On the Cauchy problem for Gross-Pitaevskii hierarchies

TL;DR

This paper establishes local existence and uniqueness of solutions for the Gross-Pitaevskii hierarchy on  spaces, clarifying the well-posedness in Sobolev-type spaces for  /2.

Contribution

It proves local well-posedness of the Gross-Pitaevskii hierarchy in Sobolev spaces with  > n/2, providing a comprehensive analysis of all such cases.

Findings

Proved local existence of solutions in  > n/2 spaces.

Established uniqueness of solutions in these Sobolev spaces.

Clarified the conditions for well-posedness of the hierarchy.

Abstract

The purpose of this paper is to investigate the Cauchy problem for the Gross-Pitaevskii infinite linear hierarchy of equations on $\mathbb{R}^n,$ $n \geq 1.$ We prove local existence and uniqueness of solutions in certain Sobolev type spaces $\mathrm{H}^{\alpha}_{\xi}$ of sequences of marginal density operators with $\alpha > n/2.$ In particular, we give a clear discussion of all cases $\alpha > n/2,$ which covers the local well-posedness problem for Gross-Pitaevskii hierarchy in this situation.