  In this paper we consider an interacting Bose gas at zero temperature, in a
finite box and in the mean field limiting regime. The N gas particles interact
through a pair potential of positive type and with an ultraviolet cut-off. Its
(nonzero) Fourier components are sufficiently large with respect to the
corresponding kinetic energies of the modes. Using the multi-scale technique in
the occupation numbers of particle states introduced in [Pi1], we provide a
convergent expansion of the ground state of the particle number preserving
Bogoliubov Hamiltonian in terms of the bare operators. In the limit N \to
\infty the expansion is up to any desired precision.
