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arXiv 1702.07974math.APmath-phmath.MP
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横观各向异性几何中磁薛定谔算子的反问题

Inverse problems for magnetic Schrödinger operators in transversally anisotropic geometries

Katya Krupchyk, Gunther Uhlmann

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摘要：

研究了具有$\ge 3$维边界的紧致黎曼流形上磁薛定谔算子的逆边界问题.本文的第一部分讨论了可容许几何的情形，即紧黎曼流形的边界共形地嵌入在欧氏直线与简单流形的乘积中。我们证明了具有$L^\infty$磁势和电势的磁薛定谔算子在流形边界上的柯西数据的知识，唯一地确定了磁场和电势。在本文的第二部分，我们解决的情况下，更一般的共形横向各向异性几何，即紧凑的黎曼流形的边界是共形嵌入产品的欧几里德线和一个紧凑的流形，这不需要简单。在这里，假设横截流形上的测地线射线变换是内射的，我们证明了具有连续势的磁薛定谔算子的流形边界上的柯西数据的知识，唯一地确定了磁场。假设电势是已知的，我们表明，柯西数据确定磁势的规范等价。

英文摘要：

We study inverse boundary problems for magnetic Schrödinger operators on a compact Riemannian manifold with boundary of dimension $\ge 3$. In the first part of the paper we are concerned with the case of admissible geometries, i.e. compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that the knowledge of the Cauchy data on the boundary of the manifold for the magnetic Schrödinger operator with $L^\infty$ magnetic and electric potentials, determines the magnetic field and electric potential uniquely. In the second part of the paper we address the case of more general conformally transversally anisotropic geometries, i.e. compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a compact manifold, which need not be simple. Here, under the assumption that the geodesic ray transform on the transversal manifold is injective, we prove that the knowledge of the Cauchy data on the boundary of the manifold for a magnetic Schrödinger operator with continuous potentials, determines the magnetic field uniquely. Assuming that the electric potential is known, we show that the Cauchy data determines the magnetic potential up to a gauge equivalence.

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