Application of the boundary control method to partial data Borg-Levinson inverse spectral problem

2017/03/26 by Yavar Kian, Kian, Yavar, Morgan Morancey +3 · 3 citations
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #Numerical methods in inverse problems

paper · doi:10.48550/arxiv.1703.08832

openalex publication_date 2017/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the multidimensional Borg-Levinson problem of determining a potential q, appearing in the Dirichlet realization of the Schrödinger operator A_q=-Δ+q on a bounded domain Ω⊂ ℝⁿ, n≥2, from the boundary spectral data of A_q on an arbitrary portion of ∂Ω. More precisely, for γ an open and non-empty subset of ∂Ω, we consider the boundary spectral data on γ given by BSD(q,γ):=\(λ_k,∂_νϕ_k|γ): k ≥1\, where \ λ_k: k ≥1\ is the non-decreasing sequence of eigenvalues of A_q, \ ϕ_k: k ≥1 \ an associated Hilbertian basis of eigenfunctions, and ν is the unit outward normal vector to ∂Ω. We prove that the data BSD(q,γ) uniquely determine a bounded potential q∈ L^∞(Ω). Previous uniqueness results, with arbitrarily small γ, assume that q is smooth. Our approach is based on the Boundary Control method, and we give a self-contained presentation of the method, focusing on the analytic rather than geometric aspects of the method.

Citations

• A Global Uniqueness Theorem for an Inverse Boundary Value Problem
• Recovery of time-dependent coefficient on Riemanian manifold for hyperbolic equations

Cited by

• Inverse spectral problems for higher-order coefficients
• Recovery of Sturm-Liouville operators from partial boundary spectral data and applications
• Hölder stability of an inverse spectral problem for the magnetic Schrödinger operator on a simple manifold

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