Validity of Bogoliubov's approximation for translation-invariant Bose gases

TL;DR

This paper rigorously confirms Bogoliubov's approximation for translation-invariant Bose gases in the mean field regime, establishing the ground state energy and Bose-Einstein condensation properties.

Contribution

It proves the validity of Bogoliubov's approximation for translation-invariant Bose gases, including energy asymptotics and existence of approximate ground states.

Findings

Ground state energy matches Bogoliubov's prediction asymptotically.

Existence of approximate ground states with Bose-Einstein condensation.

Complete validation of Bogoliubov's approximation in the mean field regime.

Abstract

We verify Bogoliubov's approximation for translation invariant Bose gases in the mean field regime, i.e. we prove that the ground state energy $E_N$ is given by $E_N=Ne_\mathrm{H}+\inf \sigma\left(\mathbb{H}\right)+o_{N\rightarrow \infty}(1)$, where $N$ is the number of particles, $e_\mathrm{H}$ is the minimal Hartree energy and $\mathbb{H}$ is the Bogoliubov Hamiltonian. As an intermediate result we show the existence of approximate ground states $\Psi_N$, i.e. states satisfying $\langle H_N\rangle_{\Psi_N}=E_N+o_{N\rightarrow \infty}(1)$, exhibiting complete Bose--Einstein condensation with respect to one of the Hartree minimizers.