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Magnetic Tunneling Between Disc-Shaped Obstacles
1Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark.
Communications in Mathematical Physics
|April 24, 2025
Summary
We derived formulas for semiclassical tunneling in 2D with magnetic fields. Our method applies to general obstacle configurations, yielding Harper's equation for lattice-placed discs.
Area of Science:
- Mathematical Physics
- Quantum Mechanics
- Condensed Matter Physics
Background:
- Semiclassical tunneling is crucial for understanding quantum phenomena.
- Magnetic fields significantly alter quantum tunneling behavior.
- Previous studies often simplified obstacle configurations.
Purpose of the Study:
- To derive formulas for semiclassical tunneling in 2D with constant magnetic fields.
- To develop a general reduction method for systems with multiple obstacles.
- To investigate spectral properties and effective operators for specific configurations.
Main Methods:
- Analysis of the magnetic Neumann Laplacian in the complement of discs.
- Development of a reduction method to an interaction matrix.
- Derivation of asymptotic formulas for spectral gaps.
- Construction of an effective operator for lattice configurations.
Main Results:
- A general reduction method applicable to various obstacle arrangements.
- An asymptotic formula for the spectral gap with two discs.
- Derivation of Harper's equation for discs arranged in a regular lattice.
- Identification of challenges including non-trivial angular momentum and eigenvalue crossings.
Conclusions:
- The developed method provides a powerful tool for studying magnetic tunneling in complex geometries.
- The connection to Harper's equation highlights the relevance to condensed matter systems.
- The findings offer new insights into quantum phenomena influenced by magnetic fields and boundary conditions.
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