Tunneling between magnetic wells in two dimensions
Søren Fournais, Yannick Guedes Bonthonneau, Léo Morin, Nicolas Raymond
TL;DR
This work delivers a general 2D magnetic tunneling formula for the pure magnetic double-well problem in the semiclassical limit. By transforming the magnetic Laplacian to a pseudodifferential operator whose leading part decouples cyclotron motion from center-guide motion, the authors construct a one-well quasimode and a WKB-type upper-well state, then quantify their exponential overlap via a magnetic Agmon distance S. A carefully built parametrix and exponential weights yield optimal Agmon-type decay for the one-well groundstate and an explicit interaction matrix between the two wells, leading to a leading-order spectral gap of the form $\\lambda_2-\\lambda_1 = c_0 h^{3/2} e^{-S/h} (1+o(1))$ with a computable $c_0$ and $S$. The result extends magnetic tunneling theory beyond radial or electric-reduction regimes, establishing a purely magnetic tunneling mechanism and providing a robust framework for future semiclassical spectral analyses of magnetic operators.
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.