The thermodynamic limit for an imperfect Boson gas

We give a rigorous treatment in the infinite volume limit of a model Hamiltonian representing an imperfect Boson gas. In particular we obtain the exact expression for the mean particle density in the infinite volume limit as a function of the chemical potential, and show that the density function has a singularity at the critical density for Bose-Einstein condensation. We prove that, unlike the ideal Boson gas, the imperfect Boson gas has the same behaviour in the infinite volume limit for the grand canonical ensemble as for the canonical ensemble, and is moreover stable under small perturbations. We finally exhibit the possibility of ordinary condensation and prove that a system in an intermediate situation between two pure phases consists of a simple mixture of the two phases involved.