Nonuniqueness properties of the physical solutions of the Lorentz-Dirac equation

The solutions of the Lorentz-Dirac equation are investigated, for the problem of a one-dimensional scattering of a charged particle by a potential barrier, and a phenomenon is found having some similarity to the quantum weak-reflection effect. Namely, there exists an energy strip, slightly above the maximum of the barrier, such that for any given initial energy in the strip there is a certain number of physical (or nonrunaway) solutions of two types, i.e. Those of mechanical type, transmitted beyond the barrier, and those of nonmechanical type, reflected by the barrier. From the mathematical point of view, the existence of this phenomenon is related to the nonuniqueness of the physical solutions of the Lorentz-Dirac equation for given initial data of position and velocity. This in turn is strictly related to a property recently pointed out, namely the asymptotic character of the relevant series expansions occurring for that equation. Correspondingly, the width of the energy strip where the phenomenon occurs is found to decrease exponentially fast, as the small parameter entering the problem tends to zero.