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

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.

References in corpus (5)

• 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 in spectral asymptotics of the magnetic Schrödinger operator
• High-Velocity Estimates for Schrödinger Operators in Two Dimensions: Long-Range Magnetic Potentials and Time-Dependent Inverse Scattering
• On inverse problems for the multidimensional relativistic Newton equation at fixed energy

Cited by in corpus (2)

• Schrödinger operators with multiple Aharonov-Bohm fluxes
• High-Momenta Estimates for the Klein-Gordon Equation: Long-Range Magnetic Potentials and Time-Dependent Inverse Scattering