通过Born近似实现Calderón问题的稳定分解
Stable factorization of the Calderón problem via the Born approximation
AI总结：
针对径向Schrödinger算子的Calderón问题，证明Born近似存在性，并展示其非线性部分具有局部性和Hölder稳定性，从而揭示问题病态性源于线性步骤。
摘要：
本文中，我们证明了在径向Schrödinger算子的Calderón问题背景下Born近似的存在性。Born近似自然作为Calderón问题分解的线性分量出现；我们表明，从Born近似获得势的非线性部分具有若干有趣性质。首先，该映射是局部的，即边界邻域内Born近似的知识等价于同一邻域内势的知识；其次，它是Hölder稳定的。这证明了Calderón问题的病态性源于线性步骤，即通过求解Hausdorff矩问题从DtN映射计算Born近似。此外，我们提出了一种从Born近似计算势的有效算法。最后，我们利用Born近似获得了径向势的DtN映射集合的部分刻画。这些结果的证明不依赖于复几何光学解或其类似物；它们基于半线上Schrödinger算子的逆谱理论结果，特别是Barry Simon引入的$A$-振幅概念。
英文摘要：
In this article, we prove the existence of the Born approximation in the context of the radial Calderón problem for Schrödinger operators. The Born approximation naturally appears as the linear component of a factorization of the Calderón problem; we show that the non-linear part, obtaining the potential from the Born approximation, enjoys several interesting properties. First, this map is local, in the sense that knowledge of the Born approximation in a neighborhood of the boundary is equivalent to knowledge of the potential in the same neighborhood, and, second, it is Hölder stable. This proves that the ill-posedness of the Calderón problem arises from the linear step, which consists in computing the Born approximation from the DtN map by solving a Hausdorff moment problem. Moreover, we present an effective algorithm to compute the potential from the Born approximation. Finally, we use the Born approximation to obtain a partial characterization of the set of DtN maps for radial potentials. The proofs of these results do not make use of Complex Geometrical Optics solutions or their analogs; they are based on results from inverse spectral theory for Schrödinger operators on the half-line, in particular on the concept of $A$-amplitude introduced by Barry Simon.
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