Dilute Fermi gas in quasi-one-dimensional traps: From weakly interacting fermions via hard core bosons to weakly interacting Bose gas

We study equilibrium properties of a cold two-component Fermi gas confined in a quasi-one-dimensional trap of the transverse size $l_{\perp }$. In the dilute limit ($n{l}_{\perp }\ll 1$, where $n$ is the 1D density) the problem is exactly solvable for an arbitrary 3D fermionic scattering length $a_F$. When $l_{\perp }\mathrm{/}{a}_F$ goes from $-\mathrm{\infty }$ to $+\mathrm{\infty }$, the system successively passes three regimes: weakly interacting Fermi gas, hard core Bose gas and weakly coupled Bose gas. The regimes are separated by two crossovers at $a_F\sim \pm n{l}_{\perp }^2$. In conclusion we discuss experimental implications of these results.