On resonances in disordered multi-particle systems

Victor Chulaevsky

doi:10.1016/j.crma.2011.12.003

Probability Theory/Mathematical Physics

On resonances in disordered multi-particle systems

Presented by: Jean-Michel Bony

Victor Chulaevsky ¹

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

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

Abstract
Résumé

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 $N > 2$ .

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 $N = 2$ .

Received: 2010-05-27
Accepted: 2011-12-05
Published online: 2011-12-21

DOI: 10.1016/j.crma.2011.12.003

Author's affiliations:

Victor Chulaevsky 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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Victor Chulaevsky. On resonances in disordered multi-particle systems. Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 81-85. doi : 10.1016/j.crma.2011.12.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.12.003/

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