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

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