The Flea on the Magnetic Elephant

TL;DR

This work analyzes a $2$-D magnetic Laplacian with two radial wells and shows that the bottom of the spectrum is governed by tunneling between the wells. By reducing the problem to a $2\times 2$ interaction matrix through quasimodes, the authors establish an explicit exponentially small coupling $w$ that determines whether the two lowest states are delocalized across both wells or localized dominantly in a single well. They also prove a magnetic version of the flea-on-the-elephant phenomenon: a tiny, exponentially small symmetry-breaking perturbation can force eigenfunctions to localize strongly in one well, with precise thresholds given by the scales $S$ and $I_0$. The results extend magnetic tunneling theory for radial wells and provide rigorous asymptotics for the bottom spectrum under small perturbations, highlighting a purely magnetic mechanism for localization switching with potential applications in quantum devices.

Abstract

We investigate a two-dimensional magnetic Laplacian with two radially symmetric magnetic wells. Its spectral properties are determined by the tunneling between them. If the tunneling is weak and the wells are mirror symmetric, the two lowest eigenfunctions are localized in both wells being distributed roughly equally. In this note we show that an exponentially small symmetry violation can in this situation have a dramatic effect, making each of the eigenfunctions localized dominantly in one well only. This is reminiscent of the `flea on the elephant' effect for Schrödinger operators; our result shows that it has a purely magnetic counterpart.

Theorems & Definitions (28)

• Theorem 2.1
• Proposition 2.2
• Theorem 2.3
• Corollary 2.4
• Remark 2.5
• proof
• Theorem 2.6
• Corollary 2.7
• proof : Proof of Theorem \ref{['thm.mag.flea']}
• proof : Proof of Corollary \ref{['cor.mgflea']}