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 Schrödinger equation are brought to the relativistic scenario. It is shown that there are important differences between relativistic and non-relativistic scattering concerning the long range. In particular, it is proven 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 the handles of the obstacle. Moreover, it is proven that for every $\hat{v}\in S^2$, $A_{\infty }\left(\hat{v}\right)+A_{\infty }(-\hat{v})$ can be reconstructed, where $A_{\infty }$ is the long-range part of the magnetic potential. 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. (paper)