An Inverse Scattering Problem for Dirac Equations with Time-Dependent Electromagnetic Potentials

Abstract

We consider an inverse scattering problem, by using a time-dependent method, for the Dirac equation with a time-dependent electromagnetic field. The Fourier transform of the field is reconstructed from the scattering operator on a Lorents invariant set
(0.1)   D:={(τ, ξ)∈R×R³;|τ|<c|ξ|}
in the dual space of the space-time. As corollaries of this result, we can reconstruct the electromagnetic field completely if it is a finite sum of fields each of which is a time-independent one by a suitable Lorentz transform, and we can also determine the field uniquely if the fields satisfies some exponential decay condition. Our assumptions and results are independent of a choice of inertial frames.