supersymmetrizing separately the two one-dimensional cartesian Hamiltonians or by setting $W = - a \ln |x| - b \ln |y|$ in equations (\[eq:Calogero-1\]) and (\[eq:Calogero-2\]).

$BC_2$ model ($k=2$)
--------------------

The Hamiltonian of the $BC_2$ model is given by [@olsha83] $$H_2 = - \partial_x^2 - \partial_y^2 + \omega^2 (x^2 + y^2) + 2a(a-1) \left(\frac{1}{(x-y)^2} + 
  \frac{1}{(x+y)^2}\right) + b(b-1) \left(\frac{1}{x^2} + \frac{1}{y^2}\right).$$

For $k=2$, equations (\[eq:Gamma\]), (\[eq:Y\]), (\[eq:super-TTW\]) and (\[eq:V-W\]) easily yield $$\begin{split}
  {\cal H}^s & = - \partial_x^2 - \partial_y^2 + \omega^2 (x^2 + y^2) + 2a^2 \left(\frac{1}{(x-y)^2} + 
       \frac{1}{(x+y)^2}\right) + b^2 \left(\frac{1}{x^2} + \frac{1}{y^2}\right) \\
  & \quad + 2\omega ({b^{\dagger}}_x b_x + {b^{\dagger}}_y b_y)  + a \left(\frac{1}{(x-y)^2} [{b^{\dagger}}_x - {b^{\dagger}}_y, b_x - b_y] + 
       \frac{1}{(x+y)^2} [{b^{\dagger}}_x + {b^{\dagger}}_y, b_x + b_y]\right) \\
  & \quad + b \left(\frac{1}{x^2} [{b^{\dagger}}_x, b_x] + \frac{1}{y^2} [{b^{\dagger}}_y, b_y]\right) - 2\omega (2a+2b+1), \\
  Q & = \left[- \partial_x + \omega x - a \left(\frac{1}{x-y} + \frac{1}{x+y}\right) - \frac{b}{x}\right] b_x \\
  & \quad + \left[- \partial_y + \omega y + a \left(\frac{1}{x-y} - \frac{1}{x+y}\right) - \frac{b}{y}\right] b_y,  
\end{split}$$ which would also result from section 2 by using $W = - a (\ln |x-y] + \ln |x+y|) - b (\ln |x| + \ln |y|)$, in agreement with [@brink98].

