The improved strong-field approximation (ISFA) is a version of the strong-field approximation which takes into account an additional interaction of the ionized electron with the parent ion within the first Born approximation. The ISFA describes well the middle- and high-energy parts of the electron spectra in the above-threshold ionization process. We show, using an example of a short-range potential, that the ISFA is able to describe the low-energy structure in the energy spectra if it is calculated without additional approximations. We introduce two different forms of the T-matrix element which are appropriate for application of two widely used approximations: the saddle-point approximation [i.e., its more advanced version, the uniform approximation (UA)] and the pole approximation (PA). We show that both the PA and UA are not able to describe the low-energy structure. Furthermore, the UA describes better the plateau of the spectrum than the PA. We also identify the origin of a very-low-energy structure; it is connected to the laser-free (i.e., without exchange of the laser photons) electron forward scattering.
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