论文标题
RadialSchrödinger操作员和量化共鸣的反向steklov和Calderón问题中的本地Hölder稳定性
Local Hölder Stability in the Inverse Steklov and Calderón Problems for Radial Schrödinger operators and Quantified Resonances
论文作者
论文摘要
我们为Schrödinger操作员的逆势型和Calderón问题获得了Hölder稳定性估计，这些估计与单位球上的特殊类别的$ L^2 $径向电势相对应。这些结果可以改善与封闭欧几里得单位球变形有关的schrödinger算子在\ cite {dkn5}中获得的对数稳定性估计值。 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 \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 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.
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 \cite{DKN5} in the case of the 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 \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 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.
