  We study a system of trapped bosonic particles interacting by model harmonic
forces. Our model allows for detailed examination of the notion of an order
parameter (a condensate wave function). By decomposing a single particle
density matrix into coherent eigenmodes we study an effect of interaction on
the condensate. We show that sufficiently strong interactions cause that the
condensate disappears even if the whole system is in its lowest energy state.
In the second part of our paper we discuss the validity of the Bogoliubov
approximation by comparing its predictions with results inferred from the
exactl y soluble model. In particular we examine an energy spectrum,
occupation, and fluctuations of the condensate. We conclude that Bogoliubov
approach gives quite accurate description of the system in the limit of weak
interactions.
