Peschl-Minda derivatives and convergent Wick star products on the disk, the sphere and beyond
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arXiv
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| Format: | Preprint |
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2023
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| author | Heins, Michael Moucha, Annika Roth, Oliver Sugawa, Toshiyuki |
| author_facet | Heins, Michael Moucha, Annika Roth, Oliver Sugawa, Toshiyuki |
| contents | We introduce and study invariant differential operators acting on the space $\mathcal{H}(Ω)$ of holomorphic functions on the complement ${Ω=\{(z,w) \in \hat{\mathbb{C}}^2 \, : \, z\cdot w \not=1\}}$ of the "complexified unit circle" $\{(z,w) \in \hat{\mathbb{C}}^2 \, : \, z\cdot w =1\}$. We obtain recursion identities, describe the behaviour under change of coordinates and find the generators of the corresponding operator algebra. We illustrate how this provides a unified framework for investigating conformally invariant differential operators on the unit disk $\mathbb{D}$ and the Riemann sphere $\hat{\mathbb{C}}$, which have been studied by Peschl, Aharonov, Minda and many others, within their conjecturally natural habitat. We apply the machinery to a problem in deformation quantization by deriving explicit formulas for the canonical Wick-type star products on $Ω$, the unit disk $\mathbb{D}$ and the Riemann sphere $\hat{\mathbb{C}}$ in terms of such invariant differential operators. These formulas are given in form of factorial series which depend holomorphically on a complex deformation parameter $\hbar$ and lead to asymptotic expansions of the star products in powers of $\hbar$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_01101 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Peschl-Minda derivatives and convergent Wick star products on the disk, the sphere and beyond Heins, Michael Moucha, Annika Roth, Oliver Sugawa, Toshiyuki Complex Variables Mathematical Physics Functional Analysis Primary 30F45, 30B50, 53D55, Secondary 53A55 We introduce and study invariant differential operators acting on the space $\mathcal{H}(Ω)$ of holomorphic functions on the complement ${Ω=\{(z,w) \in \hat{\mathbb{C}}^2 \, : \, z\cdot w \not=1\}}$ of the "complexified unit circle" $\{(z,w) \in \hat{\mathbb{C}}^2 \, : \, z\cdot w =1\}$. We obtain recursion identities, describe the behaviour under change of coordinates and find the generators of the corresponding operator algebra. We illustrate how this provides a unified framework for investigating conformally invariant differential operators on the unit disk $\mathbb{D}$ and the Riemann sphere $\hat{\mathbb{C}}$, which have been studied by Peschl, Aharonov, Minda and many others, within their conjecturally natural habitat. We apply the machinery to a problem in deformation quantization by deriving explicit formulas for the canonical Wick-type star products on $Ω$, the unit disk $\mathbb{D}$ and the Riemann sphere $\hat{\mathbb{C}}$ in terms of such invariant differential operators. These formulas are given in form of factorial series which depend holomorphically on a complex deformation parameter $\hbar$ and lead to asymptotic expansions of the star products in powers of $\hbar$. |
| title | Peschl-Minda derivatives and convergent Wick star products on the disk, the sphere and beyond |
| topic | Complex Variables Mathematical Physics Functional Analysis Primary 30F45, 30B50, 53D55, Secondary 53A55 |
| url | https://arxiv.org/abs/2308.01101 |