# A note on 2D focusing many boson systems

Lewin M, Nam P, Rougerie N. 2017. A note on 2D focusing many boson systems. Proceedings of the American Mathematical Society. 145(6), 2441–2454.

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https://arxiv.org/abs/1509.09045
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Author

Lewin, Mathieu;
Phan Thanh, Nam

^{ISTA}; Rougerie, NicolasCorresponding author has ISTA affiliation

Department

Abstract

We consider a 2D quantum system of N bosons in a trapping potential |x|s, interacting via a pair potential of the form N2β−1 w(Nβ x). We show that for all 0 < β < (s + 1)/(s + 2), the leading order behavior of ground states of the many-body system is described in the large N limit by the corresponding cubic nonlinear Schrödinger energy functional. Our result covers the focusing case (w < 0) where even the stability of the many-body system is not obvious. This answers an open question mentioned by X. Chen and J. Holmer for harmonic traps (s = 2). Together with the BBGKY hierarchy approach used by these authors, our result implies the convergence of the many-body quantum dynamics to the focusing NLS equation with harmonic trap for all 0 < β < 3/4.

Publishing Year

Date Published

2017-01-01

Journal Title

Proceedings of the American Mathematical Society

Publisher

American Mathematical Society

Volume

145

Issue

6

Page

2441 - 2454

IST-REx-ID

### Cite this

Lewin M, Nam P, Rougerie N. A note on 2D focusing many boson systems.

*Proceedings of the American Mathematical Society*. 2017;145(6):2441-2454. doi:10.1090/proc/13468Lewin, M., Nam, P., & Rougerie, N. (2017). A note on 2D focusing many boson systems.

*Proceedings of the American Mathematical Society*. American Mathematical Society. https://doi.org/10.1090/proc/13468Lewin, Mathieu, Phan Nam, and Nicolas Rougerie. “A Note on 2D Focusing Many Boson Systems.”

*Proceedings of the American Mathematical Society*. American Mathematical Society, 2017. https://doi.org/10.1090/proc/13468.M. Lewin, P. Nam, and N. Rougerie, “A note on 2D focusing many boson systems,”

*Proceedings of the American Mathematical Society*, vol. 145, no. 6. American Mathematical Society, pp. 2441–2454, 2017.Lewin, Mathieu, et al. “A Note on 2D Focusing Many Boson Systems.”

*Proceedings of the American Mathematical Society*, vol. 145, no. 6, American Mathematical Society, 2017, pp. 2441–54, doi:10.1090/proc/13468.**All files available under the following license(s):**

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