# Uniform in $N$ Global Well-posedness of the Time-Dependent Hartree-Fock-Bogoliubov Equations in $\mathbb{R}^{1+1}$

## TL;DR

This paper proves the global well-posedness of the time-dependent Hartree-Fock-Bogoliubov equations in one spatial dimension for a broad class of interaction potentials, uniformly in the particle number N.

## Contribution

It establishes uniform-in-N global well-posedness of TDHFB equations in 1+1 dimensions for all interaction scalings using dispersive PDE techniques.

## Key findings

- Global well-posedness in Strichartz spaces independent of N
- Applicability to a wide range of interaction potentials
- Extension of dispersive PDE methods to TDHFB equations

## Abstract

In this article, we prove the global well-posedness of the time-dependent Hartree-Fock-Bogoliubov (TDHFB) equations in $\mathbb{R}^{1+1}$ with two-body interaction potentials of the form $N^{-1}v_N(x) = N^{\beta-1} v(N^\beta x)$ where $v$ is a sufficiently regular radial function $v \in L^1(\mathbb{R})\cap C^\infty(\mathbb{R})$. In particular, using methods of dispersive PDEs similar to the ones used in Grillakis and Machedon, Comm. PDEs., (2017), we are able to show for any scaling parameter $\beta>0$ the TDHFB equations are globally well-posed in some Strichartz-type spaces independent of $N$, cf. (Bach et al. in arXiv:1602.05171).