Title: Hölder stability of an inverse spectral problem for the magnetic Schrödinger operator on a simple manifold Show Chinese title

Title: 磁 Schrödinger 算子在简单流形上的反谱问题的 Hölder 稳定性

Abstract: We show that on a simple Riemannian manifold, the electric potential and the solenoidal part of the magnetic potential appearing in the magnetic Schr\"odinger operator can be recovered H\"older stably from the boundary spectral data. This data contains the eigenvalues and the Neumann traces of the corresponding sequence of Dirichlet eigenfunctions of the operator. Our proof contains two parts, which we present in the reverse order. (1) We show that the boundary spectral data can be stably obtained from the Dirichlet-to-Neumann map associated with the respective initial boundary value problem for a hyperbolic equation, whose leading order terms are a priori known. (2) We construct geometric optics solutions to the hyperbolic equation, which reduce the stable recovery of the lower order terms to the stable inversion of the geodesic ray transform of one-forms and functions. Show Chinese abstract

Abstract: 我们证明，在一个简单的黎曼流形上，磁 Schrödinger 算子中出现的电势和磁势的无旋部分可以从边界谱数据中以 Hölder 稳定的方式恢复。此数据包含算子对应的一系列狄利克雷特征函数的特征值和诺伊曼迹。我们的证明包含两个部分，我们按相反的顺序进行阐述。 (1) 我们证明，从与双曲方程各自初始边值问题相关的狄利克雷到诺伊曼映射中可以稳定地获得边界谱数据，其高阶项是先验已知的。 (2) 我们构造了双曲方程的几何光学解，这将低阶项的稳定恢复归结为对 1-形式和函数的测地线射线变换的稳定逆。