The harmonic oscillator on the Moyal-Groenewold plane: an approach via Lie groups and twisted Weyl tuples
Fuente:
arXiv
Salvato in:
| Autori principali: | , , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866912322889449472 |
|---|---|
| author | Arhancet, Cédric Hagedorn, Lukas Kriegler, Christoph Portal, Pierre |
| author_facet | Arhancet, Cédric Hagedorn, Lukas Kriegler, Christoph Portal, Pierre |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_06143 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The harmonic oscillator on the Moyal-Groenewold plane: an approach via Lie groups and twisted Weyl tuples Arhancet, Cédric Hagedorn, Lukas Kriegler, Christoph Portal, Pierre Functional Analysis High Energy Physics - Theory Mathematical Physics Group Theory Operator Algebras 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. |
| title | The harmonic oscillator on the Moyal-Groenewold plane: an approach via Lie groups and twisted Weyl tuples |
| topic | Functional Analysis High Energy Physics - Theory Mathematical Physics Group Theory Operator Algebras |
| url | https://arxiv.org/abs/2312.06143 |