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

TL;DR

This paper develops a functional calculus for the harmonic oscillator on noncommutative Moyal-Groenewold planes, connecting Lie group theory with operator analysis to advance understanding of quantum phase spaces.

Contribution

It introduces twisted Weyl tuples and a twisted transference principle, linking noncommutative harmonic analysis with Lie group representations and operator semigroups.

Findings

Harmonic oscillator admits bounded H-infinity functional calculus on noncommutative spaces

Establishes a connection between noncommutative ergodic theory and R-boundedness

Provides new insights into magnetic Weyl calculus and maximal regularity

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.