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The Klein-Gordon equation in the presence of a strong electric field, takingthe form of the Mathieu equation, is studied. A novel analytical solution isderived for particles whose asymptotic energy is much lower or much higher thanthe electromagnetic field amplitude. The condition for which the new solutionrecovers the familiar Volkov wavefunction naturally follows. When notsatisfied, significant deviation from the Volkov wavefunction is demonstrated.The new condition is shown to differ by orders of magnitudes from the commonlyused one. As this equation describes (neglecting spin effects) the emissionprocesses and the particle motion in Quantum Electrodynamics (QED) cascades,our results suggest that the standard theoretical approach towards thisphenomenon should be revised.

A Novel Solution to the Klein-Gordon Equation in the Presence of a Strong Rotating Electric Field