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 a(1D) of the STG gas, a highly excited gaslike state with nodes only at interparticle separations vertical bar x(jl)vertical bar = x(node) approximate to a(1D). It is shown herein that when the coupling constant g(B) in the Lieb-Liniger interaction g(B)delta(x(jl)) is negative and vertical bar x(12)vertical bar >= 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 vertical bar x(12)vertical bar < x(node), whereas for a HSB gas with hard-sphere diameter a(hs) = x(node), the HSB wave function vanishes when vertical bar x(12)vertical bar < a(hs). 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 >= 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.