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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.05649 |
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| _version_ | 1866917563831681024 |
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| author | Liu, Yihan Yan, Jun Zhao, Jia |
| author_facet | Liu, Yihan Yan, Jun Zhao, Jia |
| contents | This paper mainly deals with the Sturm-Liouville operator \begin{equation*} \mathbf{H}=\frac{1}{w(x)}\left( -\frac{\mathrm{d}}{\mathrm{d}x}p(x)\frac{ \mathrm{d}}{\mathrm{d}x}+q(x)\right) ,\text{ }x\in Γ\end{equation*} acting in $L_{w}^{2}\left( Γ\right) ,$ where $Γ$ is a metric graph. We establish a relationship between the bottom of the spectrum and the positive solutions of quantum graphs, which is a generalization of the classical Allegretto-Piepenbrink theorem. Moreover, we prove the Persson-type theorem, which characterizes the infimum of the essential spectrum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05649 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral properties of Sturm-Liouville operators on infinite metric graphs Liu, Yihan Yan, Jun Zhao, Jia Spectral Theory This paper mainly deals with the Sturm-Liouville operator \begin{equation*} \mathbf{H}=\frac{1}{w(x)}\left( -\frac{\mathrm{d}}{\mathrm{d}x}p(x)\frac{ \mathrm{d}}{\mathrm{d}x}+q(x)\right) ,\text{ }x\in Γ\end{equation*} acting in $L_{w}^{2}\left( Γ\right) ,$ where $Γ$ is a metric graph. We establish a relationship between the bottom of the spectrum and the positive solutions of quantum graphs, which is a generalization of the classical Allegretto-Piepenbrink theorem. Moreover, we prove the Persson-type theorem, which characterizes the infimum of the essential spectrum. |
| title | Spectral properties of Sturm-Liouville operators on infinite metric graphs |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2401.05649 |