In Figs. 3.3-3.4 we present results for , corresponding to a less tight transverse confinement or, equivalently, to a smaller scattering length. By decreasing we enter more deeply in the universal regime where the theory of pseudo-potentials applies and we find no difference between HS and SS results. Further, as for the case, we find no difference between DMC results obtained starting from Hamiltonian (3.1) and from the 1D Hamiltonian (3.8). Compared to Figs. 3.1-3.2, the cross-over between mean field and Lieb-Liniger occurs for smaller values of . For the smallest values of beyond mean-field effects become evident, though one would need to decrease even further to enter the Tonks-Girardeau gas regime.
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