Eigenvalue asymptotics for the Pauli operator in strong nonconstant magnetic fields

Raikov, Georgi D.

doi:10.5802/aif.1731

Raikov, Georgi D.

Annales de l'Institut Fourier, Tome 49 (1999) no. 5, pp. 1603-1636.

Résumé
Abstract

On considère l’opérateur de Pauli $H (μ) : = {(\sum_{j = 1}^{m} σ_{j} (- i \frac{\partial}{\partial x_{j}} - μ A_{j}))}^{2} + V$ autoadjoint dans $L^{2} (ℝ^{m}; ℂ^{2})$ , $m = 2, 3$ . Ici $σ_{j}$ , $j = 1, ..., m$ , sont les matrices de Pauli, $A : = (A_{1}, ..., A_{m})$ est le potentiel magnétique, $μ > 0$ 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 $μ \to \infty$ du nombre des valeurs propres de $H (μ)$ inférieures à $λ$ , le paramètre $λ < 0$ étant fixé. De plus, si $m = 2$ , on étudie l’asymptotique lorsque $μ \to \infty$ du nombre des valeurs propres de $H (μ)$ appartenant à l’intervalle $] λ_{1}, λ_{2} [$ avec $0 < λ_{1} < λ_{2}$ .

We consider the Pauli operator $H (μ) : = {(\sum_{j = 1}^{m} σ_{j} (- i \frac{\partial}{\partial x_{j}} - μ A_{j}))}^{2} + V$ selfadjoint in $L^{2} (ℝ^{m}; ℂ^{2})$ , $m = 2, 3$ . Here $σ_{j}$ , $j = 1, ..., m$ , are the Pauli matrices, $A : = (A_{1}, ..., A_{m})$ is the magnetic potential, $μ > 0$ 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 $μ \to \infty$ of the number of the eigenvalues of $H (μ)$ smaller than $λ$ , the parameter $λ < 0$ being fixed. Furthermore, if $m = 2$ , we study the asymptotics as $μ \to \infty$ of the number of the eigenvalues of $H (μ)$ situated on the interval $(λ_{1}, λ_{2})$ with $0 < λ_{1} < λ_{2}$ .

MR Zbl

DOI : 10.5802/aif.1731

@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},
     mrnumber = {2000k:35227},
     zbl = {0935.35109},
     language = {en},
     url = {https://www.numdam.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, Tome 49 (1999) no. 5, pp. 1603-1636. doi : 10.5802/aif.1731. https://www.numdam.org/articles/10.5802/aif.1731/

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