{"ID":2882930,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.10121","arxiv_id":"2508.10121","title":"An inverse problem on a metric graph with cycle","abstract":"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 $\\{ ω_n^2, \\ \\varphi_n(x)\\}$ 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: $\\{ ω_n^2,\\partial_x \\varphi_n(v_1),\\partial_x\\varphi_n(v_2)\\}.$ Here $\\partial_x\\varphi_n(v_2)$ 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.","short_abstract":"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 $\\{ ω_n^2, \\ \\varphi_n(x)\\}$ be the eigenval...","url_abs":"https://arxiv.org/abs/2508.10121","url_pdf":"https://arxiv.org/pdf/2508.10121v1","authors":"[\"Sergei Avdonin\",\"Julian Edward\"]","published":"2025-08-13T18:36:21Z","proceeding":"math.AP","tasks":"[\"math.AP\",\"math.OC\",\"math.SP\"]","methods":"[]","has_code":false}