Harmonic and anharmonic oscillators on the Heisenberg group

Abstract

In this article, we present a notion of the harmonic oscillator on the Heisenberg group Hn, which, under a few reasonable assumptions, forms the natural analog of a harmonic oscillator on Rn: a negative sum of squares of operators on Hn, which is essentially self-adjoint on L2(Hn) with purely discrete spectrum and whose eigenvectors are Schwartz functions forming an orthonormal basis of L2(Hn). The differential operator in question is determined by the Dynin–Folland group—a stratified nilpotent Lie group—and its generic unitary irreducible representations, which naturally act on L2(Hn). As in the Euclidean case, our notion of harmonic oscillator on Hn extends to a whole class of so-called anharmonic oscillators, which involve left-invariant derivatives and polynomial potentials of order greater or equal 2. These operators, which enjoy similar properties as the harmonic oscillator, are in one-to-one correspondence with positive Rockland operators on the Dynin–Folland group. The latter part of this article is concerned with spectral multipliers. We obtain useful Lp-Lq-estimates for a large class of spectral multipliers of the sub-Laplacian LHn,2 and, in fact, of generic Rockland operators on graded groups. As a by-product, we obtain explicit hypoelliptic heat semigroup estimates and recover the continuous Sobolev embeddings on graded groups, provided 1 < p ≤ 2 ≤ q < ∞.