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
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.
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Communicated by Jens Marklof.
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Elton, D.M. Asymptotics for Erdős–Solovej Zero Modes in Strong Fields. Ann. Henri Poincaré 17, 2951–2973 (2016). https://doi.org/10.1007/s00023-016-0478-5
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DOI: https://doi.org/10.1007/s00023-016-0478-5
Keywords
- Line Bundle
- Dirac Operator
- Zero Mode
- Dual Frame
- Pauli Operator