Bose--Einstein condensation in the Luttinger--Sy model with contact interaction

Abstract

We study bosons on the real line in a Poisson random potential (Luttinger--Sy model) with contact interaction in the thermodynamic limit at absolute zero temperature. We prove that generalized Bose--Einstein condensation (BEC) occurs almost surely if the intensity ${\nu }_N$ of the Poisson potential satisfies $[\ln (N){]}^4/N^{1-2\eta }\ll {\nu }_N\lesssim 1$ for arbitrary $0<\eta \le 1/3$. We also show that the contact interaction alters the type of condensation, going from a type-I BEC to a type-III BEC as the strength of this interaction is increased. Furthermore, for sufficiently strong contact interactions and $0<\eta <1/6$ we prove that the mean particle density in the largest interval is almost surely bounded asymptotically by ${\nu }_N{N}^{3/5+\delta }$ for $\delta >0$.

35 pages