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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.10121 |
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| _version_ | 1866913990705152000 |
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| author | Avdonin, Sergei Edward, Julian |
| author_facet | Avdonin, Sergei Edward, Julian |
| contents | Consider a quantum graph consisting of a ring with two attached edges, and assume Kirchhoff-Neumann conditions hold at the internal vertices. Associated to this graph is a Schrödinger type operator $L=-Δ+q(x)$ with Dirichlet boundary conditions at the two boundary nodes. Let $\{ ω_n^2, \ φ_n(x)\}$ be the eigenvalues and associated normalized eigenfunctions. Let $v_1$ be a boundary vertex, and $v_2$ the adjacent internal vertex. Assume we know the following data: $\{ ω_n^2,\partial_x φ_n(v_1),\partial_xφ_n(v_2)\}.$ Here $\partial_xφ_n(v_2)$ refers to an outward normal derivative at $v_2$ along one of the edges incident to the other internal vertex. From this data we determine the following unknown quantities: the lengths of edges and the potential functions on each edge. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10121 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An inverse problem on a metric graph with cycle Avdonin, Sergei Edward, Julian Analysis of PDEs Optimization and Control Spectral Theory 35R30 (primary), 35L05, 93B05 (secondary) Consider a quantum graph consisting of a ring with two attached edges, and assume Kirchhoff-Neumann conditions hold at the internal vertices. Associated to this graph is a Schrödinger type operator $L=-Δ+q(x)$ with Dirichlet boundary conditions at the two boundary nodes. Let $\{ ω_n^2, \ φ_n(x)\}$ be the eigenvalues and associated normalized eigenfunctions. Let $v_1$ be a boundary vertex, and $v_2$ the adjacent internal vertex. Assume we know the following data: $\{ ω_n^2,\partial_x φ_n(v_1),\partial_xφ_n(v_2)\}.$ Here $\partial_xφ_n(v_2)$ refers to an outward normal derivative at $v_2$ along one of the edges incident to the other internal vertex. From this data we determine the following unknown quantities: the lengths of edges and the potential functions on each edge. |
| title | An inverse problem on a metric graph with cycle |
| topic | Analysis of PDEs Optimization and Control Spectral Theory 35R30 (primary), 35L05, 93B05 (secondary) |
| url | https://arxiv.org/abs/2508.10121 |