Wave functions of the super-Tonks-Girardeau gas and the trapped one-dimensional hard-sphere Bose gas

Recent theoretical and experimental results demonstrate a close connection between the super-Tonks-Girardeau (STG) gas and a one-dimensional (1D) hard-sphere Bose (HSB) gas with hard-sphere diameter nearly equal to the 1D scattering length a1D of the STG gas, a highly excited gaslike state with nodes only at interparticle separations |xjl|=xnode≅a1D. It is shown herein that when the coupling constant gB in the Lieb-Liniger interaction gBδ(xjl) is negative and |x12|≥xnode, the STG and HSB wave functions for N=2 particles are not merely similar, but identical; the only difference between the STG and HSB wave functions is that the STG wave function allows a small penetration into the region |x12|2. The STG and HSB wave functions for N=2 are given exactly in terms of a parabolic cylinder function, and for N≥2, xnode is given accurately by a simple parabola. The metastability of the STG phase generated by a sudden change of the coupling constant from large positive to large negative values is explained in terms of the very small overlap between the ground state of the Tonks-Girardeau gas and collapsed cluster states.