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

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

Mine, Takuya ; Nomura, Yuji

Annales de l'Institut Fourier, Tome 59 (2009), p. 659-689 / Harvested from Numdam

Résumé

On considère les opérateurs de Schrödinger sur $\mathbb{R}^2$ avec champ magnétique donné par un champ constant et positif ou nul plus des champs magnétiques aléatoires $\delta $ 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.

Abstract

We shall consider the Schrödinger operators on $\mathbb{R}^2$ with the magnetic field given by a nonnegative constant field plus random $\delta $ 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.

Détail

Publié le : 2009-01-01

MR 2521433 | Zbl 1161.81015

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

Bibliographie

[1] Aharonov, Y.; Bohm, D. Significance of electromagnetic potentials in the quantum theory, Phys. Rev., Tome 115 (1959), pp. 485-491 | Article | MR 110458 | Zbl 0099.43102

[2] Albeverio, S.; Gesztesy, F.; Høegh-Krohn, R.; Holden, H. Solvable models in quantum mechanics, Springer-Verlag, New York (1988) | MR 926273 | Zbl 0679.46057

[3] Ando, K.; Iwatsuka, A.; Kaminaga, M.; Nakano, F. The spectrum of Schrödinger operators with Poisson type random potential, Ann. Henri Poincaré, Tome 7 (2006), pp. 145-160 | Article | MR 2205467 | Zbl 1091.81014

[4] Avishai, Y.; Azbel, M. Ya.; Gredeskul, S. A. Electron in a magnetic field interacting with point impurities, Phys. Rev. B, Tome 48 (1993), pp. 17280-17295 | Article

[5] Avishai, Y.; Redheffer, R. M. Two dimensional disordered electronic systems in a strong magnetic field, Phys. Rev. B, Tome 47 (1993), pp. 2089-2100 | Article

[6] Avishai, Y.; Redheffer, R. M.; Band, Y. B. Electron states in a magnetic field and random impurity potential: use of the theory of entire functions, J. Phys. A, Tome 25 (1992), pp. 3883-3889 | Article | Zbl 0784.30024

[7] Borg, J. L. Private communication (2007)

[8] Borg, J. L.; Pulé, J. V. Lifshits tails for random smooth magnetic vortices, J. Math. Phys., Tome 45 (2004), pp. 4493-4505 | Article | MR 2105202 | Zbl 1064.82018

[9] Chistyakov, G.; Lyubarskii, Y.; Pastur, L. On completeness of random exponentials in the Bargmann-Fock space, J. Math. Phys., Tome 42 (2001), pp. 3754-3768 | Article | MR 1845217 | Zbl 1009.42005

[10] Desbois, J.; Furtlehner, C.; Ouvry, S. Random magnetic impurities and the Landau problem, Nuclear Physics B, Tome 453 (1995), pp. 759-776 | Article

[11] Desbois, J.; Furtlehner, C.; Ouvry, S. Density correlations of magnetic impurities and disorder, J. Phys. A: Math. Gen., Tome 30 (1997), pp. 7291-7300 | Article | Zbl 0925.82114

[12] Desbois, J.; Ouvry, S.; Texier, C. Hall conductivity for two-dimensional magnetic systems, Nuclear Physics B, Tome 500 (1997), pp. 486-510 | Article | MR 1471659 | Zbl 0934.81074

[13] Dinaburg, E. I.; Sinai, Y. G.; Soshnikov, A. B. Splitting of the low Landau levels into a set of positive Lebesgue measure under small periodic perturbations, Comm. Math. Phys., Tome 189 (1997), pp. 559-575 | Article | MR 1480033 | Zbl 0888.60055

[14] Dorlas, T. C.; Macris, N.; Pulé, J. V. Characterization of the spectrum of the Landau Hamiltonian with delta impurities, Comm. Math. Phys., Tome 204 (1999), pp. 367-396 | Article | MR 1704280 | Zbl 0937.60063

[15] Exner, P.; Šťovíček, P.; Vytřas, P. Generalized boundary conditions for the Aharonov-Bohm effect combined with a homogeneous magnetic field, J. Math. Phys., Tome 43 (2002), pp. 2151-2168 | Article | MR 1893665 | Zbl 1059.81056

[16] Geĭler, V. A. The two-dimensional Schrödinger operator with a homogeneous magnetic field and its perturbations by periodic zero-range potentials, St. Petersburg Math. J., Tome 3 (1992), pp. 489-532 | MR 1150551

[17] Geim, A. K.; Bending, S. J.; Grigorieva, I. V. Asymmetric scattering and diffraction of two-dimensional electrons at quantized tubes of magnetic flux, Phys. Rev. Lett., Tome 69 (1992), pp. 2252-2255 | Article

[18] Geim, A. K.; Bending, S. J.; Grigorieva, I. V.; Blamire, M. G. Ballistic two-dimensional electrons in a random magnetic field, Phys. Rev. B, Tome 49 (1994), pp. 5749-5752 | Article

[19] Geyler, V. A.; Grishanov, E  N. Zero Modes in a Periodic System of Aharonov-Bohm Solenoids, JETP Letters, Tome 75 (2002), pp. 354-356 | Article

[20] Geyler, V. A.; Šťovíček, P. Zero modes in a system of Aharonov-Bohm fluxes, Rev. Math. Phys., Tome 16 (2004), pp. 851-907 | Article | MR 2097362 | Zbl 1063.81054

[21] Ito, H. T.; Tamura, H. Aharonov-Bohm effect in scattering by point-like magnetic fields at large separation, Ann. Henri Poincaré, Tome 2 (2001), pp. 309-359 | Article | MR 1832970 | Zbl 0992.81084

[22] Kirsch, W. Random Schrödinger operators. A course, in Schrödinger operators, (Sønderborg, 1988), p.264–370, Springer, Berlin, Lecture Notes in Phys., Tome 345 (1989) | MR 1037323 | Zbl 0712.35070

[23] Kirsch, W.; Martinelli, F. On the spectrum of Schrödinger operators with a random potential, Comm. Math. Phys., Tome 85 (1982), pp. 329-350 | Article | MR 678150 | Zbl 0506.60058

[24] Laptev, A.; Weidl, T. Hardy inequalities for magnetic Dirichlet forms, in Mathematical results in quantum mechanics (Prague, 1998), Oper. Theory Adv. Appl., Tome 108 (1999), pp. 299-305 | MR 1708811 | Zbl 0977.26005

[25] Levin, B. Ja. Distribution of zeros of entire functions, American Mathematical Society (1964) | MR 156975 | Zbl 0152.06703

[26] Melgaard, M.; Ouhabaz, E.-M.; Rozenblum, G. Negative discrete spectrum of perturbed multivortex Aharonov-Bohm Hamiltonians, Ann. Henri Poincaré, Tome 5 (2004), pp. 979-1012 (Errata, ibid., 6 (2005), 397–398) | Article | MR 2091985 | Zbl 1059.81049

[27] Mine, T. The Aharonov-Bohm solenoids in a constant magnetic field, Ann. Henri Poincaré, Tome 6 (2005), pp. 125-154 | Article | MR 2121279 | Zbl 1062.81034

[28] Mine, T.; Nomura, Y. Periodic Aharonov-Bohm Solenoids in a Constant Magnetic Field, Rev. Math. Phys., Tome 18 (2006), pp. 913-934 | Article | MR 2273660 | Zbl 1113.81055

[29] Nambu, Y. The Aharonov-Bohm problem revisited, Nuclear Phys. B, Tome 579 (2000), pp. 590-616 | Article | MR 1769914 | Zbl 1071.81525

[30] Reed, M.; Simon, B. Methods of modern mathematical physics. I. Functional analysis. Second edition, Academic Press (1980) | MR 751959 | Zbl 0459.46001

[31] Reiss, R.-D. A course on point processes, Springer-Verlag, New York (1993) | MR 1199815 | Zbl 0771.60037

[32] Rozenblum, G.; Shirokov, N. Infiniteness of zero modes for the Pauli operator with singular magnetic field, J. Funct. Anal., Tome 233 (2006), pp. 135-172 | Article | MR 2204677 | Zbl 1088.81046

[33] Ruijsenaars, S. N. M. The Aharonov-Bohm effect and scattering theory, Ann. Physics, Tome 146 (1983), pp. 1-34 | Article | MR 701261 | Zbl 0554.47003

[34] Zak, J. Group-theoretical consideration of Landau level broadening in crystals, Phys. Rev., Tome 136 (1964), p. A776-A780 | Article | MR 177773