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

Elliott H. Lieb, Robert Seiringer, Jakob Yngvason

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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 ρ̄a2 is small, where ρ̄ 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(ρ̄a2)|. 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.

Original languageEnglish (US)
Pages (from-to)17-31
Number of pages15
JournalCommunications In Mathematical Physics
Volume224
Issue number1
DOIs
StatePublished - Dec 1 2001

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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