Bose-Einstein Condensation in the Luttinger-Sy Model
2007, v.13, №2, 441-468
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 a 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 the mathematical mechanism of 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.
Keywords: generalized Bose-Einstein condensation,random potential,density of states,Lifshitz tail,Kac-Luttinger conjecture