论文标题
一般度量的非压缩歧管上的反向散射
Inverse scattering on non-compact manifolds with general metric
论文作者
论文摘要
我们在本文中解决的问题是光谱理论和与非紧凑型riemannian歧管上的拉普拉斯人相关的逆问题，并且更通用的歧管承认圆锥形奇异性。特别是，我们研究了反向散射问题，其中人们观察了流形方程溶液的渐近行为。这些观察结果类似于量子力学中海森堡的散射基质。然后，我们表明散射矩阵的知识决定了歧管的拓扑和指标。 在论文中，我们开发了一种统一的方法，以考虑在流形类型的多种类型的流形上，例如常规双曲末端，尖端和圆柱形末端，与波浪指南研究中遇到的模型相关。我们允许歧管也具有圆锥形奇点。因此，研究的一类歧管包括Orbifolds。这种非平滑结构是在研究反问题和几何崩溃的稳定性中。
The problems we address in this paper are the spectral theory and the inverse problems associated with Laplacians on non-compact Riemannian manifolds and more general manifolds admitting conic singularities. In particular, we study the inverse scattering problem where one observes the asymptotic behavior of the solutions of the Helmholtz equation on the manifold. These observations are analogous to Heisenberg's scattering matrix in quantum mechanics. We then show that the knowledge of the scattering matrix determines the topology and the metric of the manifold. In the paper we develop a unified approach to consider scattering problems on manifolds that can have very different type of infinities, such as regular hyperbolic ends, cusps, and cylindrical ends related to models encountered in the study of wave guides. We allow the manifold to have also conic singularities. Due to this, the studied class of manifolds include orbifolds. Such non-smooth structures arise in the study of the stability of inverse problems and of the geometrical collapse.
