Travelling Times in Scattering by Obstacles in Curved Space

从其传播时间恢复障碍物

Recovering Obstacles from their Travelling Times

本文将Noakes和Stoyanov提出的从传播时间恢复严格凸平面障碍物的方法，扩展至一般曲率条件下的黎曼曲面障碍物，并要求光滑测地线最多与两个障碍物相交。

从行进时间看黎曼流形中障碍物的唯一性

Uniqueness of Obstacles in Riemannian Manifolds from Travelling Times

本文研究黎曼流形中由行进时间数据反演障碍物几何的问题，在自然曲率条件及测地线最多与两个障碍物分量相交的假设下，证明了具有相同行进时间的两组严格凸障碍物必然完全相同。

Travelling times in scattering by obstacles

The paper deals with some problems related to recovering information about an obstacle in an Euclidean space from certain measurements of lengths of generalized geodesics in the exterior of the obstacle. The main result is that if two obstacles satisfy some generic regularity conditions and have (almost) the same traveling times, then the generalized geodesic flows in their exteriors are conjugate on the non-trapping part of their phase spaces with a time preserving conjugacy. In the case of a union of two strictly convex domains in the plane, a constructive algorithm is described to recover t…

黎曼流形中严格凸障碍物旅行时间的刚性

Rigidity of Travelling Times for Strictly Convex Obstacles in Riemannian Manifolds

本文证明在维数至少为3的黎曼流形中，若两个严格凸障碍物的旅行时间集合相等，则它们必然相同，无需切线等价条件。

Lens rigidity in scattering by non-trapping obstacles

We prove that if two non-trapping obstacles in $\mathbb{R}^n$ satisfy some rather weak non-degeneracy conditions and the scattering rays in their exteriors have (almost) the same travelling times or (almost) the same scattering length spectrum, then they coincide.

Lens Rigidity in Scattering by Unions of Strictly Convex Bodies in $\R^2$

It was proved in \cite{NS1} that obstacles $K$ in $\R^d$ that are finite disjoint unions of strictly convex domains with $C^3$ boundaries are uniquely determined by the travelling times of billiard trajectories in their exteriors and also by their so called scattering length spectra. However the case $d = 2$ is not properly covered in \cite{NS1}. In the present paper we give a separate different proof of the same result in the case $d = 2$.