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Standard analytical construction of the many-body wave function ofinteracting particles in one dimension, beyond mean-field theory, is based onthe Jastrow approach. The many-body interacting ground state is build up fromthe ground state of the non-interacting system and the product of solutions ofthe corresponding interacting two-body problem. However, this is possible onlyif the center-of-mass motion is decoupled from the mutual interactions. In ourwork, based on the general constraints given by contact nature of the atom-atominteractions, we present an alternative approach to the standard constructionof the \textit{pair-correlation} wave-function. Within the proposed ansatz, westudy the many-body properties of trapped bosons as well as fermionic mixturesand we compare these predictions with the exact diagonalization approach in awide range of particle numbers, interaction strengths, and different trappingpotentials.

Variational approach for interacting ultra-cold atoms in arbitrary one-dimensional confinement