Purely magnetic tunnelling between radial magnetic wells

TL;DR

This paper analyzes the spectral gap of the magnetic Laplacian with two symmetric radial wells, providing an accurate semiclassical tunnelling estimate showing the gap is exponentially small but positive as the semiclassical parameter approaches zero.

Contribution

It introduces a precise tunnelling formula for the spectral gap in a magnetic Laplacian with symmetric radial wells, advancing semiclassical spectral analysis.

Findings

Spectral gap is exponentially small as the semiclassical parameter tends to zero.

Established an accurate one-term tunnelling estimate for the spectral gap.

Spectral gap remains positive despite being exponentially small.

Abstract

This article is devoted to the semiclassical spectral analysis of the magnetic Laplacian in two dimensions. Assuming that the magnetic field is positive and has two symmetric radial wells, we establish an accurate tunnelling formula, that is a one-term estimate of the spectral gap between the lowest two eigenvalues. This gap is exponentially small when the semiclassical parameter goes to zero, but positive.