  In this article we discuss the accuracy of effective one-dimensional theories
used to describe the behavior of ultracold atomic ensembles confined in quantum
wires by a harmonic trap. We derive within a fully many-body approach the
effective Hamiltonian describing this class of systems and we calculate the
beyond-mean field corrections to the energy of the ground state arising from
virtual transitions towards excited state of the confining potential. We find
that, due to the Pauli principle, effective finite-range corrections are one of
magnitude larger than effective three-body interactions.By comparing to exact
solutions of the purely 1D problem, we conclude that a 1D effective theory
provides a good description of the ground state of the system for a rather
large range of interaction parameters.
