Zeta-invariants of the Steklov spectrum for a planar domain

2014/04/08 by Evgeny Malkovich, Malkovich, Evgeny, V. A. Sharafutdinov +2
Mathematics · #35P10 #35R30 #47A75 #Differential Geometry (math.DG) #FOS: Mathematics #Numerical methods in inverse problems #math.DG #msc:35P10 #msc:35R30 #msc:47A75

paper · pdf · doi:10.48550/arxiv.1404.2117

arxiv created 2014/04/08 · openalex publication_date 2014/04/08 · arxiv updated 2014/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The classical inverse problem of recovering a simply connected smooth planar domain from the Steklov spectrum \citeE is equivalent to the problem of recovering, up to a conformal equivalence, a positive function a∈ C^∞(\mathbb S) on the unit circle \mathbb S=\e^iθ\ from the eigenvalue spectrum of the operator aΛ_e, where Λ_e=(-d²/dθ²)^1/2. We introduce 2k-forms Z_k(a) (k=1,2,…) in Fourier coefficients of the function a which are called zeta-invariants. They are uniquely determined by the eigenvalue spectrum of aΛ_e. We study some properties of Z_k(a), in particular, their invariance under the conformal group. Some open questions on zeta-invariants are posed at the end of the paper.

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