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The flea on the Magnetic Elephant.

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A small symmetry violation dramatically alters magnetic Laplacian eigenfunctions. Even tiny asymmetries concentrate wave functions into single wells, mirroring effects seen in Schrödinger operators.

Keywords:
Magnetic laplacianSemiclassical limitSpectral theoryTunneling

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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.