Dynamical Hartree-Fock-Bogoliubov Approximation of Interacting Bosons

TL;DR

This paper extends the global-in-time Fock space approximation for the dynamics of interacting bosons with a specific class of pairwise interactions, improving upon previous local-in-time results and including a broader set of initial states.

Contribution

It generalizes previous local-in-time approximations to global-in-time for a wider range of initial data and interaction parameters, with improved error estimates.

Findings

Achieved global-in-time norm approximation for 0<β<1

Extended initial data class to include coherent states

Provided uniform in N global well-posedness results

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 Poincar\'e, (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.