The Heisenberg Oscillator

TL;DR

This paper determines the spectrum of the Heisenberg oscillator, an operator combining the sublaplacian and a quadratic term on the 3D Heisenberg group, providing insights into its spectral properties.

Contribution

It explicitly computes the spectrum of the Heisenberg oscillator operator on the 3D Heisenberg group, a novel spectral analysis in this context.

Findings

Spectrum of the Heisenberg oscillator explicitly determined

Provides spectral properties of the operator on the Heisenberg group

Enhances understanding of harmonic analysis on the Heisenberg group

Abstract

In this short note, we determine the spectrum of the Heisenberg oscillator which is the operator defined as $L + |z|^2$ 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.