The Hamiltonian Dynamics of Magnetic Confinement and Instances of Quantum Tunneling
We consider a class of magnetic fields defined over the interior of a manifold $M$ which go to infinity at its boundary and whose direction near the boundary of $M$ is controlled by a closed 1-form $\sigma_\infty \in \Omega^1(\partial M)$. We are able to show that charged particles in the interior of $M$ under the influence of such fields can only escape the manifold through the zero locus of $\sigma_\infty$. In particular in the case where the 1-form is nowhere vanishing we conclude that the particles become confined to its interior for all time.
We also describe a class of magnetic fields defined on the unit disc which is strong enough to confine classical charged particles to the inside of the disc but fails to confine quantum particles, which provides evidence for the presence of quantum tunneling for such systems.