  We consider the Bogoliubov vacuum state in the number-conserving Bogoliubov
theory proposed by Castin and Dum [Phys. Rev. A 57, 3008 (1998)]. We show that
in the particle representation the vacuum can be written in a simple diagonal
form. The vacuum state can describe the stationary N-particle ground state of a
condensate in a trap, but it can also represent a dynamical state when, for
example, a Bose-Einstein condensate initially prepared in the stationary ground
state is subject to a time-dependent perturbation. In both cases the diagonal
form of the Bogoliubov vacuum can be obtained by basically diagonalizing the
reduced single particle density matrix of the vacuum. We compare N-body states
obtained within the Bogoliubov theory with the exact ground states in a 3-site
Bose-Hubbard model. In this example, the Bogoliubov theory fails to accurately
describe the stationary ground state in the limit when N goes to infinity but a
small fraction of depleted particles is kept constant.
