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

Iwatsuka, Akira; Tamura, Hideo

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

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.

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.

[1] Y. Aharonov and A. Casher, Ground state of a spin-1/2 charged particle in a two-dimensional magnetic field, Phys. Rev. A, 19 (1979), 2461-2462.

[2] J. Avron, I. Herbst and B. Simon, Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J., 45 (1978), 847-883. | MR | Zbl

[3] Y. Colin De Verdière, L'asymptotique de Weyl pour les bouteilles magnétiques, Commun. Math. Phys., 105 (1986), 327-335. | MR | Zbl

[4] H. Cycon, L. R. Froese, W. G. Kirsch and B. Simon, Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Springer Verlag, 1987. | Zbl

[5] L. Erdös, Magnetic Lieb-Thirring inequalities and stochastic oscillatory integrals, Operator Theory, Advances and Applications, 78 (1994), Birkhäuser Verlag, 127-134. | MR | Zbl

[6] L. Erdös, Magnetic Lieb-Thirring inequalities, Commun. Math. Phys., 170 (1995), 629-668. | MR | Zbl

[7] L. Erdös and J. P. Solovej, Semiclassical eigenvalue estimates for the Pauli operator with strong non-homogeneous magnetic fields. I. Non-asymptotic Lieb-Thirring type estimate, preprint, 1996.

[8] L. Erdös and J. P. Solovej, Semiclassical eigenvalue estimates for the Pauli operator with strong non-homogeneous magnetic fields. II. Leading order asymptotic estimates, Commun. Math. Phys., 188 (1997), 599-656. | Zbl

[9] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs, Vol. 18, A.M.S., 1969. | MR | Zbl

[10] A. Iwatsuka and H. Tamura, Asymptotic distribution of eigenvalues for Pauli operators with nonconstant magnetic fields, preprint, 1997 (to be published in Duke Math. J.). | MR | Zbl

[11] A. Mohamed and G. D. Raikov, On the spectral theory of the Schrödinger operator with electromagnetic potential, Pseudo-differential Calculus and Mathematical Physics, Adv. Partial Differ. Eq., Academic Press, 5 (1994), 298-390. | MR | Zbl

[12] I. Shigekawa, Spectral properties of Schrödinger operators with magnetic fields for a spin 1/2 particle, J. Func. Anal., 101 (1991), 255-285. | MR | Zbl

[13] A. V. Sobolev, Asymptotic behavior of the energy levels of a quantum particle in a homogeneous magnetic field, perturbed by a decreasing electric field I, J. Soviet Math., 35 (1986), 2201-2211. | Zbl

[14] A. V. Sobolev, On the Lieb-Thirring estimates for the Pauli operator, Duke Math. J., 82 (1996), 607-637. | MR | Zbl

[15] H. Tamura, Asymptotic distribution of eigenvalues for Schrödinger operators with homogeneous magnetic fields, Osaka J. Math., 25 (1988), 633-647. | MR | Zbl