Purely magnetic tunneling effect in two dimensions

TL;DR

This paper derives the first explicit tunneling formula for a purely magnetic field in a two-dimensional domain, analyzing eigenvalue splitting using advanced boundary reduction and Agmon estimates.

Contribution

It introduces the first explicit tunneling formula in a purely magnetic setting and provides optimal Agmon estimates for the magnetic Schrödinger operator.

Findings

Explicit tunneling formula established

Eigenvalue splitting analyzed for symmetric domains

Optimal magnetic Agmon estimates proved

Abstract

The semiclassical magnetic Neumann Schr\"odinger operator on a smooth, bounded, and simply connected domain $\Omega$ of the Euclidean plane is considered. When $\Omega$ 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.