Tunneling between magnetic wells in two dimensions
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.
Keywords
Cite
@article{arxiv.2502.17290,
title = {Tunneling between magnetic wells in two dimensions},
author = {Søren Fournais and Yannick Guedes Bonthonneau and Léo Morin and Nicolas Raymond},
journal= {arXiv preprint arXiv:2502.17290},
year = {2025}
}