Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Cédric Arhancet, Lukas Hagedorn, Christoph Kriegler, Pierre Portal

TL;DR

The paper develops a comprehensive framework for the functional calculus of the harmonic oscillator on noncommutative Moyal-Groenewold planes via Theta-Weyl tuples, twisted Weyl calculus, and a twisted transference principle. It connects noncommutative $L^p$-spaces to 2-step nilpotent Lie groups (via the Mackey group and universal covering) and provides bounded $\mathrm{H}^\infty(\Sigma_\omega)$ calculus for shifted sum-of-squares operators, along with a bounded Hörmander calculus, on Bochner spaces with Schatten targets. Central to the approach are $R$-boundedness, square-max decompositions, and a transference strategy that reduces noncommutative estimates to a universal model, enabling maximal regularity results for evolution equations driven by the harmonic oscillator. The results illuminate spectral multiplier behavior in noncommutative geometry, establish maximal $L^p$-regularity in this setting, and offer new transference tools and kernel estimates with potential implications for quantum field theory and noncommutative harmonic analysis.

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 $\mathrm{H}^\infty(Σ_ω)$ functional calculus for any angle $0 < ω< \fracπ{2}$ and even a bounded Hörmander functional calculus on the associated noncommutative $\mathrm{L}^p$-spaces, where $Σ_ω=\{ z \in \mathbb{C}^*: |\arg z| <ω\}$. 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 $\mathrm{L}^p$-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.

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

TL;DR

The paper develops a comprehensive framework for the functional calculus of the harmonic oscillator on noncommutative Moyal-Groenewold planes via Theta-Weyl tuples, twisted Weyl calculus, and a twisted transference principle. It connects noncommutative -spaces to 2-step nilpotent Lie groups (via the Mackey group and universal covering) and provides bounded calculus for shifted sum-of-squares operators, along with a bounded Hörmander calculus, on Bochner spaces with Schatten targets. Central to the approach are -boundedness, square-max decompositions, and a transference strategy that reduces noncommutative estimates to a universal model, enabling maximal regularity results for evolution equations driven by the harmonic oscillator. The results illuminate spectral multiplier behavior in noncommutative geometry, establish maximal -regularity in this setting, and offer new transference tools and kernel estimates with potential implications for quantum field theory and noncommutative harmonic analysis.

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 functional calculus for any angle and even a bounded Hörmander functional calculus on the associated noncommutative -spaces, where . 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 -square-max decompositions lead to new insights between noncommutative ergodic theory and -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 -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.
Paper Structure (53 sections, 66 theorems, 459 equations, 2 figures)

This paper contains 53 sections, 66 theorems, 459 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Introduction
  3. Hörmander functional calculus
  4. The harmonic oscillator on the Moyal-Groenewold plane $\mathbb{R}^d_\Theta$
  5. Application to maximal regularity of evolution equations
  6. Approach and structure of the paper
  7. Preliminaries
  8. Gaussian integrals
  9. Bochner spaces
  10. Bounded holomorphic semigroups
  11. Functional calculus of sectorial operators
  12. Commutation relations
  13. 2-cocycles
  14. Twisted group von Neumann algebras
  15. A central extension
  16. ...and 38 more sections

Key Result

Theorem 1.1.1

Suppose that $f \in \mathrm{L}^\infty(\mathbb{R}^d)$. Assume also that for every distributional derivative $\partial_\alpha$ where $\alpha = (\alpha_1,\ldots,\alpha_d) \in \{0,1\}^d$, we have If $1 < p < \infty$ then the operator $M_f \colon \mathcal{S}(\mathbb{R}^d) \to \mathcal{S}'(\mathbb{R}^d)$, $g\mapsto \mathcal{F}^{-1}( f \hat{g} )$ extends to a bounded operator on the Banach space $\mathr

Figures (2)

  • Figure :
  • Figure :

Theorems & Definitions (154)

  • Theorem 1.1.1: Mikhlin
  • Theorem 1.1.2: Hörmander
  • Corollary 1.1.3: Hörmander
  • Proposition 1.2.1
  • Proposition 1.2.2
  • Example 1.2.3
  • Example 1.2.4
  • Lemma 1.2.5
  • proof
  • Theorem 1.2.6
  • ...and 144 more