Linearized Calderón problem and exponentially accurate quasimodes for analytic manifolds
Linearized Calderón problem and exponentially accurate quasimodes for analytic manifolds
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DOI：
10.1016/j.aim.2022.108362
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发表时间：
2020-09
影响因子：
1.7
通讯作者：
Katya Krupchyk;Tony Liimatainen;M. Salo
中科院分区：
文献类型：
--
作者：
Katya Krupchyk;Tony Liimatainen;M. Salo
本文研究了具有边界的紧黎曼流形上的线性化各向异性Calderón问题。这个问题等于证明流形的调和函数对的乘积是一个完整的集合。我们假定流形是横截面各向异性的，横截面流形是实解析的，并且满足与相交测地线对几何有关的一个几何条件。在这种情况下，我们解决了线性化的各向异性Calderón问题。几何条件不涉及测地线x射线变换的注入性。证明我们的结果的关键因素是在横向流形上构造高斯光束准模，具有指数小的误差，以及解析波前集的FBI变换表征。
In this article we study the linearized anisotropic Calderón problem on a compact Riemannian manifold with boundary. This problem amounts to showing that products of pairs of harmonic functions of the manifold form a complete set. We assume that the manifold is transversally anisotropic and that the transversal manifold is real analytic and satisfies a geometric condition related to the geometry of pairs of intersecting geodesics. In this case, we solve the linearized anisotropic Calderón problem. The geometric condition does not involve the injectivity of the geodesic X-ray transform. Crucial ingredients in the proof of our result are the construction of Gaussian beam quasimodes on the transversal manifold, with exponentially small errors, as well as the FBI transform characterization of the analytic wave front set.