  We perform a systematic analysis of how nonsequential double ionization in
intense, linearly polarized laser fields is influenced by the initial states in
which both electrons are bound, and by the residual ionic potential. We assume
that the second electron is released by electron-impact ionization of the first
electron with its parent ion, using an S-Matrix approach. We work within the
Strong-Field Approximation, and compute differential electron momentum
distributions using saddle-point methods. Specifically, we consider electrons
in $1s$, $2p$, $3p$ and localized states, which are released by either a
contact or a Coulomb-type interaction. We also perform an adequate treatment of
the bound-state singularity which is present in this framework. We show that
the momentum distributions are very sensitive with respect to spatially
extended or localized wave functions, but are not considerably influenced by
their shape. Furthermore, the modifications performed in order to overcome the
bound-state singularity do not significantly alter the momentum distributions,
apart from a minor suppression in the region of small momenta. The only radical
changes occur if one employs effective form factors, which take into account
the presence of the residual ion upon rescattering. If the ionic potential is
of contact type, it outweighs the spreading caused by a long-range
electron-electron interaction, or by spatially extended bound states. This
leads to momentum distributions which exhibit a very good agreement with the
existing experiments.
