Raikov, Georgi D.
Eigenvalue asymptotics for the Pauli operator in strong nonconstant magnetic fields
Annales de l'institut Fourier, Tome 49 (1999) no. 5 , p. 1603-1636
Zbl 0935.35109 | MR 2000k:35227
doi : 10.5802/aif.1731
URL stable : http://www.numdam.org/item?id=AIF_1999__49_5_1603_0

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 ( m ; 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 ( m ; 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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