The Sub- Laplacian on The Heisenberg Group and Itʼs Spectral Properties

Abstract

In this paper we talk about the spectral theory of the sub-Laplacian on the Heisenberg group. Then we give a complete analysis of the spectrum of the unique self- adjoint extension of this sub-Laplacian on the one-dimensional Heisenberg group. The Heisenberg group is the most known example from the realm of nilpotent Lie groups and plays an important role in several branches of mathematics, such as representation theory, partial differential equations and number theory... It also offers the greatest opportunity for generalizing the remarkable results of Euclidean harmonic analysis. The results in this paper are valid for the sub-Laplacian on the n-dimensional Heisenberg group, in which the underlying space is, but we have chosen to present the results for the one-dimensional Heisenberg group ℍ for the sake of simplicity and transparency.