no. 2
Asymptotic distribution of negative eigenvalues for two dimensional Pauli operators with nonconstant magnetic fields
Annales de l'Institut Fourier, Tome 48 (1998) no. 2, pp. 479-515.

Cet article étudie le comportement asymptotique des valeurs propres négatives <-λ, quand λ→+0, des opérateurs de Pauli avec un potentiel électrique V(x) 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(λ)=(1/2π)∫ V(x)>λ b(x)dx(1+o(1)), quand V(x) diminue plus rapidement que b(x) sous des hypothèses supplémentaires.

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

@article{AIF_1998__48_2_479_0,
     author = {Iwatsuka, Akira and Tamura, Hideo},
     title = {Asymptotic distribution of negative eigenvalues for two dimensional {Pauli} operators with nonconstant magnetic fields},
     journal = {Annales de l'Institut Fourier},
     pages = {479--515},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {48},
     number = {2},
     year = {1998},
     doi = {10.5802/aif.1626},
     zbl = {0909.35100},
     mrnumber = {99e:35168},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1626/}
}
TY  - JOUR
AU  - Iwatsuka, Akira
AU  - Tamura, Hideo
TI  - Asymptotic distribution of negative eigenvalues for two dimensional Pauli operators with nonconstant magnetic fields
JO  - Annales de l'Institut Fourier
PY  - 1998
SP  - 479
EP  - 515
VL  - 48
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1626/
DO  - 10.5802/aif.1626
LA  - en
ID  - AIF_1998__48_2_479_0
ER  - 
%0 Journal Article
%A Iwatsuka, Akira
%A Tamura, Hideo
%T Asymptotic distribution of negative eigenvalues for two dimensional Pauli operators with nonconstant magnetic fields
%J Annales de l'Institut Fourier
%D 1998
%P 479-515
%V 48
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1626/
%R 10.5802/aif.1626
%G en
%F AIF_1998__48_2_479_0
Iwatsuka, Akira; Tamura, Hideo. Asymptotic distribution of negative eigenvalues for two dimensional Pauli operators with nonconstant magnetic fields. Annales de l'Institut Fourier, Tome 48 (1998) no. 2, pp. 479-515. doi : 10.5802/aif.1626. https://aif.centre-mersenne.org/articles/10.5802/aif.1626/

[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

Cité par Sources :