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Pré-Publication, Document De Travail Année : 2014

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

Résumé

We study the ground state of a trapped Bose gas, starting from the full many-body Schrödinger Hamiltonian, and derive the nonlinear Schrödinger 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 Schrödinger ground state.
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Dates et versions

hal-00990051 , version 1 (13-05-2014)
hal-00990051 , version 2 (31-07-2014)
hal-00990051 , version 3 (24-09-2015)

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Mathieu Lewin, Phan Thành Nam, Nicolas Rougerie. The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases. 2014. ⟨hal-00990051v2⟩

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