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We prove the following formula for the ground state energy density of a dilute Bose gas with density ρ in 2 dimensions in the thermodynamic limit e2D(ρ)=4πρ2Y(1-Y|logY|+(2Γ+12+log(π))Y)+o(ρ2Y2), as ρa2→0 . Here Y=|log(ρa2)|-1 and a is the scattering length of the two-body potential. This result in 2 dimensions corresponds to the famous Lee-Huang-Yang formula in 3 dimensions. The proof is valid for essentially all positive potentials with finite scattering length, in particular, it covers the crucial case of the hard core potential.
Fournais et al. (Fri,) studied this question.
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