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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 gasparticles interact through a pair potential of positive type and with anultraviolet cut-off. The (nonzero) Fourier components of the potential areassumed to be sufficiently large with respect to the corresponding kineticenergies of the modes like in the companion papers [Pi1]-[Pi2]. Using themulti-scale technique in the occupation numbers of particle states introducedin [Pi1]-[Pi2], we provide a convergent expansion of the ground state of theHamiltonian in terms of the bare operators. In the limit N \to \infty theexpansion is up to any desired precision.

Bose particles in a box III. A convergent expansion of the ground state of the Hamiltonian in the mean field limiting regime