# Magnetic Tunneling Between Disc-Shaped Obstacles

## TL;DR

This paper studies how magnetic tunneling behaves between disc-shaped obstacles in two dimensions, using mathematical models to derive new insights.

## Contribution

The paper introduces a novel reduction method for magnetic tunneling and derives asymptotic and effective models for different configurations.

## Key findings

- A reduction method to an interaction matrix is developed for general obstacle configurations.
- An asymptotic formula for the spectral gap is derived for two disc-shaped obstacles.
- An effective operator leading to Harper’s equation is derived for regularly spaced disc lattices.

## Abstract

In this paper we derive formulae for the semiclassical tunneling in the presence of a constant magnetic field in 2 dimensions. The ‘wells’ in the problem are identical discs with Neumann boundary conditions, so we study the magnetic Neumann Laplacian in the complement of a set of discs. We provide a reduction method to an interaction matrix, which works for a general configuration of obstacles. When there are two discs, we deduce an asymptotic formula for the spectral gap. When the discs are placed along a regular lattice, we derive an effective operator which gives rise to the famous Harper’s equation. Main challenges in this problem compared to recent results on magnetic tunneling are the fact that one-well ground states have non-trivial angular momentum which depends on the semiclassical parameter, and the existence of eigenvalue crossings.