A Rigorous Derivation of the Gross-Pitaevskii Energy Functional for a Two-Dimensional Bose Gas

Elliott H. Lieb; Robert Seiringer; Jakob Yngvason

doi:10.1007/s002200100533

paper

A Rigorous Derivation of the Gross-Pitaevskii Energy Functional for a Two-Dimensional Bose Gas

arXiv:cond-mat/0005026 · doi:10.1007/s002200100533

Abstract

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 $ρ ˉ a^{2}$ is small, where $ρ ˉ$ 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 $g \sim 1/∣ ln (ρ ˉ a^{2}) ∣$ . In contrast to the 3D case the coupling constant depends on $N$ through the mean density. The GP energy per particle depends only on $N g$ . 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.

14 pages, no figures, latex 2e. References, some clarifications and an appendix added. To appear in Commun. Math. Phys

A Rigorous Derivation of the Gross-Pitaevskii Energy Functional for a Two-Dimensional Bose Gas

Abstract

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