Elliott H. Lieb, Jakob Yngvason
The Ground State Energy of a Dilute Two-dimensional Bose Gas
(43K, Latex)

ABSTRACT.  The ground state energy per particle of a dilute, 
homogeneous, two-dimensional Bose gas, in the thermodynamic limit is 
shown rigorously to be $E_0/N = (2\pi \hbar^2\rho /m){|\ln 
(\rho a^2)|^{-1}}$, to leading order, with a relative error at most 
${\rm O} \left(|\ln (\rho a^2)|^{-1/5}\right)$. Here 
$N$ is the number of particles, $\rho =N/V$ is the particle density and 
$a$ is the scattering length of the two-body potential. We assume that 
the two-body potential is short range and nonnegative. The amusing 
feature of this result is that, in contrast to the three-dimensional 
case, the energy, $E_0$ is not simply $N(N-1)/2$ times the energy of two 
particles in a large box of volume (area, really) $V$. It is much 
larger.