Self-adjoint extensions of singular Sturm-Liouville operators on graphs and Weyl's law
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911225752846336 |
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| author | Falbel, Elisha |
| author_facet | Falbel, Elisha |
| contents | We study self-adjoint extensions of a second order differential operator of Sturm-Liouville type on a graph. We relate self-adjointness of the operator to the existence of non-complete trajectories of the Hamiltonian vector field defined by its principal symbol outside the vertices. We define Kirchhoff conditions at the vertices which guarantee a self-adjoint extension analogous to the case of quantum graphs. The singular vertices may be interpreted as introducing a singular potential at those points. We also establish a Weyl's law for the spectrum asymptotics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18922 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Self-adjoint extensions of singular Sturm-Liouville operators on graphs and Weyl's law Falbel, Elisha Spectral Theory We study self-adjoint extensions of a second order differential operator of Sturm-Liouville type on a graph. We relate self-adjointness of the operator to the existence of non-complete trajectories of the Hamiltonian vector field defined by its principal symbol outside the vertices. We define Kirchhoff conditions at the vertices which guarantee a self-adjoint extension analogous to the case of quantum graphs. The singular vertices may be interpreted as introducing a singular potential at those points. We also establish a Weyl's law for the spectrum asymptotics. |
| title | Self-adjoint extensions of singular Sturm-Liouville operators on graphs and Weyl's law |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2510.18922 |