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

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 $[\ln (N)]^4/N^{1 - 2\eta} \ll \nu_N \lesssim 1$forarbitrary$0 < \eta \leq 1/3$. 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 ${\nu }_N{N}^{3\mathrm{/}5+\delta }$ for $\delta > 0$.