Asymptotics for Erdos-Solovej Zero Modes in Strong Fields

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.