Bose particles in a box I. A convergent expansion of the ground state of a three-modes Bogoliubov Hamiltonian

Abstract

In this paper we introduce a novel multi-scale technique to study many-body quantum systems where the total number of particles is kept fixed. The method is based on Feshbach map and the scales are represented by occupation numbers of particle states. Here, we consider a three-modes (including the zero mode) Bogoliubov Hamiltonian for a sufficiently small ratio between the kinetic energy and the Fourier component of the (positive type) potential corresponding to the two nonzero modes. For any space dimension d\geq 1 and in the mean field limiting regime (i.e., at fixed box volume |\Lambda| and for a number of particles, N, sufficiently large) this method provides the construction of the ground state and its expansion in terms of the bare operators that in the limit N \to \infty is up to any desired precision. In space dimension d \geq 3 the method provides similar results for an arbitrarily large (finite) box and a large but fixed particle density \rho, i.e.,\rho is independent of the size of the box.