or a Coulomb interaction, and we do or do not
incorporate the mutual Coulomb repulsion of the two electrons in their final
state. In particular, we investigate the correlation of the momentum components
parallel to the laser-field polarization, with the transverse momentum
components either restricted to certain finite ranges or entirely summed over.
In the latter case, agreement with experimental data is best for the contact
interaction and without final-state repulsion. In the former, if the transverse
momenta are restricted to small values, comparison of theory with the data
shows evidence of Coulomb effects. We propose that experimental data selecting
events with small transverse momenta of \textit{both} electrons are
particularly promising in the elucidation of the dynamics of NSDI. Also, a
classical approximation of the quantum-mechanical $S$ matrix is formulated and
shown to work very well inside the classically allowed region.
