Electron scattering from atoms in the presence of a laser field

The formal development of the problem of electron-atom scattering in the presence of a laser field is considered. A time-dependent close-coupling method and a time-dependent optical-potential method are presented. With hydrogen as an example, and a weak laser tuned near the $1S-2P$ resonance, the effective long-range potentials are explicitly obtained. These potentials are time dependent, noncentral, and fall off as $r^{-n}$, where we limit ourselves to terms with $n=2, 3, \mathrm{and} 4$. They are then used to find the elastic, inelastic, and superelastic cross sections for small momentum transfer.