Rigorous Derivation of the Gross-pitaevskii Equation with a Large Interaction Potential

We consider a bosonic system of N particles with a repulsive interaction. The states of the system are given by elements of the Hilbert space Ls(R 3N ), the subspace of L(R ) consisting of permutation symmetric wave functions. We are interested in describing the time evolution of special initial wave functions ψN ∈ Ls(R ) that exhibit complete Bose-Einstein condensation. For a given wave function ψN , we define the density matrix γN = ∣ψN ⟩⟨ψN ∣ associated with ψN as the orthogonal projection onto ψN . Moreover, for k = 1, . . . , N , we define the k-particle marginal density γ (k) N , associated with ψN , by taking the partial trace of γN over the last (N − k) variables. In other words, γ N is defined as a positive trace-class operator on L(R) with kernel given by