Dynamical Hartree-Fock-Bogoliubov Approximation of Interacting Bosons

Abstract

We consider a many-body Boson system with pairwise particle interaction given by $N^{3\beta -1}v(N^{\beta }x)$ for $0<\beta <1$ and $v$ a non-negative spherically-symmetric function. Our main result is the extension of the local-in-time Fock space (norm) approximation of the exact dynamics of squeezed states proved in Grillakis and Machedon, Comm. PDEs, (2017) for $0<\beta <\frac{2}{3}$ to a global-in-time norm approximation for $0<\beta <1$. Our work can also be viewed as a generalization of the results of Boccato, Cenatiempo, and Schlein, Ann. Henri Poincaré, (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 norm approximation are the work of Grillakis and Machedon on the local wellposedness theory, Grillakis and Machedon, Comm. PDEs, (2019), some recent established global estimates in Chong, Grillakis, Machedon, and Zhao, arXiv:2008.01753 (2020), and our quantitative result on the uniform in $N$ global wellposedness of the time-dependent Hartree-Fock-Bogoliubov (TBHFB) system.