High-Momenta Estimates for the Klein-Gordon Equation: Long-Range Magnetic Potentials and Time-Dependent Inverse Scattering

The study of obstacle scattering for the Klein-Gordon equation in the presence of long-range magnetic potentials is addressed. Previous results of the authors are extended to the long-range case and the results the authors previously proved for high-momenta long-range scattering for the Schr\"odinger equation are brought to the relativistic scenario. It is shown that there are important differences between relativistic and non-relativistic scattering concerning long-range. In particular, it is proved that the electric potential can be recovered without assuming the knowledge of the long-range part of the magnetic potential, which has to be supposed in the non-relativistic case. The electric potential and the magnetic field are recovered from the high momenta limit of the scattering operator, as well as fluxes modulo $2 \pi$ around handles of the obstacle. Moreover, it is proved that, for every $\hat{\mathbf v} \in \mathbb{S}^2$, $A_\infty(\hat{\mathbf v}) + A_\infty(-\hat{\mathbf v})$ can be reconstructed, where $A_\infty$ is the long-range part of the magnetic potential. A a simple formula for the high momenta limit of the scattering operator is given, in terms of magnetic fluxes over handles of the obstacle and long-range magnetic fluxes at infinity, that are introduced in this paper. The appearance of these long-range magnetic fluxes is a new effect in scattering theory.