The Harmonic Oscillator on the Heisenberg Group

In this note we present a notion of harmonic oscillator on the Heisenberg group H-n which forms the natural analogue of the harmonic oscillator on R-n under a few reasonable assumptions: the harmonic oscillator on H-n should be a negative sum of squares of operators related to the sub-Laplacian on H-n, essentially self-adjoint with purely discrete spectrum, and its eigenvectors should be smooth functions and form an orthonormal basis of L-2(H-n). This approach leads to a differential operator on 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.