The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases

We study the ground state of a trapped Bose gas, starting from the full many-body Schr{ö}dinger Hamiltonian, and derive the nonlinear Schr{ö}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{ö}dinger ground state.

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