# ANNALES DE L'INSTITUT FOURIER

Eigenvalue asymptotics for the Pauli operator in strong nonconstant magnetic fields
Annales de l'Institut Fourier, Volume 49 (1999) no. 5, pp. 1603-1636.

We consider the Pauli operator $\mathbf{H}\left(\mu \right):={\left({\sum }_{j=1}^{m}{\sigma }_{j}\left(-i\frac{\partial }{\partial {x}_{j}}-\mu {A}_{j}\right)\right)}^{2}+V$ selfadjoint in ${L}^{2}\left({ℝ}^{m};{ℂ}^{2}\right)$, $m=2,3$. Here ${\sigma }_{j}$, $j=1,...,m$, are the Pauli matrices, $A:=\left({A}_{1},...,{A}_{m}\right)$ is the magnetic potential, $\mu >0$ is the coupling constant, and $V$ is the electric potential which decays at infinity. We suppose that the magnetic field generated by $A$ satisfies some regularity conditions; in particular, its norm is lower-bounded by a positive constant, and, in the case $m=3$, its direction is constant. We investigate the asymptotic behaviour as $\mu \to \infty$ of the number of the eigenvalues of $\mathbf{H}\left(\mu \right)$ smaller than $\lambda$, the parameter $\lambda <0$ being fixed. Furthermore, if $m=2$, we study the asymptotics as $\mu \to \infty$ of the number of the eigenvalues of $\mathbf{H}\left(\mu \right)$ situated on the interval $\left({\lambda }_{1},{\lambda }_{2}\right)$ with $0<{\lambda }_{1}<{\lambda }_{2}$.

On considère l’opérateur de Pauli $\mathbf{H}\left(\mu \right):={\left({\sum }_{j=1}^{m}{\sigma }_{j}\left(-i\frac{\partial }{\partial {x}_{j}}-\mu {A}_{j}\right)\right)}^{2}+V$ autoadjoint dans ${L}^{2}\left({ℝ}^{m};{ℂ}^{2}\right)$, $m=2,3$. Ici ${\sigma }_{j}$, $j=1,...,m$, sont les matrices de Pauli, $A:=\left({A}_{1},...,{A}_{m}\right)$ est le potentiel magnétique, $\mu >0$ est la constante de couplage, et $V$ est le potentiel électrique qui décroît à l’infini. On suppose que le champ magnétique engendré par $A$ satisfait à certaines conditions de régularité; en particulier, sa norme est minorée par une constante strictement positive et, dans le cas $m=3$, sa direction est constante. On analyse le comportement asymptotique quand $\mu \to \infty$ du nombre des valeurs propres de $\mathbf{H}\left(\mu \right)$ inférieures à $\lambda$, le paramètre $\lambda <0$ étant fixé. De plus, si $m=2$, on étudie l’asymptotique lorsque $\mu \to \infty$ du nombre des valeurs propres de $\mathbf{H}\left(\mu \right)$ appartenant à l’intervalle $\right]{\lambda }_{1},{\lambda }_{2}\left[$ avec $0<{\lambda }_{1}<{\lambda }_{2}$.

@article{AIF_1999__49_5_1603_0,
author = {Raikov, Georgi D.},
title = {Eigenvalue asymptotics for the {Pauli} operator in strong nonconstant magnetic fields},
journal = {Annales de l'Institut Fourier},
pages = {1603--1636},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {49},
number = {5},
year = {1999},
doi = {10.5802/aif.1731},
zbl = {0935.35109},
mrnumber = {2000k:35227},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1731/}
}
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Raikov, Georgi D. Eigenvalue asymptotics for the Pauli operator in strong nonconstant magnetic fields. Annales de l'Institut Fourier, Volume 49 (1999) no. 5, pp. 1603-1636. doi : 10.5802/aif.1731. https://aif.centre-mersenne.org/articles/10.5802/aif.1731/

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