  It is well-known that the {\it Gross-Pitaevskii} variational formula
describes the the ground state energy of of $N$-indistinguishable trapped
particles (bosons) in a dilute state in the large system size $N\to\infty$. The
goal of the present article is to prove that the Gross-Pitaevskii formula also
appears in the {\it iterative limit} of zero temperature and large system size
of the {\it product ground state energy} of the $N$-particle Hamiltonian
operator.
