Harmonic and Anharmonic Oscillators on the Heisenberg Group
David Rottensteiner, Michael Ruzhansky
Abstract
Although there is no canonical version of the harmonic oscillator on the Heisenberg group $\mathbf{H}_n$ so far, we make a strong case for a particular choice of operator by using the representation theory of the Dynin-Folland group $\mathbf{H}_{n, 2}$, a $3$-step stratified Lie group, whose generic representations act on $L^2(\mathbf{H}_n)$. Our approach is inspired by the connection between the harmonic oscillator on $\mathbb{R}^n$ and the sum of squares in the first stratum of $\mathbf{H}_n$ in the sense that we define the harmonic oscillator on $\mathbf{H}_n$ as the image of the sub-Laplacian $\mathcal{L}_{\mathbf{H}_{n, 2}}$ under the generic unitary irreducible representation $π$ of the Dynin-Folland group which has formal dimension $d_π= 1$. This approach, more generally, permits us to define a large class of so-called anharmonic oscillators by employing positive Rockland operators on $\mathbf{H}_{n, 2}$. By using the methods developed in ter Elst and Robinson [tERo], we obtain spectral estimates for the harmonic and anharmonic oscillators on $\mathbf{H}_n$. Moreover, we show that our approach extends to graded $SI/Z$-groups of central dimension $1$, i.e., graded groups which possess unitary irreducible representations which are square-integrable modulo the $1$-dimensional center $Z(G)$. The latter part of the article is concerned with spectral multipliers. By combining ter Elst and Robinson's techniques with recent results in [AkRu18], we obtain useful $L^\mathbf{p}$-$L^\mathbf{q}$-estimates for spectral multipliers of the sub-Laplacian $\mathcal{L}_{\mathbf{H}_{n, 2}}$ and, in fact more generally, of general Rockland operators on general graded groups. As a by-product, we recover the Sobolev embeddings on graded groups established in [FiRu17], and obtain explicit hypoelliptic heat semigroup estimates.