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

TL;DR

This paper develops a convergent expansion for the ground state of the Bogoliubov Hamiltonian describing an interacting Bose gas in a finite box under mean field conditions, valid in the large particle number limit.

Contribution

It introduces a multi-scale technique to obtain a convergent expansion of the ground state in terms of bare operators, enhancing understanding of Bose gases at zero temperature.

Findings

Convergent expansion valid as N approaches infinity

Expansion accurately describes the ground state in the mean field regime

Method applicable to particle number preserving Hamiltonians

Abstract

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