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arXiv 1401.5960math-phmath.MP
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n >3维稀玻色气体的基态能量

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

Anders Aaen

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摘要：

我们考虑空间维度上的玻色气体$n>3$，具有排斥的、径向对称的两体势$V$。在低密度$ρ$的极限下，热力学极限下每个粒子的基态能量为$(n-2)|\mathbb S^{n-1}|a^{n-2}ρ$，其中$|\mathbb S^{n-1}|$表示$\mathbb{R}^n$中单位球的表面测度，$a$表示$V$的散射长度.此外，对于光滑和紧支撑的两体势，我们导出了基态能量的上界，其中修正项为$(1+Cγ)8π^4a^6ρ^2|\ln(a^4ρ)|$在维度$n=4$中，其中$γ:=\int V(x)|x|^{-2}\, dx$，和修正项是$\mathcal{O}(ρ^2)$在更高的维度。

英文摘要：

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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