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dc.contributor.authorRottensteiner, Den_US
dc.contributor.authorRuzhansky, Men_US
dc.date.accessioned2020-06-17T09:45:13Z
dc.date.available2020-05-25en_US
dc.identifier.issn0764-4442en_US
dc.identifier.urihttps://qmro.qmul.ac.uk/xmlui/handle/123456789/65045
dc.description.abstractAlthough 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.publisherElsevieren_US
dc.relation.ispartofComptes Rendus de l'Académie des Sciences - Series I - Mathematicsen_US
dc.rightsThis 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.subjectmath.FAen_US
dc.subjectmath.FAen_US
dc.subjectmath.SPen_US
dc.subject35R03, 35P20en_US
dc.titleHarmonic and Anharmonic Oscillators on the Heisenberg Groupen_US
dc.typeArticle
dc.rights.holder© Elsevier 2020
pubs.author-urlhttp://arxiv.org/abs/1812.09620v1en_US
pubs.notesNot knownen_US
pubs.publication-statusAccepteden_US
dcterms.dateAccepted2020-05-25en_US
rioxxterms.funderDefault funderen_US
rioxxterms.identifier.projectDefault projecten_US
rioxxterms.funder.project483cf8e1-88a1-4b8b-aecb-8402672d45f8en_US


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