A Rigorous Derivation¶of the Gross–Pitaevskii Energy Functional¶for a Two-dimensional Bose Gas
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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 ρ̅a2 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(ρ̅a2)|. 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.
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