Global uniform in $N$ estimates for solutions of a system of Hartree-Fock-Bogoliubov type in the case $\beta<1$

TL;DR

This paper extends previous results to establish uniform in N estimates for solutions of a coupled Hartree-Fock-Bogoliubov system with a specific interaction potential, advancing understanding of many-body quantum dynamics.

Contribution

It provides the first global-in-time uniform estimates for solutions of a Hartree-Fock-Bogoliubov system with non-small interaction potential for eta<1.

Findings

Uniform in N estimates for solutions , ,  in the system

Development of expected correlations in  dynamically over time

Potential satisfies technical conditions but is not small

Abstract

We extend the results of the 2019 paper by the third and fourth author globally in time. More precisely, we prove uniform in $N$ estimates for the solutions $\phi$, $\Lambda$ and $\Gamma$ of a coupled system of Hartree-Fock-Bogoliubov type with interaction potential $V_N(x-y)=N^{3 \beta}v(N^{\beta}(x-y))$ with $\beta<1$. The potential satisfies some technical conditions, but is not small. The initial conditions have finite energy and the "pair correlation" part satisfies a smallness condition, but are otherwise general functions in suitable Sobolev spaces, and the expected correlations in $\Lambda$ develop dynamically in time. The estimates are expected to improve the Fock space bounds from the 2021 paper of the first and fifth author. This will be addressed in a different paper.