Asymptotics for Erdos-Solovej Zero Modes in Strong Fields

TL;DR

This paper analyzes the asymptotic behavior of zero modes in certain Weyl-Dirac operators under strong magnetic fields, revealing a quadratic growth rate related to the magnetic field's geometric properties.

Contribution

It provides the first asymptotic formula for the count of zero modes of Weyl-Dirac operators with magnetic fields derived from sphere pullbacks, connecting spectral properties to geometric integrals.

Findings

Zero mode count grows quadratically with field strength T.

Asymptotic formula involves integrals of the magnetic field form over the sphere.

Results depend on the magnetic field's topological and geometric characteristics.

Abstract

We consider the strong field asymptotics for the occurrence of zero modes of certain Weyl-Dirac operators on $\mathbb{R}^3$. In particular we are interested in those operators $\mathcal{D}_{B}$ for which the associated magnetic field $B$ is given by pulling back a $2$-form $\beta$ from the sphere $\mathbb{S}^2$ to $\mathbb{R}^3$ using a combination of the Hopf fibration and inverse stereographic projection. If $\int_{\mathbb{S}^2}\beta\neq0$ we show that \[ \sum_{0\le t\le T}\mathrm{dim}\,\mathrm{Ker}\,\mathcal{D}_{tB} =\frac{T^2}{8\pi^2}\,\biggl\lvert\int_{\mathbb{S}^2}\beta\biggr\rvert\,\int_{\mathbb{S}^2}\lvert{\beta}\rvert+o(T^2) \] as $T\to+\infty$. The result relies on Erd\H{o}s and Solovej's characterisation of the spectrum of $\mathcal{D}_{tB}$ in terms of a family of Dirac operators on $\mathbb{S}^2$, together with information about the strong field localisation of the Aharonov-Casher zero modes of the latter.