The Heisenberg Oscillator

In this short note, we determine the spectrum of the Heisenberg oscillator which is the operator defined as $L+\mathrm{\mid }z{\mathrm{\mid }}^2$ on the 3-dimensional Heisenberg group $H_1=\mathbb{C}\times \mathbb{R}$ where $L$ stands for the positive sublaplacian and $z$ for the complex variable of a point of the group.