Scattering on the resonance state of a Bose gas

In the case of the Hamiltonian (4.9) we use Gaussian construction for the one-body Bijl-Jastrow term

$$f_1(z) = \exp\left\{-\frac{z^2}{2\alpha_z^2}\right\}$$ (2.64)

where the Gaussian width $\alpha _z$ is treated as a variational parameter. The two-body correlation term $f_2(z)$ (2.37) is chosen as

$$f_2(z) = \left\{ \begin{array}{cl} \cos[k_z(\vert z\vert-\ba... ...le \bar{Z} \\ 1, &\vert z\vert>\bar{Z} \;. \end{array}\right.$$ (2.65)

The cut-off length $\bar{Z}$ is fixed at $\bar{Z}=500 a_{1D}$, while the wave vector $k_z$ is chosen such that the boundary condition at $z=0$ imposed by the $\delta$-function potential is satisfied: $-k_z\mathop{\rm tg}\nolimits (k_z \bar{Z})=1/a_{1D}$. For negative $a_{1D}$ ($g_{1D}>0$) the correlation function, Eq. 2.65, is positive everywhere. For positive $a_{1D}$ ($g_{1D}<0$), in contrast, $f_2(z)$ changes sign at $\vert z\vert=a_{1D}$. The parameterization given by Eq. 2.65 is used in our stability analysis performed within a VMC framework (see Sec. 4.5) and in our DMC calculations for $g_{1D}>0$. To perform the FN-DMC calculations for negative $g_{1D}$, we need to construct a trial wave function that is positive definite everywhere. In the FN-DMC calculations, we thus use an alternative parameterization, which imposes the constraint $f_2=0$ for $a_{1D}\le z$,

$$f_2(z)= \left\{ \begin{array}{cl} 0,& z \le a_{1D} \\ \cos[... ...,& a_{1D} < z \le \bar{Z} \\ 1,& z>\bar{Z} \end{array}\right.$$ (2.66)