The relativistic low-frequency approximation is derived for the potential scattering of electrons in an ultrastrong elliptically polarised laser pulse. It has a correct on-shell non-relativistic limit and is highly accurate in the high-intensity limit (I>>IA) even for the optical frequencies.

Potential and electron-atom scattering in the presence of an intense low-frequency laser field is considered. Forms of the T matrices convenient for low-frequency treatment are developed. Unambiguous answers are obtained on convergence and on the upper bounds of the common, on-shell low-frequency results in terms of the expansion parameters, which are dependent either on the laser field intensity or on the frequency. The compact form of the T matrix, which includes corrections in the laser electric field amplitude to the low-frequency approximation, is derived. It has the same form for both potential and electron-atom scattering and is applicable up to laser intensities of 1014W cm-2 in the low-frequency range.

The total T matrix for the potential scattering in a laser field is decomposed into resonant and non-resonant parts using the projection operator formalism. The general case of several resonances is considered. It is shown that the resonant contribution to the total T matrix can be written as the sum over resonant states of the product of three factors: initial and final laser-modified vertex functions and a factor which contains the level shift matrix. A new method is used for deriving the low-frequency expansion for these three factors. The zeroth-order term gives the generalisation of the non-resonant Kroll-Watson (1973) result to the case of several resonances. Finally, the corrections in the laser electric field amplitude, to these nearly on-shell terms, are also obtained.