Bijl-Jastrow term

Now let us specify the Bijl-Jastrow term which will take care of the interactions between spin up and spin down particles. We consider an attractive interaction potential which supports a bound state. Thus we can describe resonant scattering with very large scattering lengths $a_{3D}$. It also means that for the unit of length it is preferable to take instead the range of potential $R$ instead of $a_{3D}$ which can be even diverging (the unitary regime).

We consider scattering on the square well (SW) potential (1.89). The scattering problem was studied in Sec. 1.3.5.1. Here we only summarize the construction of the Bijl-Jastrow term:

1)

the equation for the scattering momentum is

$$\frac{1}{k}\left[\mathop{\rm arctg}\nolimits \frac{kL}{2}-\mathop... ...mathop{\rm tg}\nolimits \sqrt{\varkappa ^2+k^2} R\right)\right] = \frac{L}{2}-R$$ (2.111)

2)

the shift phase $\delta$ is defined as

$$\delta = \mathop{\rm arctg}\nolimits \left(\frac{kL}{2}\right)-\frac{kL}{2}$$ (2.112)

3)

normalization factor $B$

$$B = \frac{L/2}{\sin(kL/2+\delta)}$$ (2.113)

4)

normalization factor $A$

$$A = B\frac{\sin(k R+\delta)}{\sin(\sqrt{\varkappa ^2+k^2} R)}$$ (2.114)

The Bijl-Jastrow contribution to the force and the local energy are given by following expressions:

$$\frac{f'(r)}{f(r)} =\left\{ {\begin{array}{ll} {\cal K}\mathop{\r... ...\mathop{\rm ctg}\nolimits (kr+\delta)-\frac{1}{r}, &r\ge R \end{array}} \right.$$ (2.115)

$$E_{loc}^{3D} =\left\{ {\begin{array}{ll} {\cal K}^2-({\cal K}\mat... ...thop{\rm ctg}\nolimits (kr+\delta)-\frac{1}{r})^2, &r\ge R \end{array}} \right.$$ (2.116)