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arXiv:2511.12748

Dispersive estimates and long-time validity for Bogoliubov dynamics of interacting Bose gases

Phan Thành Nam, Simone Rademacher, Avy Soffer

Published Nov 16, 2025• 0 citations
arXivHTMLPDF

Abstract

We consider the Bogoliubov approximation for the many-body quantum dynamics of weakly interacting Bose gases and establish a uniform-in-time validity of the Bogoliubov theory. The proof relies on a detailed analysis of the dispersive behavior of the symplectic Bogoliubov dynamics, which allows for a rigorous derivation of the Bogoliubov theory as an effective description of quantum fluctuations around the Bose-Einstein condensate on all time scales.

DOI
10.48550/arxiv.2511.12748

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We consider the Bogoliubov approximation for the many-body quantum dynamics of weakly interacting Bose gases and establish a uniform-in-time validity of the Bogoliubov theory. The proof relies on a detailed analysis of the dispersive behavior of the symplectic Bogoliubov dynamics, which allows for a rigorous derivation of the Bogoliubov theory as an effective description of quantum fluctuations around the Bose-Einstein condensate on all time scales.
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Bose-Einstein condensates have been experimentally realized in cold Bose gases for the first time in 1995.This discovery has renewed interest in the theory of interacting Bose gases; however, the mathematical understanding of Bose-Einstein condensation is still incomplete.In this thesis we derive effective theories for the dynamical and spectral properties of Bose gases.Our theoretical description is based on scaling regimes where the number of particles N becomes large and the two-body interaction potential scales as N 3β-1 V (N β •), for a fixed β ∈ (0, 1].The case β = 1 is known as Gross-Pitaevskii regime.Our first result concerns the time evolution of Bose-Einstein condensates after the trapping potential is removed.In previous works, it was shown that the many-body dynamics can be approximated by a one-body nonlinear Schrödinger equation at the level of reduced density matrices.We provide a more precise approximation of the time evolution: in a second-quantized setting, we construct a unitary effective dynamics, which, in the limit of large N , approximates the fully evolved state in Fock space norm.Our result comes with an explicit bound on the rate of convergence.It is valid for β < 1.Our second result addresses the properties of the ground state wave function.It has been shown before that the ground state wave function exhibits condensation in the limit N → ∞, i.e. the one-particle density matrix associated to the ground state approaches a projection.We prove the optimal bound for the convergence rate, for a translation invariant system.To do this, we estimate uniformly in N the number of excited particles.This result is valid for β = 1.Our last result concerns the ground state energy and the excitation spectrum of the translation invariant system.We prove the validity of Bogoliubov theory; this allows us to compute the ground state energy and the low-lying excitation spectrum up to an error that vanishes for large N .This result holds for β < 1.This thesis is based on the articles [11], [9] and [10].i
Abstract We introduce the map of dynamics of quantum Bose gases into dynamics of quasifree states, which we call the “nonlinear quasifree approximation”. We use this map to derive the time-dependent Hartree–Fock–Bogoliubov (HFB) equations describing the dynamics of quantum fluctuations around a Bose–Einstein condensate. We prove global well-posedness of the HFB equations for pair potentials satisfying suitable regularity conditions, and we establish important conservation laws. We show that the space of solutions of the HFB equations has a symplectic structure reminiscent of a Hamiltonian system. This is then used to relate the HFB equations to the HFB eigenvalue equations discussed in the physics literature. We also construct Gibbs equilibrium states at positive temperature associated with the HFB equations, and we establish criteria for the appearance of Bose–Einstein condensation.
The thesis provides rigorous quantitative analyses for studying quantum fluctuations of a non-relativistic Bose gas about a Bose-Einstein condensate. In recent years, the dynamics of the condensate and the excitations of a Bose gas was shown to be well approximated by the quasifree dynamics with governing equations given by a system of coupled nonlinear dispersive PDEs called the time-dependent Hartree-Fock-Bogoliubov (HFB) system (c.f. Grillakis and Machedon, J. Fix. Point Theory A., (2013), Grillakis and Machedon, Comm. PDEs, (2017), Bach et al. in arXiv:1602.05171v2, (2018), Benedikter et al., Ann. Henri Poincare, (2018)). However, both the quantitative and qualitative analysis of the time-dependent HFB system are still in their early stages of development. Thus, the primary purposes of the thesis are to further the development of some analytic tools necessary for studying the time-dependent HFB system and use these effective equations to provide quantitative estimates for the true dynamics of the Bose gas at absolute zero temperature in Fock space norm. The thesis comprises the entirety of the author's current and past projects on the time-dependent HFB system. Each project falls into one or more of two categories: studying the local or global well-posedness of the time-dependent HFB system, or obtaining global-in-time Fock space estimates for the error terms of the quasifree approximation to the dynamics of a system of interacting bosons. In the former category, the author employs techniques of dispersive PDE theory developed in the past three decades along with classical methods of harmonic analysis to study lower regularity solutions to the time-dependent HFB system. The lower regularity well-posedness of solutions for the time-dependent HFB system is necessary for studying norm approximation of the dynamics of a dilute Bose gas with strong interactions. In the latter category, we prove global a-priori estimates for the solutions to the HFB system and use them to obtain estimates for the error terms of the Fock space approximation.