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## Asymptotics for Erdos-Solovej zero modes in strong fields

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Journal publication date 10/2016 Annales Henri Poincaré 10 17 23 2951-2973 Published 9/04/16 English

### Abstract

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

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The final publication is available at Springer via http://dx.doi.org/10.1007/s00023-016-0478-5