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The flea on the Magnetic Elephant
Pavel Exner1,2, Léo Morin3
1Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University, Břehová 7, 11519 Prague, Czechia.
Letters in Mathematical Physics
|March 2, 2026
Summary
A small symmetry violation dramatically alters magnetic Laplacian eigenfunctions. Even tiny asymmetries concentrate wave functions into single wells, mirroring effects seen in Schrödinger operators.
Area of Science:
- Quantum mechanics
- Mathematical physics
- Spectral theory
Background:
- The spectral properties of magnetic Laplacians are influenced by tunneling between magnetic wells.
- In symmetric systems with weak tunneling, eigenfunctions are typically delocalized across wells.
Purpose of the Study:
- To investigate the impact of symmetry violation on the eigenfunctions of a 2D magnetic Laplacian with two wells.
- To demonstrate a magnetic counterpart to the 'flea on the elephant' effect.
Main Methods:
- Analysis of a two-dimensional magnetic Laplacian.
- Perturbation theory for spectral properties.
Main Results:
- An exponentially small symmetry violation leads to eigenfunctions localizing in a single well.
- This localization occurs even when tunneling is weak and wells are nearly symmetric.
Conclusions:
- Symmetry breaking has a profound effect on the localization of eigenfunctions in magnetic systems.
- The study reveals a purely magnetic analog of a known phenomenon in quantum mechanics.
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