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تسجيل مستخدم جديدAsymptotics for Erdos-Solovej Zero Modes in Strong Fields
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Daniel M. Elton
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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 $int_{mathbb{S}^2}beta eq0$ we show that [ sum_{0le tle T}mathrm{dim},mathrm{Ker},mathcal{D}_{tB} =frac{T^2}{8pi^2},biggllvertint_{mathbb{S}^2}betabiggrrvert,int_{mathbb{S}^2}lvert{beta}rvert+o(T^2) ] as $Tto+infty$. The result relies on ErdH{o}s and Solovejs 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.
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