The spectrum of Schrödinger operators with random δ magnetic fields
[Le spectre des opérateurs de Schrödinger avec un champ magnétique de Dirac aléatoire]
Annales de l'Institut Fourier, Tome 59 (2009) no. 2, pp. 659-689.

On considère les opérateurs de Schrödinger sur 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 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.

DOI : https://doi.org/10.5802/aif.2445
Classification : 81Q10,  30D15,  47F05,  47N50,  82B44
Mots clés : opérateur de Schrödinger, champ magnétique aléatoire, champ magnétique singulier, effet d’Aharonov-Bohm, niveau Landau, fonction entière
@article{AIF_2009__59_2_659_0,
     author = {Mine, Takuya and Nomura, Yuji},
     title = {The spectrum of Schr\"odinger operators with random $\delta $ magnetic fields},
     journal = {Annales de l'Institut Fourier},
     pages = {659--689},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {2},
     year = {2009},
     doi = {10.5802/aif.2445},
     mrnumber = {2521433},
     zbl = {1161.81015},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2445/}
}
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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