dc.contributor.author Rottensteiner, D en_US dc.contributor.author Ruzhansky, M en_US dc.date.accessioned 2020-06-17T09:45:13Z dc.date.available 2020-05-25 en_US dc.identifier.issn 0764-4442 en_US dc.identifier.uri https://qmro.qmul.ac.uk/xmlui/handle/123456789/65045 dc.description.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 $\pi$ of the Dynin-Folland group which has formal dimension $d_\pi = 1$. 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. en_US dc.publisher Elsevier en_US dc.relation.ispartof Comptes Rendus de l'Académie des Sciences - Series I - Mathematics en_US dc.rights This is a pre-copyedited, author-produced version of an article accepted for publication in Comptes Rendus de l'Académie des Sciences - Series I – Mathematics following peer review. dc.subject math.FA en_US dc.subject math.FA en_US dc.subject math.SP en_US dc.subject 35R03, 35P20 en_US dc.title Harmonic and Anharmonic Oscillators on the Heisenberg Group en_US dc.type Article dc.rights.holder © Elsevier 2020 pubs.author-url http://arxiv.org/abs/1812.09620v1 en_US pubs.notes Not known en_US pubs.publication-status Accepted en_US dcterms.dateAccepted 2020-05-25 en_US rioxxterms.funder Default funder en_US rioxxterms.identifier.project Default project en_US rioxxterms.funder.project 483cf8e1-88a1-4b8b-aecb-8402672d45f8 en_US
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