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Asymptotics for Erdos-Solovej zero modes in strong fields

Research output: Contribution to journalJournal articlepeer-review

Published
<mark>Journal publication date</mark>10/2016
<mark>Journal</mark>Annales Henri Poincaré
Issue number10
Volume17
Number of pages23
Pages (from-to)2951-2973
Publication StatusPublished
Early online date9/04/16
<mark>Original language</mark>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.

Bibliographic note

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