Scattering resonances in the vanishing curvature limit

在闭黎曼流形上，拉普拉斯的特征值和特征函数编码关于流形几何的信息。在开放流形上，拉普拉斯量的谱一般不是离散的，然而在某些情况下，可能存在一类被称为“散射共振”的物体，其作用与闭合环境中的本征值有些相似。特别地，Mazzeo-Melrose证明了，对于开口端渐近于常负曲率空间的流形，可以通过解析延拓引理来定义散射共振。最近，Vasy利用微局域分析的工具，发展了一个稳健的Fredholm型散射共振理论。在无限远曲率趋于零的极限处，这些方法失效了，但众所周知，至少有一些散射共振在极限处表现良好。这个项目的目标是看看是否可以开发出符合维希工作精神的方法，允许研究无穷远处曲率消失的极限。

On a closed Riemannian manifold, the eigenvalues and eigenfunctions of the Laplacian encode information about the geometry of the manifold. On an open manifold the spectrum of the Laplacian is not discrete in general, however in certain circumstances there may exist a class of objects called "scattering resonances" which play a somewhat analogous role to the eigenvalues in the closed setting. In particular, Mazzeo-Melrose showed that scattering resonances can be defined for manifolds whose open end is asymptotic to a space of constant negative curvature by means of an analytic continuation argument. More recently Vasy has developed a robust Fredholm theory for the scattering resonances, making use of the tools of microlocal analysis. In the limit where the curvature at infinity tends to zero these methods break down, but nevertheless it is known that at least some of the scattering resonances are well behaved in the limit. The goal of this project is to see whether methods in the spirit of Vasy's work can be developed that permit the limit where the curvature at infinity vanishes to be studied.