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 $(1+C\gamma )8{\pi }^4{a}^6{\rho }^2|\ln (a^4\rho )|$ 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.

References in corpus (4)

• The Second Order Upper Bound for the Ground Energy of a Bose Gas
• Ground state energy of the two-dimensional weakly interacting Bose gas: First correction beyond Bogoliubov theory
• The ground state energy of a low density Bose gas: a second order upper bound
• The ground state energy of the weakly interacting Bose gas at high density