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

Abstract

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 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.

Published version, it has been edited to improve the presentation

References in corpus (6)

• A macroscopic test of the Aharonov-Bohm effect
• Aharonov-Bohm Effect and High-Velocity Estimates of Solutions to the Schrödinger Equation
• Scattering theory for Klein-Gordon equations with non-positive energy
• The Aharonov-Bohm effect: the role of tunneling and associated forces
• Aharonov-Bohm Effect and High-Momenta Inverse Scattering for the Klein-Gordon Equation
• High-Velocity Estimates for Schrödinger Operators in Two Dimensions: Long-Range Magnetic Potentials and Time-Dependent Inverse Scattering