The Harmonic Oscillator on the Heisenberg Group

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.