  We study a noncommutative nonrelativistic theory in 2+1 dimensions of a
scalar field coupled to the Chern-Simons field. In the commutative situation
this model has been used to simulate the Aharonov-Bohm effect in the field
theory context. We verified that, contrarily to the commutative result, the
inclusion of a quartic self-interaction of the scalar field is not necessary to
secure the ultraviolet renormalizability of the model. However, to obtain a
smooth commutative limit the presence of a quartic gauge invariant
self-interaction is required. For small noncommutativity we fix the corrections
to the Aharonov-Bohm scattering and prove that up to one-loop the model is free
from dangerous infrared/ultraviolet divergences.
