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The harmonic oscillator on the Moyal-Groenewold plane: an approach via Lie groups and twisted Weyl tuples (2312.06143v8)

Published 11 Dec 2023 in math.FA, hep-th, math-ph, math.GR, math.MP, and math.OA

Abstract: 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 $\mathrm{H}\infty(\Sigma_\omega)$ functional calculus for any angle $0 < \omega < \frac{\pi}{2}$ and even a bounded H\"ormander functional calculus on the associated noncommutative $\mathrm{L}p$-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 $\mathrm{L}p$-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 $\mathrm{L}p$-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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