I will present a rigorous study of the perfect Bose-gas in the
presence of a homogeneous ergodic random potential. It is
demonstrated that the Lifshitz tail behaviour of the one-particle
spectrum reduces the critical dimensionality of the (generalized)
Bose-Einstein Condensation (BEC) to $d=1$. To tackle the
Off-Diagonal Long-Range Order (ODLRO) I will introduce the
space averaged one-body reduced density matrix. For a one
dimensional Poisson-type random potential we proved that
randomness enhances the exponential decay of this matrix in domain
free of the BEC.
These general results will then be applied to the Luttinger-Sy model in
which I can explicitely compute any of the physical quantities
(pressure, density, type of condensation, ODLRO...).