  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.
