# Reconstruction and stability in Gel'fand's inverse interior spectral problem

## TL;DR

This paper develops a stable method to reconstruct a Riemannian manifold from approximate spectral data, providing explicit stability estimates and advancing the understanding of Gel'fand's inverse spectral problem.

## Contribution

It introduces a new stable reconstruction approach for manifolds from spectral data with explicit log-log stability estimates, improving previous inverse spectral results.

## Key findings

- Constructs a stable approximation of the manifold from spectral data with small error.
- Provides explicit log-log stability estimates for the reconstruction.
- Extends stability analysis to Gel'fand's inverse problem.

## 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 $\Delta_g$ on $M$ as well as the corresponding eigenfunctions restricted on an open set in $M$. We then construct a stable approximation to the manifold $(M,g)$. 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 Gel'fand's 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.