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

TL;DR

This work establishes a rigorous, uniform-in-time validation of the Bogoliubov approximation for weakly interacting Bose gases by analyzing quantum fluctuations around a Bose–Einstein condensate. It develops a fluctuation framework that maps N-body dynamics to a Fock space of excitations and proves dispersive decay for the symplectic Bogoliubov dynamics, enabling a sharp, norm-based comparison between the many-body evolution and a quadratic Bogoliubov generator for all times. The authors show that the fluctuation dynamics are accurately described by a Bogoliubov dynamics with a quadratic Hamiltonian, with error O(1/N), and further demonstrate that, at long times, the Bogoliubov dynamics itself can be approximated by free evolution, yielding a robust, long-time description of quantum depletion. These results significantly strengthen prior time-scale limitations and provide a solid mathematical foundation for the long-time validity of Bogoliubov theory in mean-field Bose gases, with implications for the connection to kinetic descriptions.

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.

Theorems & Definitions (14)

• Theorem 1.1: Dispersive estimate for Bogoliubov dynamics
• Theorem 1.2: Uniform-in-time validity of Bogoliubov approximation
• Corollary 1.3
• Proposition 2.1: Proposition 3.3 in GM1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of Theorem \ref{['thm:bogo-disp']}
• Lemma 3.1