The mean-field approximation and the non-linear Schr\"odinger functional   for trapped Bose gases
  We study the ground state of a trapped Bose gas, starting from the full many-body Schr{\"o}dinger Hamiltonian, and derive the nonlinear Schr{\"o}dinger energy functional in the limit of large particle number, when the interaction potential converges slowly to a Dirac delta function. Our method is based on quantitative estimates on the discrepancy between the full many-body energy and its mean-field approximation using Hartree states. These are proved using finite dimensional localization and a quantitative version of the quantum de Finetti theorem. Our approach covers the case of attractive interactions in the regime of stability. In particular, our main new result is a derivation of the 2D attractive nonlinear Schr{\"o}dinger ground state. 
Mathieu Lewin (CEREMADE), Phan Th\`anh Nam (IST Austria), Nicolas   Rougerie (LPMMC)
math-ph cond-mat.quant-gas math.AP math.MP