Local H\"older Stability in the Inverse Steklov and Calder\'on Problems for Radial Schr\"odinger operators and Quantified Resonances

TL;DR

This paper establishes H"older stability estimates for inverse Steklov and Calderón problems involving radial Schr"odinger operators, improving upon previous logarithmic stability results by explicitly quantifying resonances and eigenvalue perturbations.

Contribution

It introduces a novel approach linking spectral differences to Laplace transforms of amplitude functions, leading to explicit stability estimates for radial potentials.

Findings

H"older stability estimates for inverse problems

Explicit quantification of negative eigenvalues and resonances

Improved stability bounds over previous logarithmic estimates

Abstract

We obtain H\"older stability estimates for the inverse Steklov and Calder\'on problems for Schr\"odinger operators corresponding to a special class of $L^2$ radial potentials on the unit ball. These results provide an improvement on earlier logarithmic stability estimates obtained in \cite{DKN5} in the case of the the Schr\"odinger operators related to deformations of the closed Euclidean unit ball. The main tools involve: i) A formula relating the difference of the Steklov spectra of the Schr\"odinger operators associated to the original and perturbed potential to the Laplace transform of the difference of the corresponding amplitude functions introduced by Simon \cite{Si1} in his representation formula for the Weyl-Titchmarsh function, and ii) A key moment stability estimate due to Still \cite{St}. It is noteworthy that with respect to the original Schr\"odinger operator, the type of perturbation being considered for the amplitude function amounts to the introduction of a finite number of negative eigenvalues and of a countable set of negative resonances which are quantified explicitly in terms of the eigenvalues of the Laplace-Beltrami operator on the boundary sphere.