Scattering Theory for the Dirac Operator with a Long-Range Electromagnetic Potential

Abstract

We consider the Dirac operator with a long-range potential V(x). Scalar, pseudo-scalar and vector components of V(x) may have arbitrary power-like decay at infinity. We introduce wave operators with time-independent modifiers. These modifiers are pseudo-differential operators whose symbols are, roughly speaking, constructed in terms of approximate eigenfunctions of the stationary problem. We derive and solve eikonal and transport equations for the corresponding phase and amplitude functions. From an analytical point of view, our proof of the existence and completeness of the wave operators relies on the limiting absorption principle and radiation estimates established in the paper. This allows us to fit the long-range scattering theory for the Dirac operator into the framework of smooth perturbations. Finally, we find the asymptotics for large times t of solutions u(x, t) of the time-dependent Dirac equation.