Super-Tonks trial wave function (attractive $\delta$-potential)

In this section we will describe the construction of the trial wave function $\psi _T$ used fir the investigation of the super-Tonks system (see Chapter 6).

We use Bijl-Jastrow construction (2.37) with the two-body term which is chosen to be similar to (2.51)

$$f_2(z)=\left\{ \begin{array}{cr} \cos[k(\vert z\vert-R_{m})]... ...z\vert \le R_{m}\\ 1, & \vert z\vert>R_{m} \end{array}\right.$$ (2.63)

The cut-off length $R_{m}$ is a variational parameter, while the wave vector $\k$ (for a given $R_{m}$) is chosen in a such way that the boundary condition imposed by the $\delta$-function potential (2.52) at $z=0$ is satisfied: $-k\mathop{\rm tg}\nolimits (kR_{m})=1/a_{1D}$. For distances smaller than the cut-off length, $\vert z\vert\le R_{m}$, the above wave function corresponds to the exact solution with positive energy of the two-body problem with the interaction potential $g_{1D}\delta(z)$ (see, formula 1.65).