CMW model ($k=3$)
-----------------

In contrast with the two previous cases, the comparison of the results obtained in section 3 for $k=3$ with those of the supersymmetrized CMW model is more involved because the latter is a three-particle system, which can only be interpreted as a planar problem after eliminating the centre-of-mass motion. On starting from [@wolfes] $$H_{\rm CMW} = \sum_i (- \partial_i^2 + \omega^2 x_i^2) + a(a-1) \sum_{\substack{i,j \\ i\ne j}} 
  \frac{1}{x_{ij}^2} + 3b(b-1) \sum_{\substack{i,j \\ i\ne j}} \frac{1}{y_{ij}^2},$$ where $x_{ij} = x_i - x_j$ ($i \ne j$), $y_{ij} = x_i + x_j - 2x_k$ ($i \ne j \ne k \ne i$) and all indices run over 1, 2, 3, we indeed get $$H_{\rm CMW} = H_{\rm rel} + H_{\rm cm}, \qquad H_{\rm rel} = H_3, \qquad H_{\rm cm} = -\partial_X^2
  + \omega^2 X^2,$$ by setting $r \cos \varphi = x_{12}/\sqrt{2}$, $r \sin \varphi = y_{12}/\sqrt{6}$ and $X = (x_1 + x_2 + x_3)/\sqrt{3}$.

The supersymmetrized CMW Hamiltonian and the corresponding supercharge are obtained by inserting $$W = - \frac{a}{2} \sum_{\substack{i,j \\ i\ne j}} \ln |x_{ij}| - \frac{b}{2} \sum_{\substack{i,j \\ i\ne j}} 
  \ln |y_{ij}|$$ in (\[eq:Calogero-1\]) and (\[eq:Calogero-2\]) and they are given by $$\begin{split}
  {\cal H}^s_{\rm CMW} & = 