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

Lewin, Mathieu; Nam, Phan; Rougerie, Nicolas

doi:10.1090/tran/6537

The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases
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by Mathieu Lewin, Phan Thành Nam and Nicolas Rougerie PDF

Trans. Amer. Math. Soc. 368 (2016), 6131-6157

Abstract:

We study the ground state of a trapped Bose gas, starting from the full many-body Schrödinger Hamiltonian, and derive the non-linear Schrödinger energy functional in the limit of a 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 non-linear Schrödinger ground state.

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