The spectrum of Schrödinger operators with random $\delta $ magnetic fields

Mine, Takuya; Nomura, Yuji

doi:10.5802/aif.2445

The spectrum of Schrödinger operators with random

δ

magnetic fields
[Le spectre des opérateurs de Schrödinger avec un champ magnétique de Dirac aléatoire]

Mine, Takuya ¹ ; Nomura, Yuji ²

¹ Kyoto Institute of Technology Department of Comprehensive Sciences Matsugasaki Sakyo-ku Kyoto 606-8585 (Japan)
² Ehime University Department of Computer Science Graduate School of Science and Engineering 3 Bunkyo-cho Matsuyama, Ehime 790-8577 (Japan)

Annales de l'Institut Fourier, Tome 59 (2009) no. 2, pp. 659-689.

Résumé
Abstract

On considère les opérateurs de Schrödinger sur $ℝ^{2}$ avec champ magnétique donné par un champ constant et positif ou nul plus des champs magnétiques aléatoires $δ$ du type d’Anderson ou du type de Poisson-Anderson. On étudie le spectre de ces opérateurs par la méthode des potentiels admissibles par Kirsch-Martinelli. De plus, on démontre que les niveaux inférieurs de Landau sont infiniment dégénérés lorsque le champ constant est suffisamment grand en évaluant l’ordre de croissance, utilisant la théorie de la fonction entière de Levin.

We shall consider the Schrödinger operators on $ℝ^{2}$ with the magnetic field given by a nonnegative constant field plus random $δ$ magnetic fields of the Anderson type or of the Poisson-Anderson type. We shall investigate the spectrum of these operators by the method of the admissible potentials by Kirsch-Martinelli. Moreover, we shall prove the lower Landau levels are infinitely degenerated eigenvalues when the constant field is sufficiently large, by estimating the growth order of the eigenfunctions using the entire function theory by Levin.

MR Zbl

DOI : 10.5802/aif.2445

Classification : 81Q10, 30D15, 47F05, 47N50, 82B44
Keywords: Schrödinger operator, random magnetic field, singular magnetic field, Aharonov-Bohm effect, Landau level, entire function
Mot clés : opérateur de Schrödinger, champ magnétique aléatoire, champ magnétique singulier, effet d’Aharonov-Bohm, niveau Landau, fonction entière

Affiliations des auteurs :

Mine, Takuya ¹ ; Nomura, Yuji ²

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     title = {The spectrum of {Schr\"odinger} operators with random $\delta $ magnetic fields},
     journal = {Annales de l'Institut Fourier},
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     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
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Mine, Takuya; Nomura, Yuji. The spectrum of Schrödinger operators with random $\delta $ magnetic fields. Annales de l'Institut Fourier, Tome 59 (2009) no. 2, pp. 659-689. doi : 10.5802/aif.2445. http://www.numdam.org/articles/10.5802/aif.2445/

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