Asymptotic distribution of negative eigenvalues for two dimensional Pauli operators with nonconstant magnetic fields

Iwatsuka, Akira; Tamura, Hideo

Annales de l'Institut Fourier, Volume 48 (1998) no. 2, pp. 479-515.

This article studies the asymptotic behavior of the number $N\left(\lambda \right)$ of the negative eigenvalues $<-\lambda$ as $\lambda \to +0$ of the two dimensional Pauli operators with electric potential $V\left(x\right)$ decaying at $\infty$ and with nonconstant magnetic field $b\left(x\right)$, which is assumed to be bounded or to decay at $\infty$. In particular, it is shown that $N\left(\lambda \right)=(1/2\pi ){\int }_{V\left(x\right)>\lambda }b\left(x\right)dx(1+o\left(1\right))$, when $V\left(x\right)$ decays faster than $b\left(x\right)$ under some additional conditions.

Cet article étudie le comportement asymptotique des valeurs propres négatives $<-\lambda$, quand $\lambda \to +0$, des opérateurs de Pauli avec un potentiel électrique $V\left(x\right)$ qui tend vers $0$ à l’infini et avec un champ magnétique non constant, qui est supposé borné ou tendant vers $0$ à l’infini. Il est montré, en particulier, que $N\left(\lambda \right)=(1/2\pi ){\int }_{V\left(x\right)>\lambda }b\left(x\right)dx(1+o\left(1\right))$, quand $V\left(x\right)$ diminue plus rapidement que $b\left(x\right)$ sous des hypothèses supplémentaires.

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