Comptes Rendus
Probability Theory/Mathematical Physics
On resonances in disordered multi-particle systems
Comptes Rendus. Mathématique, Volume 350 (2012) no. 1-2, pp. 81-85.

We assess the probability of resonances between sufficiently distant states x=(x1,,xN) and y=(y1,,yN) in the configuration space of an N-particle disordered quantum system on the lattice Zd, d1. This includes the cases where the transition xy “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(,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=(x1,,xN) et y=(y1,,yN) dans Zd, d1, pour un système de N3 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.

Published online:
DOI: 10.1016/j.crma.2011.12.003

Victor Chulaevsky 1

1 Département de mathématiques, université de Reims, moulin de la Housse, B.P. 1039, 51687 Reims cedex 2, France
     author = {Victor Chulaevsky},
     title = {On resonances in disordered multi-particle systems},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {81--85},
     publisher = {Elsevier},
     volume = {350},
     number = {1-2},
     year = {2012},
     doi = {10.1016/j.crma.2011.12.003},
     language = {en},
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PB  - Elsevier
DO  - 10.1016/j.crma.2011.12.003
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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.

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