Abstract
We obtain Hölder stability estimates for the inverse Steklov and Calderón problems for Schrödinger 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 Daudé et al. (J Geom Anal 31(2):1821–1854, 2021) in the case of the Schrödinger 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 Schrödinger operators associated to the original and perturbed potential to the Laplace transform of the difference of the corresponding amplitude functions introduced by Simon (Ann Math 150:1029–1057, 1999) in his representation formula for the Weyl-Titchmarsh function, and (ii) a key moment stability estimate due to Still (J Approx Theory 45:26–54, 1985). It is noteworthy that with respect to the original Schrödinger 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.
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Notes
Since we are concerned with the Dirichlet-to-Neumann map, we are putting a Dirichlet boundary condition at \(x=0\), which represents the boundary sphere \({\partial M}=S^{d-1}\) in the new radial coordinate \(x=-\log r\). Had we put a Neumann condition at \(x=0\), the corresponding Weyl-Titchmarsh function would have then corresponded to the Neumann-to-Dirichlet map, in which case the multiplication operators defined below in (1.15) would have involved the Weyl-Titchmarsh function associated to a Neumann boundary condition at \(x=0\).
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The authors would like to warmly thank the anonymous referees for their valuable comments and suggestions.
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Communicated by Jan Derezinski.
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Thierry Daudé: Research supported by the French National Research Projects AARG, No. ANR-12-BS01-012-01, and Iproblems, No. ANR-13-JS01-0006. Niky Kamran: Research supported by NSERC grant RGPIN 105490-2018. François Nicoleau: Research supported by the French GDR Dynqua.
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Daudé, T., Kamran, N. & Nicoleau, F. Local Hölder Stability in the Inverse Steklov and Calderón Problems for Radial Schrödinger Operators and Quantified Resonances. Ann. Henri Poincaré (2023). https://doi.org/10.1007/s00023-023-01391-1
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DOI: https://doi.org/10.1007/s00023-023-01391-1