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 ${\int }_{{\mathbb{S}}^2}\beta \ne 0$ we show that \[ \sum_{0\le t\le T}\mathrm{dim}\,\mathrm{Ker}\,\mathcal{D}_{tB} =\frac{T^2}{8π^2}\,\biggl\lvert\int_{\mathbb{S}^2}β\biggr\rvert\,\int_{\mathbb{S}^2}\lvertβ\rvert+o(T^2) \] 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.

24 pages, typos corrected, some minor rewording

References in corpus (2)

• The kernel of Dirac operators on ${\S }^3$ and ${\mathbb{R}}^3$
• Degeneracy of zero modes of the Dirac operator in three dimensions