Derivation of the dipolar Gross-Pitaevskii energy

Abstract

We consider N trapped bosons in R 3 interacting via a pair potential w which has a long range of dipolar type. We show the convergence of the energy and of the minimizers for the many-body problem towards those of the dipolar Gross-Pitaevskii functional, when N tends to infinity. In addition to the usual cubic interaction term, the latter has the long range dipolar interaction. Our results hold under the assumption that the two-particle interaction is scaled in the form N 3$\beta$--1 w(N $\beta$ x) for some 0 $\le$ $\beta$ \textless{} $\beta$max with $\beta$max = 1/3 + s/(45 + 42s) where s is related to the growth of the trapping potential.