Exponential localization of 2d Magnetic Schr\"odinger eigenfunctions via Brownian flux

Abstract

We study solutions of $\tfrac12 (-i\nabla - A(x))^2f=\lambda f$ on domains $\Omega\subset \mathbb{R}^2$ with Dirichlet boundary conditions and prove exponential decay estimates in terms of an Agmon type distance to a classically allowed region. This metric depends only on the eigenvalue and associated magnetic field. In fact the main quantity in the weight for this distance function can be interpreted as an `average magnetic flux' of the magnetic field along all domains whose boundary is the closed curved formed by joing the `minimal path' to another random path. Our estimates are based on an analysis of the associated heat kernel using the Feynman-Kac-It\'o formula.