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

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^∞(Σ_ω) functional calculus for any angle 0 < ω < π/2 and even a bounded Hörmander functional calculus on the associated noncommutative L^p-spaces, where Σ_ω={ z ∈ℂ^*: | z| <ω}. 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 accommodate 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.