  We consider the scattering of high energy leptons off bound atomic electrons
focusing primarily on final state interactions i.e., the exchange of virtual
photons between the outgoing energetic electron, and the heavy residual charged
"debris" in the final state. These effects are inherently absent from
calculations for a free electron at rest. Coulomb exchanges are enhanced by the
large number of electrons in the atomic debris, and are unsuppressed by
non-relativistic velocities in the debris. We find that these exchanges can be
resummed using operator methods, and cancel at the level of the cross section
until at least $O(\alpha^3)$. Furthermore, we argue that both final {\it and}
initial state Coulomb exchanges (enhanced by the number of electrons in the
atom) do not affect the cross section until at least $O(\alpha^3)$. Transverse
photon couplings to non relativistic electrons are proportional to their small
velocities, and rotational invariance suppresses their contribution to
$O(\alpha^3)$. Our results are relevant for precision experiments involving
neutrinos, electrons, positrons, and muons scattering off of atomic electrons
in a fixed target.
