no. 5
Eigenvalue asymptotics for the Pauli operator in strong nonconstant magnetic fields
Annales de l'Institut Fourier, Tome 49 (1999) no. 5, pp. 1603-1636.

On considère l’opérateur de Pauli H(μ):=∑ j=1 m σ j - i ∂ ∂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→∞ 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→∞ 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):=∑ j=1 m σ j - i ∂ ∂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→∞ 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→∞ of the number of the eigenvalues of H(μ) situated on the interval (λ 1 ,λ 2 ) with 0<λ 1 <λ 2 .

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     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},
     zbl = {0935.35109},
     mrnumber = {2000k:35227},
     language = {en},
     url = {https://aif.centre-mersenne.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://aif.centre-mersenne.org/articles/10.5802/aif.1731/

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

[B] M.Š. Birman, On the spectrum of singular boundary value problems, Mat. Sbornik, 55 (1961) 125-174 (Russian); Engl. transl. in Amer. Math. Soc. Transl., (2) 53 (1966), 23-80. | MR | Zbl

[E] L. Erdös, Ground state density of the Pauli operator in the large field limit, Lett.Math.Phys., 29 (1993), 219-240. | MR | Zbl

[Hö] L. Hörmander, The Analysis of Linear Partial Differential Operators. IV, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1985. | Zbl

[IT1] A. Iwatsuka, H. Tamura, Asymptotic distribution of eigenvalues for Pauli operators with nonconstant magnetic fields, Duke J.Math., 93 (1998), 535-574. | MR | Zbl

[IT2] A. Iwatsuka, H. Tamura, Asymptotic distribution of negative eigenvalues for two dimensional Pauli operators with nonconstant magnetic fields, Ann. Inst. Fourier, 48-2 (1998), 479-515. | Numdam | MR | Zbl

[KMSz] M. Kac, W.L. Murdock, G. Szegö, On the eigenvalues of certain hermitian forms, Journ. Rat. Mech. Analysis, 2 (1953), 767-800. | MR | Zbl

[R1] G.D. Raikov, Eigenvalue asymptotics for the Schrödinger operator in strong constant magnetic fields, Commun. P.D.E., 23 (1998), 1583-1619. | MR | Zbl

[R2] G.D. Raikov, Eigenvalue asymptotics for the Dirac operator in strong constant magnetic fields, Math. Phys. Electron. J., 5, n° 2 (1999), 22 p. http://www.ma.utexas.edu/mpej/. | MR | Zbl

[Sh] 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

[W] H. Widom, Eigenvalue distribution in certain homogeneous spaces, J. Func. Anal., 71 (1979), 139-147. | MR | Zbl

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