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

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