The Ground State Energy of a Dilute Bose Gas in Dimension n >3

Abstract

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 $\rho$, the ground state energy per particle in the thermodynamic limit is shown to be $(n-2)|\mathbb S^{n-1}|a^{n-2}\rho$, 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 $(1+C\gamma)8\pi^4a^6\rho^2|\ln(a^4\rho)|$ in dimension $n=4$, where $\gamma:=\int V(x)|x|^{-2}\, dx$, and a correction term which is $\mathcal{O}(\rho^2)$ in higher dimensions.