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

Lewin M, Nam P, Rougerie N. 2016. The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases. Transactions of the American Mathematical Society. 368(9), 6131–6157.

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*English*

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Author

Lewin, Mathieu;
Phan Thanh, Nam

^{IST Austria}; Rougerie, NicolasDepartment

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.

Publishing Year

Date Published

2016-01-01

Journal Title

Transactions of the American Mathematical Society

Acknowledgement

The authors acknowledge financial support from the European Research Council (FP7/2007-2013 Grant Agreement MNIQS 258023) and the ANR (Mathostaq project, ANR-13-JS01-0005-01). The second and third authors have benefited from the hospitality of the Institute for Mathematical Science of the National University of Singapore.

Volume

368

Issue

9

Page

6131 - 6157

IST-REx-ID

### Cite this

Lewin M, Nam P, Rougerie N. The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases.

*Transactions of the American Mathematical Society*. 2016;368(9):6131-6157. doi:10.1090/tran/6537Lewin, M., Nam, P., & Rougerie, N. (2016). The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases.

*Transactions of the American Mathematical Society*. American Mathematical Society. https://doi.org/10.1090/tran/6537Lewin, Mathieu, Phan Nam, and Nicolas Rougerie. “The Mean-Field Approximation and the Non-Linear Schrödinger Functional for Trapped Bose Gases.”

*Transactions of the American Mathematical Society*. American Mathematical Society, 2016. https://doi.org/10.1090/tran/6537.M. Lewin, P. Nam, and N. Rougerie, “The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases,”

*Transactions of the American Mathematical Society*, vol. 368, no. 9. American Mathematical Society, pp. 6131–6157, 2016.Lewin, Mathieu, et al. “The Mean-Field Approximation and the Non-Linear Schrödinger Functional for Trapped Bose Gases.”

*Transactions of the American Mathematical Society*, vol. 368, no. 9, American Mathematical Society, 2016, pp. 6131–57, doi:10.1090/tran/6537.**All files available under the following license(s):**

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