Partial Differential Equations/Mathematical Physics
On two-particle Anderson localization at low energies
[Localisation d'Anderson pour un système à deux particules, à basses énergies]
Comptes Rendus. Mathématique, Tome 349 (2011) no. 3-4, pp. 167-170.

On démontre la localisation spectrale exponentielle pour un modèle d'Anderson discret, avec interaction à courte portée dans un champ de potentiel aléatoire i.i.d., à basses énergies. La démonstration utilise l'analyse multi-échelle multi-particule développée dans Chulaevsky et Suhov (2009) [4] dans le cas de grand désordre. Cette méthode s'applique à une classe de potentiels aléatoires plus large que dans Aizenman et Warzel (2009) [2], où la localisation dynamique a été démontrée par la méthode des moments fractionnaires.

We prove exponential spectral localization in a two-particle lattice Anderson model, with a short-range interaction and an external i.i.d. random potential, at sufficiently low energies. The proof is based on the multi-particle multi-scale analysis developed earlier in Chulaevsky and Suhov (2009) [4] in the case of high disorder. Our method applies to a larger class of random potentials than in Aizenman and Warzel (2009) [2] where dynamical localization was proved with the help of the fractional moment method.

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Accepté le :
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DOI : 10.1016/j.crma.2010.11.003
Ekanga, Trésor 1

1 Institut de mathématiques de Jussieu, université Paris Diderot, 175, rue du Chevaleret, 75013 Paris, France
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Ekanga, Trésor. On two-particle Anderson localization at low energies. Comptes Rendus. Mathématique, Tome 349 (2011) no. 3-4, pp. 167-170. doi : 10.1016/j.crma.2010.11.003. http://www.numdam.org/articles/10.1016/j.crma.2010.11.003/

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[2] Aizenman, M.; Warzel, S. Localization bounds for multi-particle systems, Comm. Math. Phys., Volume 290 (2009), pp. 903-934

[3] Chulaevsky, V.; Suhov, Y. Wegner bounds for a two-particle tight binding model, Comm. Math. Phys., Volume 283 (2008), pp. 479-489

[4] Chulaevsky, V.; Suhov, Y. Eigenfunctions in a two-particle Anderson tight binding model, Comm. Math. Phys., Volume 289 (2009), pp. 701-723

[5] Fröhlich, J.; Martinelli, F.; Scoppola, E.; Spencer, T. Constructive proof of localization in the Anderson tight binding model, Comm. Math. Phys., Volume 101 (1985), pp. 21-46

[6] Kirsch, W. A Wegner estimate for multi-particle random Hamiltonians, Zh. Mat. Fiz. Anal. Geom., Volume 4 (2008), pp. 121-127

[7] Stollmann, P. Caught by Disorder, Birkhäuser Inc., Boston, MA, 2001

[8] von Dreifus, H.; Klein, A. A new proof of localization in the Anderson tight binding model, Comm. Math. Phys., Volume 124 (1989), pp. 285-299

[9] Wegner, F. Bounds on the density of states in disordered systems, Z. Phys. B Condens. Matter, Volume 44 (1981), pp. 9-15

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