The Ground State Energy of a Dilute Bose Gas in Dimension n >3
We consider a Bose gas in spatial dimension $n>3$ with a repulsive, radially symmetric two-body potential $V$. In the limit of low density $ρ$, the ground state energy per particle in the thermodynamic limit is shown to be $(n-2)|\mathbb S^{n-1}|a^{n-2}ρ$, where $|\mathbb S^{n-1}|$ denotes the surface measure of the unit sphere in $\mathbb{R}^n$ and $a$ is the scattering length of $V$. Furthermore, for smooth and compactly supported two-body potentials, we derive upper bounds to the ground state energy with a correction term $(1+Cγ)8π^4a^6ρ^2|\ln(a^4ρ)|$ in dimension $n=4$, where $γ:=\int V(x)|x|^{-2}\, dx$, and a correction term which is $\mathcal{O}(ρ^2)$ in higher dimensions.
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