Aharonov-Bohm Effect and High-Momenta Inverse Scattering for the Klein-Gordon Equation

TL;DR

This paper studies high-momentum scattering of relativistic charged particles around obstacles with magnetic fluxes, providing methods to reconstruct electric and magnetic fields and analyzing the effects of the Aharonov-Bohm phenomenon.

Contribution

It introduces a novel reconstruction method for electric potentials and magnetic fluxes from high-momenta scattering data in the presence of obstacles with handles.

Findings

Reconstruction of electric potential and magnetic fluxes from scattering data.

High-momenta estimates with error bounds for the scattering operator.

Explicit formula for the scattering operator's high-momenta limit in the absence of electric potential.

Abstract

We analyze spin-0 relativistic scattering of charged particles propagating in the exterior, $\Lambda \subset \mathbb{R}^3$, of a compact obstacle $K \subset \mathbb{R}^3$. The connected components of the obstacle are handlebodies. The particles interact with an electro-magnetic field in $\Lambda$ and an inaccessible magnetic field localized in the interior of the obstacle (through the Aharonov-Bohm effect). We obtain high-momenta estimates, with error bounds, for the scattering operator that we use to recover physical information: We give a reconstruction method for the electric potential and the exterior magnetic field and prove that, if the electric potential vanishes, circulations of the magnetic potential around handles (or equivalently, by Stokes' theorem, magnetic fluxes over transverse sections of handles) of the obstacle can be recovered, modulo $2 \pi$. We additionally give a simple formula for the high-momenta limit of the scattering operator in terms of certain magnetic fluxes, in the absence of electric potential. If the electric potential does not vanish, the magnetic fluxes on the handles above referred can be only recovered modulo $\pi$ and the simple expression of the high-momenta limit of the scattering operator does not hold true.