The ground state energy per particle of a dilute, homogeneous,
two-dimensional Bose gas, in the thermodynamic limit is shown rigorously to be
E0β/N=(2Οβ2Ο/m)β£ln(Οa2)β£β1, to leading order, with
a relative error at most O(β£ln(Οa2)β£β1/5). Here N is the
number of particles, Ο=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, E0β is not
simply N(Nβ1)/2 times the energy of two particles in a large box of volume
(area, really) V. It is much larger