Gibbs states of a quantum crystal: uniqueness by small particle mass
[États de Gibbs de crystaux quantiques: unicité dans le cas d'une petite masse]
Comptes Rendus. Mathématique, Tome 335 (2002) no. 8, pp. 693-698.

On considère un modèle de particules quantiques en intéraction effectuant des oscillations anharmoniques uni-dimensionelles autour de leur positions d'équilibre sur le réseau d . Pour ce modèle, nous énonçons deux résultats décrivant ses propriétés d'équilibre. Le premier théorème affirme l'existence de m * >0 tel que pour toutes les valeurs de la masse m de la particule inférieures à m * , l'ensemble des mesures euclidiennes tempérées de Gibbs consiste en un seul élément, à toute température β−1. Cela résoud un problème qui est resté ouvert pour longtemps et améliore essentiellement un résultat analogue obtenu par les mêmes auteurs, lorsque m * dépendait de β de sorte que m * (β)0 si β→+∞. Le deuxième théorème dit que la fonction de corrélation a une décroissance exponentielle si m<m * .

A model of interacting quantum particles performing one-dimensional anharmonic oscillations around their unstable equilibrium positions, which form the lattice d , is considered. For this model, two statements describing its equilibrium properties are given. The first theorem states that there exists m * >0 such that for all values of the particle mass m<m * , the set of tempered Euclidean Gibbs measures consists of exactly one element at all values of the temperature β−1. This settles a problem that was open for a long time and is an essential improvement of a similar result proved before by the same authors [1] where the boundary m * depended on β in such a way that m * (β)0 for β→+∞. The second theorem states that the two-point correlation function has an exponential decay if m<m * .

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DOI : 10.1016/S1631-073X(02)02545-1
Albeverio, Sergio 1, 2, 3 ; Kondratiev, Yuri 4, 2, 5 ; Kozitsky, Yuri 6 ; Röckner, Michael 4, 2

1 Institut für Angewandte Mathematik, Universität Bonn, 53115 Bonn, Germany
2 Forschungszentrum BiBoS, Universität Bielefeld, 33615 Bielefeld, Germany
3 CERFIM, Locarno and USI, Switzerland
4 Fakultät für Mathematik, Universität Bielefeld, 33615 Bielefeld, Germany
5 Institute of Mathematics, Kiev, Ukraine
6 Instytut Matematyki, Uniwersytet Marii Curie-Skłodowskiej, 20-031 Lublin, Poland
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Albeverio, Sergio; Kondratiev, Yuri; Kozitsky, Yuri; Röckner, Michael. Gibbs states of a quantum crystal: uniqueness by small particle mass. Comptes Rendus. Mathématique, Tome 335 (2002) no. 8, pp. 693-698. doi : 10.1016/S1631-073X(02)02545-1. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02545-1/

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