The Flea on the Magnetic Elephant

TL;DR

This paper explores how tiny symmetry-breaking perturbations in a magnetic Laplacian with two wells can cause eigenfunctions to localize predominantly in one well, revealing a magnetic analogue of the 'flea on the elephant' effect.

Contribution

It demonstrates that an exponentially small symmetry violation can dramatically alter eigenfunction localization in magnetic Laplacians, extending the 'flea on the elephant' phenomenon to magnetic operators.

Findings

Weak symmetry violation causes eigenfunctions to localize in one well.

Magnetic Laplacians exhibit a magnetic 'flea on the elephant' effect.

Eigenfunction localization is highly sensitive to small perturbations.

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 Schr\"odinger operators; our result shows that it has a purely magnetic counterpart.