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Analytical model for laser-assisted recombination of hydrogenic atoms

We introduce a new method that allows one to obtain an analytical cross section for the laser-assisted electron-ion collision in a closed form. As an example we perform a calculation for the hydrogen laser-assisted recombination. The $S$-matrix element for the process is constructed from an exact electron Coulomb-Volkov wave function and an approximate laser modified hydrogen state. An explicit expression for the field-enhancement coefficient of the process is expressed in terms of the dimensionless parameter $κ= |{eε_{0}}/{qω_{0}}|^{2}$, where $e$ and $q$ are the electron charge and momentum respectively, and $ε_{0}$ and $ω_{0}$ are the amplitude and frequency of the laser field respectively. The simplified version of the cross section of the process is derived and analyzed within a soft photon approximation.