  Strongly confining an ultracold atomic gas in one direction to create a
quasi-2D system alters the scattering properties of this gas. We investigate
the effects of confinement on Feshbach scattering resonances and show that
strong confinement results in a shift in the position of the Feshbach resonance
as a function of the magnetic field. This shift, as well as the change of the
width of the resonance, are computed. We find that the resonance is strongly
damped in the thermal gas, but in the condensate the resonance remains sharp
due to many-body effects. We introduce a 2D model system, suited for the study
of resonant superfluidity, and having the same scattering properties as the
tightly confined real system near a Feshbach resonance. Exact relations are
derived between measurable quantities and the model parameters.
