Tunneling between magnetic wells in two dimensions

TL;DR

This paper analyzes quantum tunneling between magnetic wells in a 2D magnetic Laplacian, deriving the eigenvalue splitting in the semiclassical limit using microlocal analysis, revealing a purely magnetic Agmon distance.

Contribution

First to analyze tunneling between purely magnetic wells in 2D, providing a microlocal analysis approach and identifying the magnetic Agmon distance.

Findings

Calculated eigenvalue splitting in semiclassical limit

Identified a purely magnetic Agmon distance

Showed exponential decay inside the characteristic manifold

Abstract

The two-dimensional magnetic Laplacian is considered. We calculate the leading term of the splitting between the first two eigenvalues of the operator in the semiclassical limit under the assumption that the magnetic field does not vanish and has two symmetric magnetic wells with respect to the coordinate axes. This is the first result of quantum tunneling between purely magnetic wells under generic assumptions. The proof, which strongly relies on microlocal analysis, reveals a purely magnetic Agmon distance between the wells. Surprisingly, it is discovered that the exponential decay of the eigenfunctions away from the magnetic wells is not crucial to derive the tunneling formula. The key is a microlocal exponential decay inside the characteristic manifold, with respect to the variable quantizing the classical center guide motion.