On resonances in disordered multi-particle systems

Chulaevsky, Victor

doi:10.1016/j.crma.2011.12.003

Probability Theory/Mathematical Physics

On resonances in disordered multi-particle systems
[Sur les résonances dans un système à plusieurs particules en milieu désordonné]

Présenté par : Jean-Michel Bony

Chulaevsky, Victor ¹

¹ Département de mathématiques, université de Reims, moulin de la Housse, B.P. 1039, 51687 Reims cedex 2, France

Comptes Rendus. Mathématique, Tome 350 (2012) no. 1-2, pp. 81-85.

Résumé
Abstract

On établit une estimation de la probabilité de résonance entre deux états quantiques $x = (x_{1}, \dots, x_{N})$ et $y = (y_{1}, \dots, y_{N})$ dans $Z^{d}$ , $d ⩾ 1$ , pour un système de $N ⩾ 3$ particules quantiques en milieu désordonné. Cette estimation généralise lʼanalogue de lʼestimation de Wegner pour N particules, analogue démontrée précédemment dans (Chulaevsky et Suhov (2008, 2009) [6,7]). Ce résultat permet dʼobtenir des estimations optimales de décroissance de fonctions propres pour les systèmes de $N > 2$ particules dans les milieux désordonnés, déjà démontrées dans (Chulaevsky et Suhov (2008) [6]) pour $N = 2$ .

We assess the probability of resonances between sufficiently distant states $x = (x_{1}, \dots, x_{N})$ and $y = (y_{1}, \dots, y_{N})$ in the configuration space of an N-particle disordered quantum system on the lattice $Z^{d}$ , $d ⩾ 1$ . This includes the cases where the transition $x ⇝ y$ “shuffles” the particles in x, like the transition $(a, a, b) ⇝ (a, b, b)$ in a 3-particle system. In presence of a random external potential $V (\cdot, ω)$ such pairs of configurations $(x, y)$ give rise to strongly coupled random local Hamiltonians, so that eigenvalue concentration bounds are difficult to obtain (cf. Aizenman and Warzel (2009) [2]; Chulaevsky and Suhov (2009) [8]). This results in eigenfunction decay bounds weaker than expected. We show that more optimal bounds obtained so far only for 2-particle systems (Chulaevsky and Suhov (2008) [6]) can be extended to any $N > 2$ .

Reçu le : 2010-05-27
Accepté le : 2011-12-05
Publié le : 2011-12-21

DOI : 10.1016/j.crma.2011.12.003

Affiliations des auteurs :

Chulaevsky, Victor ¹

¹ Département de mathématiques, université de Reims, moulin de la Housse, B.P. 1039, 51687 Reims cedex 2, France

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     title = {On resonances in disordered multi-particle systems},
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Chulaevsky, Victor. On resonances in disordered multi-particle systems. Comptes Rendus. Mathématique, Tome 350 (2012) no. 1-2, pp. 81-85. doi : 10.1016/j.crma.2011.12.003. http://www.numdam.org/articles/10.1016/j.crma.2011.12.003/

Bibliographie
Cité par

[1] Aizenman, M.; Molchanov, S.A. Localization at large disorder and at extreme energies: an elementary derivation, Comm. Math. Phys., Volume 157 (1993), pp. 245-278

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

[3] Anderson, P. Absence of diffusion in certain random lattices, Phys. Rev., Volume 109 (1958), pp. 1492-1505

[4] Boutet de Monvel, A.; Chulaevsky, V.; Stollmann, P.; Suhov, Y. Wegner-type bounds for a multi-particle continuous Anderson model with an alloy-type external potential, J. Stat. Phys., Volume 138 (2010), pp. 553-566

[5] Chulaevsky, V.; Suhov, Y. Anderson localisation for an interacting two-particle quantum system on $Z$ , 2007 | arXiv

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

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

[8] Chulaevsky, V.; Suhov, Y. Multi-particle Anderson localisation: induction on the number of particles, Math. Phys. Anal. Geom., Volume 12 (2009), pp. 117-139

[9] 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

[10] Fröhlich, J.; Spencer, T. Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Comm. Math. Phys., Volume 88 (1983), pp. 151-184

[11] 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

[12] M. Gaume, An extension of the multi-particle Wegner-type bound for weakly decoupled Hamiltonians, preprint, Univ. Paris 7, in preparation.

[13] Goldsheid, I.Y.; Molchanov, S.A.; Pastur, L.A. A pure point spectrum of the one-dimensional Schrödinger operator, Funct. Anal. Appl., Volume 11 (1977), pp. 1-10

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

[15] Kunz, H.; Souillard, B. Sur le spectre des opérateurs aux différences finies aléatoires, Comm. Math. Phys., Volume 78 (1980), pp. 201-246

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

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

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