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We investigate the Bose gas with repulsive or attractive interactions betweenatoms in the scheme of Bethe Ansatz equation in a hard wall trap. Three typicalquantum phases in the current research of 1D interacting cold atoms areclarified in terms of the energy spectrum, single particle density distributionand two-particle correlation function. We identify two matching points in thephase diagram, i.e. the TG and STG gas show the same profiles at the stronglyinteracting point $-1/\gamma=0$, and in the weakly interacting limit $\gamma=0$the ground states TG and BS join to each other smoothly.

Tonks-Girardeau gas, super-Tonks-Girardeau gas, and bound states of one-dimensional bosons in a hard-wall trap