Observable sets for Schrödinger equations on combinatorial graphs
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910203811725312 |
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| author | Wan, Zhiqiang Zhang, Heng |
| author_facet | Wan, Zhiqiang Zhang, Heng |
| contents | We study observable sets for Schrödinger equations on combinatorial graphs. For one-dimensional lattice Schrödinger operators \(H=-Δ_{\mathrm{disc}}+V\) with \(V(n)\to c\in\mathbb R\) as \(|n|\to\infty\), we prove that a set \(E\subset\mathbb Z\) is observable at some time, equivalently at any time, if and only if it satisfies a local arithmetic condition. This reveals an arithmetic obstruction absent from the Euclidean theory, where thickness is the decisive condition. The same criterion also characterizes observability for the corresponding heat equation on \(\mathbb Z\). In higher-dimensional lattices, we prove observability from the complement of any finite set. We further obtain arithmetic criteria on discrete tori, showing that positive density alone does not ensure observability. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_10358 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Observable sets for Schrödinger equations on combinatorial graphs Wan, Zhiqiang Zhang, Heng Analysis of PDEs Combinatorics We study observable sets for Schrödinger equations on combinatorial graphs. For one-dimensional lattice Schrödinger operators \(H=-Δ_{\mathrm{disc}}+V\) with \(V(n)\to c\in\mathbb R\) as \(|n|\to\infty\), we prove that a set \(E\subset\mathbb Z\) is observable at some time, equivalently at any time, if and only if it satisfies a local arithmetic condition. This reveals an arithmetic obstruction absent from the Euclidean theory, where thickness is the decisive condition. The same criterion also characterizes observability for the corresponding heat equation on \(\mathbb Z\). In higher-dimensional lattices, we prove observability from the complement of any finite set. We further obtain arithmetic criteria on discrete tori, showing that positive density alone does not ensure observability. |
| title | Observable sets for Schrödinger equations on combinatorial graphs |
| topic | Analysis of PDEs Combinatorics |
| url | https://arxiv.org/abs/2511.10358 |