On the Cauchy problem for Gross-Pitaevskii hierarchies

Abstract

The purpose of this paper is to investigate the Cauchy problem for the Gross-Pitaevskii infinite linear hierarchy of equations on ${\mathbb{R}}^n,$ $n\ge 1.$ We prove local existence and uniqueness of solutions in certain Sobolev type spaces ${\mathrm{H}}_{\xi }^{\alpha }$ 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.

arXiv comment17 pages. The referee's comments and suggestions have been incorporated into this version of the paper

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• Randomization and the Gross-Pitaevskii hierarchy
• Local existence of solutions to Randomized Gross-Pitaevskii hierarchies