## Abstract

We consider a particle bound to a two-dimensional plane and a double-well potential, subject to a perpendicular uniform magnetic field. The energy difference between the lowest two eigenvalues—the eigenvalue splitting—is related to the tunneling probability between the two wells. We obtain upper and lower bounds on this splitting in the regime where both the magnetic field strength and the depth of the wells are large. The main step is a lower bound on the hopping amplitude between the wells, a key parameter in tight binding models of solid state physics, given by an oscillatory integral, whose phase has no critical point and which is exponentially small.

Original language | English (US) |
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Pages (from-to) | 1105-1130 |

Number of pages | 26 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 54 |

Issue number | 1 |

DOIs | |

State | Published - 2022 |

## All Science Journal Classification (ASJC) codes

- Analysis
- Computational Mathematics
- Applied Mathematics

## Keywords

- eigenvalue splitting
- hopping coefficient
- magnetic field
- Schrödinger equation
- tight binding approximation
- tunneling