On concrete spectral properties of a twisted-Laplacian associated to a central extension of the real Heisenberg group

We consider the magnetic Laplacian $\Delta_{\nu,\mu}$ on $\mathbb{R}^{2n}=\mathbb{C}^n$ given by $$ \Delta_{\nu,\mu}= 4\sum\limits_{j=1}\limits^{n}\frac{\partial^2 }{\partial z_j \partial \overline{z_j}} +2i\nu (E+ \overline{E} +n) +2\mu (E- \overline{E} ) -(\nu^2+\mu^2)|z|^2. $$ We show that $\Delta_{\nu,\mu}$ is connected to the sub-Laplacian of a group of Heisenberg type given by $\mathbb{C}\times_\omega \mathbb{C}^n$ realized as a central extension of the real Heisenberg group $H_{2n+1}$. We also discuss invariance properties of $\Delta_{\nu,\mu}$ and give some of their explicit spectral properties.