An inverse problem on a metric graph with cycle

2025/08/13 by Avdonin, Sergei, Edward, Julian · 1 citation
#35L05 #35R30 (primary) #93B05 (secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2508.10121

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_n², φ_n(x)\ be the eigenvalues and associated normalized eigenfunctions. Let v₁ be a boundary vertex, and v₂ the adjacent internal vertex. Assume we know the following data: \ ω_n²,∂_x φ_n(v₁),∂_xφ_n(v₂)\. Here ∂_xφ_n(v₂) refers to an outward normal derivative at v₂ 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.

Citations

• On the inverse problem of the two-velocity tree-like graph
• The boundary control approach to inverse spectral theory
• On an inverse problem for tree-like networks of elastic strings

Cited by

• Stability of the inverse Sturm-Liouville problem on a graph with a cycle

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