Approximate eigen-data on an open subset determine a closed Riemannian manifold up to Lipschitz distance epsilon^(1/12) and a Lipschitz potential up to epsilon^(1/(80n)), giving double-logarithmic stability.
Inverse Spectral Problems for Collapsing Manifolds II: Quantitative Stability of Reconstruction for Orbifolds
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abstract
We consider the inverse problem of determining the metric-measure structure of collapsing manifolds from local measurements of spectral data. In the part I of the paper, we proved the uniqueness of the inverse problem and a continuity result for the stability in the closure of Riemannian manifolds with bounded diameter and sectional curvature in the measured Gromov-Hausdorff topology. In this paper we show that when the collapse of dimension is $1$-dimensional, it is possible to obtain quantitative stability of the inverse problem for Riemannian orbifolds. The proof is based on an improved version of the quantitative unique continuation for the wave operator on Riemannian manifolds by removing assumptions on the covariant derivatives of the curvature tensor.
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Stability of Gel'fand's inverse interior spectral problem for Schr\"odinger operators
Approximate eigen-data on an open subset determine a closed Riemannian manifold up to Lipschitz distance epsilon^(1/12) and a Lipschitz potential up to epsilon^(1/(80n)), giving double-logarithmic stability.