Explicit Formulas for the One-Parameter Group Generated by the Dunkl Operator on $\mathbb{R}$
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915918697725952 |
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| author | Aoyama, Temma |
| author_facet | Aoyama, Temma |
| contents | Let $T_{b}$ be the Dunkl operator for the reflection group $G=\mathbb{Z}/2\mathbb{Z}$, and $D_{b}:=|x|^{b}\,T_{b}\,|x|^{-b}$. We compute explicitly the unitary one-parameter group $e^{tD_{b}}$ generated by $D_{b}$. We obtain two representations: a boundary value representation from the upper and lower half-planes, and a real-variable formula consisting of a translation term and a principal value integral term with an explicit kernel expressed in terms of Legendre functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04053 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Explicit Formulas for the One-Parameter Group Generated by the Dunkl Operator on $\mathbb{R}$ Aoyama, Temma Functional Analysis Classical Analysis and ODEs Representation Theory Let $T_{b}$ be the Dunkl operator for the reflection group $G=\mathbb{Z}/2\mathbb{Z}$, and $D_{b}:=|x|^{b}\,T_{b}\,|x|^{-b}$. We compute explicitly the unitary one-parameter group $e^{tD_{b}}$ generated by $D_{b}$. We obtain two representations: a boundary value representation from the upper and lower half-planes, and a real-variable formula consisting of a translation term and a principal value integral term with an explicit kernel expressed in terms of Legendre functions. |
| title | Explicit Formulas for the One-Parameter Group Generated by the Dunkl Operator on $\mathbb{R}$ |
| topic | Functional Analysis Classical Analysis and ODEs Representation Theory |
| url | https://arxiv.org/abs/2604.04053 |