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

Abstract

We consider the magnetic Laplacian ${\Delta }_{\nu ,\mu }$ on ${\mathbb{R}}^{2n}={\mathbb{C}}^n$ given by $${\Delta }_{\nu ,\mu }=4\sum _{j=1}^n\frac{{\partial }^2}{\partial z_j\partial \overline{z_j}}+2i\nu (E+\overline{E}+n)+2\mu (E-\overline{E})-({\nu }^2+{\mu }^2)|z{|}^2.$$ 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.

14 pages