  We consider a two-component gas of fermionic atoms confined to a
quasi-two-dimensional (quasi-2D) geometry by a harmonic trapping potential in
the transverse direction. We construct a mean field theory of the BCS-BEC
crossover at zero temperature that allows us to extrapolate to an infinite
number of transverse harmonic oscillator levels. In the extreme BEC limit,
where the confinement length exceeds the dimer size, we recover 3D dimers
confined to 2D with weak repulsive interactions. However, even when the
interactions are weak and the Fermi energy is less than the confinement
frequency, we find that the higher transverse levels can substantially modify
fermion pairing. We argue that recent experiments on pairing in quasi-2D Fermi
gases [Y. Zhang et al., Phys. Rev. Lett. 108, 235302 (2012)] have already
observed the effects of higher transverse levels.
