Purely magnetic tunneling effect in two dimensions

The semiclassical magnetic Neumann Schr"odinger operator on a smooth, bounded, and simply connected domain $\mathrm{\Omega }$ of the Euclidean plane is considered. When $\mathrm{\Omega }$ has a symmetry axis, the semiclassical splitting of the first two eigenvalues is analyzed. The first explicit tunneling formula in a pure magnetic field is established. The analysis is based on a pseudo-differential reduction to the boundary and the proof of the first known optimal purely magnetic Agmon estimates.