Bose-Einstein Condensation in the Luttinger-Sy Model

Olivier Lenoble (1, 2) , Valentin A. Zagrebnov (1, 2)

1 CPT - Centre de Physique Théorique - UMR 6207
2 CPT - Centre de Physique Théorique - UMR 7332

We present a rigorous study of the Bose-Einstein condensation in the Luttinger-Sy model. We prove the existence of the condensation in this one-dimensional model of the perfect boson gas placed in the Poisson random potential of singular point impurities. To tackle the off-diagonal long-range order we calculate explicitly the corresponding space-averaged one-body reduced density matrix. We show that mathematical mechanism of the Bose-Einstein condensation in this random model is similar to condensation in a one-dimensional nonrandom hierarchical model of scaled intervals. For the Luttinger-Sy model we prove the Kac-Luttinger conjecture, i.e., that this model manifests a type I BEC localized in a single "largest" interval of logarithmic size.

Mots clés: generalized Bose-Einstein condensation; random potential; density of states; Lifshitz tail; Kac-Luttinger conjecture

Domaines: Physique mathématique [math-ph]

Olivier Lenoble, Valentin A. Zagrebnov. Bose-Einstein Condensation in the Luttinger-Sy Model. Markov Processes And Related Fields, 2007, 13 (2), pp.441-468. ⟨hal-00023443⟩