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arXiv:2111.13864 · Probability and Mathematical Physics

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

Morris Brooks, Robert Seiringer

Published Dec 31, 2022• 3 citations
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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 $E_N=Ne_\mathrm{H}+\inf σ\left(\mathbb{H}\right)+o_{N\rightarrow \infty}(1)$, 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 $Ψ_N$, i.e. states satisfying $\langle H_N\rangle_{Ψ_N}=E_N+o_{N\rightarrow \infty}(1)$, exhibiting complete Bose--Einstein condensation with respect to one of the Hartree minimizers.

DOI
10.2140/pmp.2022.3.939

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We verify Bogoliubov's approximation for translation invariant Bose gases in the mean field regime, i.e. we prove that the ground state energy ENE_NEN​ is given by EN=NeH+inf⁡σ(H)+oN→∞(1)E_N=Ne_\mathrm{H}+\inf σ\left(\mathbb{H}\right)+o_{N\rightarrow \infty}(1)EN​=NeH​+infσ(H)+oN→∞​(1), where NNN is the number of particles, eHe_\mathrm{H}eH​ is the minimal Hartree energy and H\mathbb{H}H is the Bogoliubov Hamiltonian. As an intermediate result we show the existence of approximate ground states ΨNΨ_NΨN​, i.e. states satisfying ⟨HN⟩ΨN=EN+oN→∞(1)\langle H_N\rangle_{Ψ_N}=E_N+o_{N\rightarrow \infty}(1)⟨HN​⟩ΨN​​=EN​+oN→∞​(1), exhibiting complete Bose--Einstein condensation with respect to one of the Hartree minimizers.
This article is a review of recent results with Phan Thành Nam, Nicolas Rougerie, Sylvia Serfaty and Jan Philip Solovej. We consider a system of N bosons with an interaction of intensity 1 / N (mean-field regime). In the limit N → ∞ , we prove that the first order in the expansion of the eigenvalues of the many-particle Hamiltonian is given by the nonlinear Hartree theory, whereas the next order is predicted by the Bogoliubov Hamiltonian. We also discuss the occurrence of Bose-Einstein condensation in these systems.
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We study the large‐N limit of a system of N bosons interacting with a potential of intensity 1/N. When the ground state energy is to the first order given by Hartree's theory, we study the next order, predicted by Bogoliubov's theory. We show the convergence of the lower eigenvalues and eigenfunctions towards that of the Bogoliubov Hamiltonian (up to a convenient unitary transform). We also prove the convergence of the free energy when the system is sufficiently trapped. Our results are valid in an abstract setting, our main assumptions being that the Hartree ground state is unique and nondegenerate, and that there is complete Bose‐Einstein condensation on this state. Using our method we then treat two applications: atoms with “bosonic” electrons on one hand, and trapped two‐dimensional and three‐dimensional Coulomb gases on the other hand. © 2015 Wiley Periodicals, Inc.