  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.
