Magnetic tunneling between disc-shaped obstacles

TL;DR

This paper develops semiclassical tunneling formulae for magnetic Laplacians with disc-shaped obstacles, revealing spectral gaps and effective models like Harper's equation, with novel analysis of angular momentum effects and eigenvalue crossings.

Contribution

It introduces a reduction method to an interaction matrix for magnetic tunneling problems with arbitrary obstacle configurations, and derives asymptotic formulas for spectral gaps and effective operators.

Findings

Derived semiclassical tunneling formulae in magnetic fields

Established an interaction matrix approach for obstacle configurations

Connected the problem to Harper's equation in lattice arrangements

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.