Dynamical Hartree–Fock–Bogoliubov Approximation of Interacting Bosons

Abstract

We consider a many-body Bosonic system with pairwise particle interaction given by $$N^{3\beta -1}v(N^\beta x)$$ where $$0<\beta <1$$ and v a non-negative spherically symmetric function. Our main result is the extension of the local-in-time Fock space approximation of the exact dynamics of squeezed states proved in Grillakis and Machedon (Commun Partial Differ Equ 42(1):24–67, 2017) for $$0<\beta <\frac{2}{3}$$ to a global-in-time approximation for $$0<\beta <1$$ . Our work can also be viewed as a generalization of the results in Boccato et al. (Ann Henri Poincaré 18(1):113–191, 2017) to a more general set of initial data that includes coherent states along with an improved error estimate. The key ingredients in establishing the Fock space approximation are the work of Grillakis and Machedon on the the local well-posedness theory (Grillakis and Machedon in Commun Partial Differ Equ 44(12):1431–1465, 2019), some recent established global estimate in Chong et al. (Commun Partial Differ Equ 56:1–41, 2021), and our quantitative results on the uniform in N global well-posedness of the time-dependent Hartree–Fock–Bogoliubov system.