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 ${\mathrm{\Delta }}_{g}$ on $M$ as well as the corresponding eigenfunctions restricted on an open set in $M$. We then construct a stable approximation to the manifold $\left(M,g\right)$. 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 $\mathrm{log}$-$\mathrm{log}$-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.

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