The Ground State Energy of a Dilute Bose Gas in Dimension $n>3$
by Anders Aaen
PhD Dissertations

March 2014
We consider a Bose gas in spatial dimension $n\geq3$ with a repulsive, radially symmetric twobody potential $V$. In the limit of low density $\rho$, the ground state energy per particle in the thermodynamic limit is shown to be $(n2)\mathbb S^{n1}a^{n2}\rho$, where $\mathbb S^{n1}$ 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 twobody potentials, we derive an upper bound to the ground state energy with a correction term $(1+\gamma)8\pi^4a^6\rho^2\ln(a^4\rho)$ in $4$ dimensions, where $0<\gamma\leq C\V\_{\infty}^{1/2}\V\_1^{1/2}$, and a correction term which is $\mathcal{O}(\rho^2)$ in higher dimensions. Finally, we use a grand canonical construction to give a simplified proof of the second order upper bound to the LeeHuangYang formula, a result first obtained by Yau and Yin. We also test this method in $4$ dimensions, but with a negative outcome.
Thesis advisor: Søren Fournais