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

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Recent theoretical and experimental results demonstrate a close connection between the super Tonks-Girardeau (sTG) gas and a 1D hard sphere Bose (HSB) gas with hard sphere diameter nearly equal to the 1D scattering length $a_{1D}$ of the sTG gas, a highly excited gas-like state with nodes only at interparticle separations $|x_{j\ell}|=x_{node}\approx a_{1D}$. It is shown herein that when the coupling constant $g_B$ in the Lieb-Liniger interaction $g_B\delta(x_{j\ell})$ is negative and $|x_{12}|\ge x_{node}$, 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 $|x_{12}|<x_{node}$, whereas for a HSB gas with hard sphere diameter $a_{h.s.}=x_{node}$, the HSB wave function vanishes when all $|x_{12}|<a_{h.s.}$. Arguments are given suggesting that the same theorem holds also for $N>2$. The sTG and HSB wave functions for N=2 are given exactly in terms of a parabolic cylinder function, and for $N\ge 2$, $x_{node}$ 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.