Global uniform in $N$ estimates for solutions of a system of Hartree-Fock-Bogoliubov type in the Gross-Pitaveskii regime

TL;DR

This paper establishes global, uniform-in-N estimates for solutions to a Hartree-Fock-Bogoliubov system in the Gross-Pitaevskii regime, extending previous work to the critical case with small potentials.

Contribution

It extends recent results to the critical case, providing uniform estimates for solutions of a coupled Hartree-Fock-Bogoliubov system with small interaction potential.

Findings

Proved global in time, uniform in N estimates for solutions.

Developed a sharp estimate for the linear Schrödinger equation with potential in 6+1 dimensions.

Extended the analysis to the critical regime with small potentials.

Abstract

We extend the recent work of Chong et al., (2022) to the critical case. More precisely, we prove global in time, uniform in $N$ estimates for the solutions $\phi$, $\Lambda$ and $\Gamma$ of a coupled system of Hartree--Fock--Bogoliubov type with interaction potential $\frac1NV_N(x-y)=N^{2}v(N(x-y))$. We assume that the potential $v$ is small which satisfies some technical conditions, and the initial conditions have finite energy. The main ingredient is a sharp estimate for the linear Schr\"odinger equation with potential in 6+1 dimension, which may be of interest in its own right.