Localisation pour des opérateurs de Schrödinger aléatoires dans L 2 ( d ) : un modèle semi-classique
Annales de l'Institut Fourier, Volume 45 (1995) no. 1, p. 265-316

In L 2 ( d ), we prove exponential localization for a semi-classical periodic Schrödinger operator perturbated by small independant identically distributed random perturbations put in each well of the periodic potential. To do this, we first show that our operator, restricted to some suitably chosen energy interval, is unitarily equivalent to an infinite random matrix with coefficients we can control. Then, for this type of random matrices, we prove an Anderson localization theorem. We also apply this result to prove localization at large energy or large disorder, for long range discrete Anderson models.

Dans L 2 ( d ), nous démontrons un résultat de localisation exponentielle pour un opérateur de Schrödinger semi-classique à potentiel périodique perturbé par de petites perturbations aléatoires indépendantes identiquement distribuées placées au fond de chaque puits. Pour ce faire, on montre que notre opérateur, restreint à un intervalle d’énergie convenable, est unitairement équivalent à une matrice aléatoire infinie dont on contrôle bien les coefficients. Puis, pour ce type de matrices, on prouve un résultat de type localisation d’Anderson. On applique aussi ce résultat pour prouver la localisation à grande énergie ou grand désordre, pour des modèles d’Anderson discrets à longue portée.

@article{AIF_1995__45_1_265_0,
     author = {Klopp, Fr\'ed\'eric},
     title = {Localisation pour des op\'erateurs de Schr\"odinger al\'eatoires dans $L^2({\mathbb {R}}^d)$ : un mod\`ele semi-classique},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {45},
     number = {1},
     year = {1995},
     pages = {265-316},
     doi = {10.5802/aif.1456},
     zbl = {0817.35088},
     mrnumber = {96c:35203},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_1995__45_1_265_0}
}
Klopp, Frédéric. Localisation pour des opérateurs de Schrödinger aléatoires dans $L^2({\mathbb {R}}^d)$ : un modèle semi-classique. Annales de l'Institut Fourier, Volume 45 (1995) no. 1, pp. 265-316. doi : 10.5802/aif.1456. http://www.numdam.org/item/AIF_1995__45_1_265_0/

[Be] Yu. M. Berezanskii, On expansion in eigenfunctions of self-adjoint operators, Ukrain. Math. Zh., 11 (1959), 16-24, English transl. in Amer. Math. Soc. Trans. (2), 93 (1970). | Zbl 0206.13404

[Ca] U. Carlsson, An infinite number of wells in the semi-classical limit, Asympt. Anal., vol. 3 (1990), 189-214. | MR 91i:35134 | Zbl 0727.35094

[C] R. Carmona, Exponential localization in one dimensional disordered systems, Duke Math. Jour., 49 (1982), 191-213. | MR 84j:82082 | Zbl 0491.60058

[C-L] R. Carmona, J. Lacroix, Spectral theory of random Schrödinger operators, Birkhäuser, Boston Basel Berlin, 1990. | Zbl 0717.60074

[Co-His] J.M. Combes, P.D. Hislop, Some transport and spectral properties of disordered media, Schrödinger operators: the quantum mechanical many-body problem, LNP 403 (E. Balslev, eds.) Proceedings, Aarhus, Denmark 1991, Springer, Berlin-Heidelberg-New-York (1992), 16-47. | MR 93k:81047 | Zbl 0833.47058

[vDK1] H. Von Dreyfus, A. Klein, A new proof of localization in the Anderson tight binding model, Commun. Math. Phys., 124 (1989), 285-299. | MR 90k:82056 | Zbl 0698.60051

[vDK2] H. Von Dreyfus, A. Klein, Localization for random Schrödinger operators with correlated potentials, Commun. Math. Phys., 140 (1991), 133-147. | Zbl 0734.60070

[FMSS] J. Fröhlich, F. Martinelli, E. Scoppola, T. Spencer, Constructive proof of localization in the Anderson tight binding model, Commun. Math. Phys., 101 (1985), 21-46. | MR 87a:82047 | Zbl 0573.60096

[FS] J. Fröhlich, T. Spencer, Absence of diffusion in the Anderson tight binding model, Commun. Math. Phys., 88 (1983), 151-184. | Zbl 0519.60066

[GMP] Ya. Gol'Dsheid, S. Molchanov, L. Pastur, Pure point spectrum of stochastic one dimensional Schrödinger operators, Funct. Anal. Appl., 11 (1977) 1. | Zbl 0368.34015

[Gr] V. Grinshpun, Point spectrum of random infinite order operators acting on l2(ℤd), Dok. Akad. Nauk Ukraïni, 8 (1992), 18-21 (en russe).

[HM] H. Holden, F. Martinelli, A remark on the absence of diffusion near the bottom of the spectrum for a random Schrödinger operator in L2(Rv), Commun. Math. Phys., 93, (1984) 197-217. | MR 85m:82103 | Zbl 0546.60063

[He-Sj] B. Helffer, J. Sjöstrand, Multiple wells in the semi-classical limit 1, Comm. P.D.E, 9 (1984), 337-408. | Zbl 0546.35053

[Kl] F. Klopp, Étude semi-classique d'une perturbation d'un opérateur de Schrödinger périodique, Ann. Inst. Henri Poincaré, sér. Phys. Théor., 55 (1991), 459-509. | Numdam | MR 93c:35027 | Zbl 0754.35100

[KoSi] S. Kotani, B. Simon, Localization in general one dimensional systems, II, Commun. Math. Phys., 112 (1987), 103-119. | MR 89d:81034 | Zbl 0637.60080

[KuSo] H. Kunz, B. Souillard, Sur le spectre des opérateurs aux différences finies aléatoires, Commun. Math. Phys., 78 (1980), 201-246. | MR 83f:39003 | Zbl 0449.60048

[MS1] F. Martinelli, E. Scoppola, Introduction to the mathematical theory of Anderson localization, Riv. Nuovo Cim., 10 (1987), N10.

[MS2] F. Martinelli, E. Scoppola, Remark on the absence of absolutely continuous spectrum for d-dimensional Schrödinger operators with random potential for large disorder and low energy, Commun. Math. Phys., 97 (1985), 465-471. | MR 87b:81029a | Zbl 0603.60060

[O] A. Outassourt, Comportement semi-classique pour l'opérateur de Schrödinger à potentiel périodique, J. Funct. Anal., 72 (1987), 65-93. | MR 88k:35049 | Zbl 0662.35023

[P] L. Pastur, Spectra of random self-adjoint operators, Russ. Math. Surv., 28 (1973), 1. | MR 53 #10042 | Zbl 0277.60049

[PFi] L. Pastur, A. Figotin, Spectra of random and almost-periodic operators, Springer, Berlin-Heidelberg-New-York, 1992. | MR 94h:47068 | Zbl 0752.47002

[Si1] B. Simon, Semi-classical analysis of low lying eigenvalues III. Width of the ground state band in strongly coupled solids, Ann. Phys., 158 (1984), 415-420. | MR 87h:81045b | Zbl 0596.35028

[Si2] B. Simon, Schrödinger semigroups, Bull. Am. Math. Soc., 7 (1982), 447-526. | MR 86b:81001a | Zbl 0524.35002

[Wa] W.-M. Wang, Exponential decay of green's functions for a class of long range hamiltonians, Commun. Math. Phys., 136 (1991), 35-41. | MR 92b:81034 | Zbl 0726.58032

[We] F. Wegner, Bounds on the density of states in disordered systems, 1981, Z. Phys., B44 (1981), 9-15.