H\"{o}lder Stability in the Inverse Steklov Problem for Radial Schr\"{o}dinger operators and Quantified Resonances
Abstract
In this paper, we obtain H\"{o}lder stability estimates for the inverse Steklov problem for Schr\"{o}dinger 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 [8] in the case of the Schr\"{o}dinger operators related to deformations of the closed unit ball. The main tools involve a formula relating the difference of the Steklov spectra of the Schr\"{o}dinger operators associated to the original and perturbed potential to the Laplace transform of the difference of the corresponding amplitude functions introduced by B. Simon and a key moment stability estimate due to Still. It is noteworthy that with respect to the original Schr\"{o}dinger 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.
Keywords
Inverse Steklov problem Steklov spectrum Weyl-Titchmarsh functions moment problems Hölder stability. 2010 Mathematics Subject Classication. Primaries 81U40 35P25
Secondary 58J50. Contents
Inverse Steklov problem
Steklov spectrum
Weyl-Titchmarsh functions
moment problems
Hölder stability. 2010 Mathematics Subject Classication. Primaries 81U40
35P25
Origin : Files produced by the author(s)