We study a non-commutative non-relativistic scalar field theory in 2+1 dime
nsions. The theory shows the UV-IR mixing typical of QFT on non-commutative
spaces. The one-loop correction to the two-point function turns out to be
given by a delta function in momentum space. The one-loop correction to the
four-point function is of logarithmic type. We also evaluate the thermodyn
amic potential at two-loop order. To avoid an IR singularity we have to int
roduce a chemical potential. The suppression of the non-planar contribution
with respect to the planar one turns out to depend crucially on the value
of the chemical potential.