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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Asymptotics for Erdos-Solovej zero modes in strong fields
AU - Elton, Daniel Mark
N1 - The final publication is available at Springer via http://dx.doi.org/10.1007/s00023-016-0478-5
PY - 2016/10
Y1 - 2016/10
N2 - 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.
AB - 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.
KW - Weyl-Dirac operator
KW - zero modes
U2 - 10.1007/s00023-016-0478-5
DO - 10.1007/s00023-016-0478-5
M3 - Journal article
VL - 17
SP - 2951
EP - 2973
JO - Annales Henri Poincaré
JF - Annales Henri Poincaré
SN - 1424-0637
IS - 10
ER -