Multi-scale analysis in the occupation numbers of particle states: an application to three-modes Bogoliubov Hamiltonians

In this paper, we describe a multi-scale technique introduced in Pizzo (Bose particles in a box I. A convergent expansion of the ground state of a three-modes Bogoliubov Hamiltonian, [32]) to study many-body quantum systems. The method is based on the Feshbach–Schur map and the scales are represented by occupation numbers of particle states. The main purpose of this method is to implement singular perturbation theory to deal with large field problems. A simple model to apply our method is the three-modes (including the zero mode) Bogoliubov Hamiltonian that here we consider for a sufficiently small ratio between the kinetic energy and the Fourier component of the (positive type) potential corresponding to the two nonzero modes. In space dimension d≥3, for an arbitrarily large box and at fixed, large particle density ρ (i.e., ρ is independent of the size of the box), this method provides the construction of the ground state vector of the system and its expansion, up to any desired precision, in terms of the bare operators and the ground state energy. In the mean field limiting regime (i.e., at fixed box volume |Λ| and for a number of particles, N, sufficiently large), this method provides the same results in any dimension d≥1. Furthermore, in the mean field limit, we can replace the ground state energy with the Bogoliubov energy in the expansion of the ground state vector.