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Journal of Mathematical Physics

Bose gases, Bose–Einstein condensation, and the Bogoliubov approximation

Robert Seiringer

Published Jun 26, 2014• 14 citations
Publisher

Abstract

We review recent progress towards a rigorous understanding of the Bogoliubov approximation for bosonic quantum many-body systems. We focus, in particular, on the excitation spectrum of a Bose gas in the mean-field (Hartree) limit. A list of open problems will be discussed at the end.
DOI
10.1063/1.4881536

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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 thesis is devoted to a rigorous study of interacting quantum gases. The main objects of interest are the closely related concepts of excitation spectrum and quasiparticles. The immediate motivation of this work is to propose a spectral point of view concerning these two concepts. In the first part of this thesis we discuss the concepts of excitation spectrum and quasiparticles. We provide an overview of physical motivations and based on that we propose a spectral and Hamiltonian-based approach towards these terms. Based on that, we formulate definitions and propositions related to these concepts. In the second part we recall the Bogoliubov and Hartree-Fock-Bogoliubov approximations, which in the physics literature are used to obtain the quasiparticle picture. We show how these two approaches fit into a universal scheme which allows us to arrive at a quasiparticle picture in a more general setup. This scheme is based on the minimization of Hamiltonians over the so-called Gaussian states. Its abstract formulation is the content of Beliaev's Theorem. In the last part we present a rigorous result concerning the justification of the Bogoliubov approximation. This justification employs the concept of the mean-field and infinite-volume limit. We show that for a large number of particles, a large volume and a sufficiently high density, the low-lying energy-momentum spectrum of the homogeneous Bose gas is well described by the Bogoliubov approximation. This result, which is formulated in the form of a theorem, can be seen as the main result of this thesis.