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Pré-Publication, Document De Travail Année : 2024

The harmonic oscillator on the Moyal-Groenewold plane: an approach via Lie groups and twisted Weyl tuples

Résumé

This paper investigates the functional calculus of the harmonic oscillator on each Moyal-Groenewold plane, the noncommutative phase space which is a fundamental object in quantum mechanics. Specifically, we show that the harmonic oscillator admits a bounded $H^\infty(\Sigma_\omega)$ functional calculus for any angle $0 < \omega < \frac{\pi}{2}$ and even a bounded H\"ormander functional calculus on the associated noncommutative $L^p$-spaces, where $\Sigma_\omega = \{z \in \mathbb{C}^* :\: |\arg z|< \omega \}$. To achieve these results, we develop a connection with the theory of 2-step nilpotent Lie groups by introducing a notion of twisted Weyl tuple and connecting it to some semigroups of operators previously investigated by Robinson via group representations. Along the way, we demonstrate that $L^p$-square-max decompositions lead to new insights between noncommutative ergodic theory and $R$-boundedness, and we prove a twisted transference principle, which is of independent interest. Our approach accommodates the presence of a constant magnetic field and they are indeed new even in the framework of magnetic Weyl calculus on classical $L^p$-spaces. Our results contribute to the understanding of functional calculi on noncommutative spaces and have implications for the maximal regularity of the most basic evolution equations associated to the harmonic oscillator.
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Dates et versions

hal-04681886 , version 1 (30-08-2024)

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Cédric Arhancet, Lukas Hagedorn, Christoph Kriegler, Pierre Portal. The harmonic oscillator on the Moyal-Groenewold plane: an approach via Lie groups and twisted Weyl tuples. 2024. ⟨hal-04681886⟩
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