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arXiv · 2111.13864

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

Morris Brooks·Robert Seiringer
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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.

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Morris Brooks, Robert Seiringer. 2021-11-27. Validity of Bogoliubov's approximation for translation-invariant Bose gases. https://doi.org/10.2140/pmp.2022.3.939

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