Asymptotics for Erdős–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 two-form ${\beta}$ from the sphere ${\mathbb{S}^2}$ to ${\mathbb{R}^3}$ using a combination of the Hopf fibration and inverse stereographic projection. If ${\int_{\mathbb{s}^2} \beta \neq 0}$, we show that

$$\sum_{0 \leq t \leq T} {\rm dim Ker} \mathcal{D}{tB}=\frac{T^2}{8\pi^2}\,\Big| \int_{\mathbb{S}^2}\beta\Big|\,\int_{\mathbb{S}^2}|{\beta}| +o(T^2)$$

as ${T\to+\infty}$. The result relies on Erdős 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.