Subspaces of $L^2(\mathbb{R}^n)$ Invariant Under Crystallographic Shifts
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917206899556352 |
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| author | Potter, Tom |
| author_facet | Potter, Tom |
| contents | In this thesis we consider crystal groups in dimension $n$ and their natural unitary representation on $L^2(\mathbb{R}^n)$. We show that this representation is unitarily equivalent to a direct integral of factor representations, and use this to characterize the subspaces of $L^2(\mathbb{R}^n)$ invariant under crystal symmetry shifts. Finally, by giving an explicit unitary equivalence of the natural crystal group representation, we find the \textit{central decomposition} guaranteed by direct integral theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_02130 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Subspaces of $L^2(\mathbb{R}^n)$ Invariant Under Crystallographic Shifts Potter, Tom Functional Analysis In this thesis we consider crystal groups in dimension $n$ and their natural unitary representation on $L^2(\mathbb{R}^n)$. We show that this representation is unitarily equivalent to a direct integral of factor representations, and use this to characterize the subspaces of $L^2(\mathbb{R}^n)$ invariant under crystal symmetry shifts. Finally, by giving an explicit unitary equivalence of the natural crystal group representation, we find the \textit{central decomposition} guaranteed by direct integral theory. |
| title | Subspaces of $L^2(\mathbb{R}^n)$ Invariant Under Crystallographic Shifts |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2501.02130 |