  The Klein-Gordon equation in the presence of a strong electric field, taking
the form of the Mathieu equation, is studied. A novel analytical solution is
derived for particles whose asymptotic energy is much lower or much higher than
the electromagnetic field amplitude. The condition for which the new solution
recovers the familiar Volkov wavefunction naturally follows. When not
satisfied, significant deviation from the Volkov wavefunction is demonstrated.
The new condition is shown to differ by orders of magnitudes from the commonly
used one. As this equation describes (neglecting spin effects) the emission
processes and the particle motion in Quantum Electrodynamics (QED) cascades,
our results suggest that the standard theoretical approach towards this
phenomenon should be revised.
