The answer is crucially dependent on whether the atoms are charged or neutral.

The paper by A. Luther and I. Poeschel, Phys. Rev. B 1974 on 1D correlated electron gas with all possible types of Coulomb interactions gives a good insight . The role of quantum or classical description in this case is not basically relevant. For the sake of the Luttinger or the Thirring model especially electron-electron of the back-scattering type are determining. Similar should be valid also for the charged atomic Fermi gas, for neutral gas the situation is apparently physically different.

Do u mean plamonoic gases if trapped in cigar type trap will show luttinger behavior?

It is a !D model of electron gas on a lattice with short range Coulomb interactions. In phase representation it transforms onto the Thirring model, in a special case it reduces to the Luttinger model.

what are mechanism to make long range qoulomb interaction to become short ranged, known one is screening by othe electrons. dont u think short range need to be defined. I think simplest way to understand is if the Fourier transform of real space interaction is a constant. Only this give rise peaks in pair correlation function at 2kF and continuous momentum distributuion at k_ kF in contrast to fermi liquid.

Could it be seen in trapped 1D quantum gases? so for Luttinger liquid behavior

has been seen only in computer simulation data ,only signature in lab experiments.

The assumption of short ranged CI is a model which is exactly solvable in 1D.

For details, see review by J. Solyom, 1979.

It can be applied to dilute gases with contact interaction.Does model simulate any real system?

The first principles: electron Hamiltonian on a 1Dlattice, with interactions parametrized by constants. The basic property of the model is SU(2) symmetry (electrons moving in the directions +k and -k).The physical origin of the interactions is not so important (it can be applied for electron-phonon interactions), the model is universal if having SU(2) symmetry. It was applied as an efficient model for Peierls instability in, e.g. polyacetylene, as the famous Su-Schrieffer-Heeger model (1979) and other polymers. It has been also a basis for 1D Froehlich superconductivity. (sliding Goldstone mode) experimentally observed in some quasi-one-dimensional structures .