# A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas

Lieb É, Seiringer R, Yngvason J. 2001. A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas. Communications in Mathematical Physics. 224(1), 17–31.

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Author

Lieb, Élliott;
Seiringer, Robert

^{ISTA}^{}; Yngvason, JakobAbstract

We consider the ground state properties of an inhomogeneous two-dimensional Bose gas with a repulsive, short range pair interaction and an external confining potential. In the limit when the particle number N is large but ρ̅a 2 is small, where ρ̅ is the average particle density and a the scattering length, the ground state energy and density are rigorously shown to be given to leading order by a Gross–Pitaevskii (GP) energy functional with a coupling constant g~1/|1n(ρ̅a 2)|. In contrast to the 3D case the coupling constant depends on N through the mean density. The GP energy per particle depends only on Ng. In 2D this parameter is typically so large that the gradient term in the GP energy functional is negligible and the simpler description by a Thomas–Fermi type functional is adequate.

Publishing Year

Date Published

2001-11-01

Journal Title

Communications in Mathematical Physics

Volume

224

Issue

1

Page

17 - 31

ISSN

IST-REx-ID

### Cite this

Lieb É, Seiringer R, Yngvason J. A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas.

*Communications in Mathematical Physics*. 2001;224(1):17-31. doi:10.1007/s002200100533Lieb, É., Seiringer, R., & Yngvason, J. (2001). A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas.

*Communications in Mathematical Physics*. Springer. https://doi.org/10.1007/s002200100533Lieb, Élliott, Robert Seiringer, and Jakob Yngvason. “A Rigorous Derivation of the Gross-Pitaevskii Energy Functional for a Two-Dimensional Bose Gas.”

*Communications in Mathematical Physics*. Springer, 2001. https://doi.org/10.1007/s002200100533.É. Lieb, R. Seiringer, and J. Yngvason, “A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas,”

*Communications in Mathematical Physics*, vol. 224, no. 1. Springer, pp. 17–31, 2001.Lieb, Élliott, et al. “A Rigorous Derivation of the Gross-Pitaevskii Energy Functional for a Two-Dimensional Bose Gas.”

*Communications in Mathematical Physics*, vol. 224, no. 1, Springer, 2001, pp. 17–31, doi:10.1007/s002200100533.**All files available under the following license(s):**

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arXiv cond-mat/0005026