Vol. 15, No. 2, 2022

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Reconstruction and stability in Gelfand's inverse interior spectral problem

Roberta Bosi, Yaroslav Kurylev and Matti Lassas

Vol. 15 (2022), No. 2, 273–326
Abstract

Assume that M is a compact Riemannian manifold of bounded geometry given by restrictions on its diameter, Ricci curvature and injectivity radius. Assume we are given, with some error, the first eigenvalues of the Laplacian Δg on M as well as the corresponding eigenfunctions restricted on an open set in M. We then construct a stable approximation to the manifold (M,g). Namely, we construct a metric space and a Riemannian manifold which differ, in a proper sense, just a little from M when the above data are given with a small error. We give an explicit log - log -type stability estimate on how the constructed manifold and the metric on it depend on the errors in the given data. Moreover a similar stability estimate is derived for the Gelfand inverse problem. The proof is based on methods from geometric convergence, a quantitative stability estimate for the unique continuation and a new version of the geometric boundary control method.

Keywords
inverse spectral problems, Riemannian manifolds
Mathematical Subject Classification 2010
Primary: 35R30, 58J05
Milestones
Received: 4 October 2018
Revised: 3 August 2020
Accepted: 6 October 2020
Published: 12 April 2022
Authors
Roberta Bosi
Department of Mathematics and Statistics
University of Helsinki
Helsinki
Finland
Yaroslav Kurylev
Department of Mathematics
University College London
London
United Kingdom
Matti Lassas
Department of Mathematics and Statistics
University of Helsinki
Helsinki
Finland