渐近双曲流形上的spectral theory及其反问题
Introduction to spectral theory and inverse problem on asymptotically hyperbolic manifolds
摘要：
研究了渐近双曲流形的谱理论和反问题。主要内容如下：（1）本质谱的定位。（2）没有嵌入在连续谱中的本征值。（3）分解物的极限吸收原理和连续谱的绝对连续性。（4）广义傅里叶变换的构造。（5）含时波算子的渐近完备性。（6）Helmhotz方程散射解空间的广义傅里叶变换刻画。(7)Helmholtz方程和S-矩阵散射解的渐近展开。（8）上半空间模型中波动方程的基本解的表示。（9）Radon变换与波动方程的奇性传播。最后，我们将讨论逆问题。即（10）从散射矩阵识别黎曼度量。
英文摘要：
We study the spectral theory and inverse problem on asymptotically hyperbolic manifolds. The main subjects are as follows: (1)Location of the essential spectrum. (2)Absence of eigenvalues embedded in the continuous spectrum. (3)Limiting absorption principle for the resolvent and the absolute continuity of the continuous spectrum. (4)Construction of the generalized Fourier transform. (5)symptotic completeness of time-dependent wave operators. (6)Characterization of the space of scattering solutions to the Helmhotz equation in terms of the generalized Fourier transform. (7)Asymptotic expansion of scattering solutions to the Helmholtz equation and the S-matrix. (8)Representation of the fundamental solution to the wave equation in the upper-half space model. (9)Radon transform and the propagation of singularities for the wave equation. Finally, we shall discuss the inverse problem. Namely (10)Identification of the Riemannian metric from the scattering matrix.
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