  In this paper we consider an interacting Bose gas at zero temperature,
constrained to 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. The (nonzero) Fourier components of the potential are
assumed to be sufficiently large with respect to the corresponding kinetic
energies of the modes like in the companion papers [Pi1]-[Pi2]. Using the
multi-scale technique in the occupation numbers of particle states introduced
in [Pi1]-[Pi2], we provide a convergent expansion of the ground state of the
Hamiltonian in terms of the bare operators. In the limit N \to \infty the
expansion is up to any desired precision.
