Mean-field limit of bose-einstein condensates with attractive interactions in

Starting with the many-body Schrödinger Hamiltonian in , we prove that the ground state energy of a two-dimensional interacting Bose gas with the pairwise attractive interaction approaches to the minimum of the Gross-Pitaevskii energy functional in the mean-field regime, as the particle number N → ∞ and however the scattering length κ → 0. By fixing N|κ|, this leads to the mean-field approximation of Bose-Einstein condensates with attractive interactions in .