$$\begin{split}
  \Gamma & = \frac{k}{2\omega r^2} \bigl\{a \bigl[{\bar{b}^{\dagger}}_x {\bar{b}}_x - \tan k\varphi \bigl({\bar{b}^{\dagger}}_x {\bar{b}}_y 
       + {\bar{b}^{\dagger}}_y {\bar{b}}_x\bigr) + (k \sec^2 k\varphi - 1) {\bar{b}^{\dagger}}_y {\bar{b}}_y\bigr] \\
  & \quad + b \bigl[{\bar{b}^{\dagger}}_x {\bar{b}}_x + \cot k\varphi \bigl({\bar{b}^{\dagger}}_x {\bar{b}}_y + {\bar{b}^{\dagger}}_y {\bar{b}}_x\bigr)
       + (k \csc^2 k\varphi - 1) {\bar{b}^{\dagger}}_y {\bar{b}}_y\bigr]\bigr\},
\end{split}$$ $$Y = \frac{1}{2} \bigl[{\bar{b}^{\dagger}}_x {\bar{b}}_x + {\bar{b}^{\dagger}}_y {\bar{b}}_y - k(a+b) - 1\bigr]$$ and $$\begin{split}
  V_{\pm} & = \frac{1}{2\sqrt{\omega}} \biggl[{\bar{b}^{\dagger}}_x \biggl(\mp \partial_r + \omega r \pm \frac{k(a+b)}{r}
         \biggr) \mp {\bar{b}^{\dagger}}_y \frac{1}{r} (\partial_{\varphi} + ka \tan k\varphi - kb \cot k\varphi)\biggr], \\
  W_{\pm} & = \frac{1}{2\sqrt{\omega}} \biggl[{\bar{b}}_x \biggl(\mp \partial_r + \omega r \mp \frac{k(a+b)}{r}
         \biggr) \mp {\bar{b}}_y \frac{1}{r} (\partial_{\varphi} - ka \tan k\varphi + kb \cot k\varphi)\biggr].
\end{split}  \label{eq:V-W-bis}$$

In the next section, we will proceed to determine the action of the ${\rm osp}(2/2, {{\mathbb{R}}})$ generators on the TTW Hamiltonian eigenstates after extending the latter with fermionic degrees of freedom.

Irreducible representations of ${\rm osp}(2/2, {{\mathbb{R}}})$ for the supersymmetrized TTW Hamiltonians
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In [@tremblay09], it has been shown that $H_k$ is exactly solvable and satisfies the eigenvalue equation $$H_k \Psi_{N,n}(r, \varphi) = E_{N,n} \Psi_{N,n}(r, \varphi), \qquad E_{N,n} = 2\omega [2N + (2n+a+b)k + 1],$$ where $N$, $n=0$, 1, 2, …. The wavefunctions can 