\qquad K_+ = \frac{1}{2} \sum_i 
        a^{\dagger2}_i, \qquad K_- = \frac{1}{2} \sum_i a_i^2,  \\
  & Y = \frac{1}{2} \biggl(\sum_i {b^{\dagger}}_i b_i - \frac{\nu}{2}\biggr), \qquad V_+ = \frac{1}{\sqrt{2}} 
        \sum_i {a^{\dagger}}_i {b^{\dagger}}_i, \qquad V_- = \frac{1}{\sqrt{2}} \sum_i a_i {b^{\dagger}}_i,  \\ 
  & W_+ = \frac{1}{\sqrt{2}} \sum_i {a^{\dagger}}_i b_i, \qquad W_- = \frac{1}{\sqrt{2}} \sum_i a_i b_i, 
\end{split}  \label{eq:oscillator}$$ where all summations run over 1, 2, …, $\nu$. It is related to the superoscillator (see, e.g., [@bagchi]), for which $${\cal H}^s = 4 \omega (K_0 + Y), \qquad Q = 2 \sqrt{\omega}\, W_+, \qquad Q^{\dagger} = 2 \sqrt{\omega}\,
  V_-  \label{eq:SUSY}$$ provide a realization of the ${\rm sl}(1/1)$ superalgebra of standard supersymmetric quantum mechanics, $$[{\cal H}^s, Q] = [{\cal H}^s, Q^{\dagger}] = 0, \qquad \{Q, Q^{\dagger}\} = {\cal H}^s,$$ and the operators (\[eq:oscillator\]) generate a dynamical superalgebra.

The boson-fermion realization (\[eq:oscillator\]) can be generalized by including an additional contribution in the bosonic operators ${a^{\dagger}}_i = (- \partial_i + \omega x_i)/\sqrt{2\omega}$, $a_i = (\partial_i + \omega x_i)/\sqrt{2\omega}$ (with $\partial_i \equiv \partial/\partial x_i$) appearing in the odd generators $V_{\pm}$, $W_{\pm}$. The latter become $$V_{\pm} = \frac{1}{2\sqrt{\omega}} \sum_i (\mp \partial_i + \omega x_i \mp \partial_i W) {b^{\dagger}}_i, \qquad
  W_{\pm} = \frac{1}{2\sqrt{\omega}} \sum_i (\mp \partial_i + \omega x_i \pm \partial_i W) 