On the measure of spectra for discrete Schrödinger operators on periodic graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911558095863808 |
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| author | Saburova, Natalia |
| author_facet | Saburova, Natalia |
| contents | We consider discrete Schrödinger operators $H_{μQ}=Δ+μQ$ with real periodic potentials $Q$ on periodic graphs, where $Δ$ is the adjacency operator and $μ\in\mathbb R$ is a coupling constant. The spectra of the operators consist of a finite number of closed intervals (bands). In the large coupling regime, we obtain an asymptotic upper bound for the measure of the spectrum of $H_{μQ}$ which depends essentially on a "degeneracy degree" of the potential $Q$. This result extends the result of Y. Last obtained for the one-dimensional lattice $\mathbb Z$ to the case of general periodic graphs. It also may serve as a certain quantitative complement to the recent criterion of J. Fillman for the measure of the spectrum of $H_{μQ}$ to go to zero as $μ\to\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29898 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the measure of spectra for discrete Schrödinger operators on periodic graphs Saburova, Natalia Spectral Theory 35J10 We consider discrete Schrödinger operators $H_{μQ}=Δ+μQ$ with real periodic potentials $Q$ on periodic graphs, where $Δ$ is the adjacency operator and $μ\in\mathbb R$ is a coupling constant. The spectra of the operators consist of a finite number of closed intervals (bands). In the large coupling regime, we obtain an asymptotic upper bound for the measure of the spectrum of $H_{μQ}$ which depends essentially on a "degeneracy degree" of the potential $Q$. This result extends the result of Y. Last obtained for the one-dimensional lattice $\mathbb Z$ to the case of general periodic graphs. It also may serve as a certain quantitative complement to the recent criterion of J. Fillman for the measure of the spectrum of $H_{μQ}$ to go to zero as $μ\to\infty$. |
| title | On the measure of spectra for discrete Schrödinger operators on periodic graphs |
| topic | Spectral Theory 35J10 |
| url | https://arxiv.org/abs/2603.29898 |