Scattering resonances in the vanishing curvature limit

曲率消失极限中的散射共振

• 批准号: 2746963
• 金额: --
• 依托单位: University of Cambridge
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2746963
• 关键词: Scattering; resonances; vanishing; curvature; limit

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.

在闭黎曼流形上，拉普拉斯算子的特征值和特征函数编码了流形的几何信息。在开流形上，拉普拉斯算子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”的对象，它们在封闭环境中扮演着与本征值类似的角色。特别是，Mazzeo-Melrose表明，散射共振可以定义为流形的开放端是渐近的空间常数负曲率的手段，分析延续的论点。最近Vasy已经开发了一个强大的Fredholm理论的散射共振，利用工具的microlocal分析。在无穷远曲率趋于零的极限中，这些方法失效了，但是已知至少一些散射共振在极限中表现良好。这个项目的目标是要看看是否方法的精神，瓦西的工作可以开发，允许限制在无穷大曲率消失进行研究。