  The feasibility of obtaining exact analytical results in the realm of QED in
the presence of a background electromagnetic field is almost exclusively
limited to a few tractable cases, where the Dirac equation in the corresponding
background field can be solved analytically. This circumstance has restricted,
in particular, the theoretical analysis of QED processes in intense laser
fields to within the plane-wave approximation even at those high intensities,
achievable experimentally only by tightly focusing the laser energy in space.
Here, within the Wentzel-Kramers-Brillouin (WKB) or eikonal approximation, we
construct analytically single-particle electron states in the presence of a
background electromagnetic field of general space-time structure in the
realistic assumption that the initial energy of the electron is the largest
dynamical energy scale in the problem. The relatively compact expression of
these states opens, in particular, the possibility of investigating
analytically strong-field QED processes in the presence of spatially focused
laser beams, which is of particular relevance in view of the upcoming
experimental campaigns in this field.
