Eigenvalue asymptotics for the Pauli operator in strong nonconstant magnetic fields

Raikov, Georgi D.

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

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 $L^2({ℝ}^m;{ℂ}^2)$, $m=2,3$. Ici ${\sigma }_j$, $j=1,...,m$, sont les matrices de Pauli, $A:=(A_1,...,A_m)$ 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 $]{\lambda }_1,{\lambda }_2[$ avec $0<{\lambda }_1<{\lambda }_2$.

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 $L^2({ℝ}^m;{ℂ}^2)$, $m=2,3$. Here ${\sigma }_j$, $j=1,...,m$, are the Pauli matrices, $A:=(A_1,...,A_m)$ 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 $({\lambda }_1,{\lambda }_2)$ with $0<{\lambda }_1<{\lambda }_2$.

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