The Harmonic Oscillator on the Heisenberg Group

TL;DR

This paper defines a harmonic oscillator on the Heisenberg group, extending the classical concept to a non-commutative setting, and provides explicit formulas and eigenvalue estimates for this operator.

Contribution

It introduces a natural analogue of the harmonic oscillator on the Heisenberg group, based on the sub-Laplacian and Lie algebra structure, with explicit operator expression and spectral analysis.

Findings

Explicit differential operator on $ extbf{H}_n$ derived

Eigenvalue asymptotics established

Operator has a discrete spectrum with smooth eigenfunctions

Abstract

In this note we present a notion of harmonic oscillator on the Heisenberg group $\mathbf{H}_n$ which forms the natural analogue of the harmonic oscillator on $\mathbb{R}^n$ under a few reasonable assumptions: the harmonic oscillator on $\mathbf{H}_n$ should be a negative sum of squares of operators related to the sub-Laplacian on $\mathbf{H}_n$, essentially self-adjoint with purely discrete spectrum, and its eigenvectors should be smooth functions and form an orthonormal basis of $L^2(\mathbf{H}_n)$. This approach leads to a differential operator on $\mathbf{H}_n$ which is determined by the (stratified) Dynin-Folland Lie algebra. We provide an explicit expression for the operator as well as an asymptotic estimate for its eigenvalues.