  We use a BCS-type variational wavefunction to study attractively-interacting
quasi one-dimensional (1D) fermionic atomic gases, motivated by cold-atom
experiments that access the 1D regime using an anisotropic harmonic trapping
potential (with trapping frequencies $\omega_x = \omega_y \gg \omega_z$) that
confines the gas to a cigar-shaped geometry. To handle the presence of the trap
along the $z$-direction, we construct our variational wavefunction from the
harmonic oscillator Hermite functions that are the eigenstates of the
single-particle problem. Using an analytic determination of the effective
interaction among harmonic oscillator states along with a numerical solution of
the resulting variational equations, we make specific experimental predictions
for how pairing correlations would be revealed in experimental probes like the
local density and the momentum correlation function.
