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.