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

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