标题： 双曲空间上的散射理论 显示英文标题

标题： A Scattering theory on hyperbolic spaces

摘要： 在本文中，我们开发了一个用于双曲空间上时谐波散射的理论框架。 利用极限吸收原理（LAP），我们推导了双曲空间的Helmholtz算子的入射和出射格林函数的显式形式，并验证了它们是基本解。 随后，我们建立了格林函数渐近行为的准确表征，并利用它们建立入射和出射辐射条件，这些条件类似于欧几里得设置中的Sommerfeld辐射条件。 此外，我们证明了一个类似Rellich的定理，该定理保证了散射场及其远场模式是唯一定义的。 在该框架下，我们分别考虑了来自源和势的散射。 据我们所知，该理论框架是文献中新的，并为双曲空间上的波散射的许多后续发展铺平了道路。 显示英文摘要

摘要： In this paper, we develop a theoretical framework for time-harmonic wave scattering on hyperbolic spaces. Using the limiting absorption principle (LAP), we derive the explicit forms of the ingoing and outgoing Green functions of the Helmholtz operator of hyperbolic spaces, and verify that they are the fundamental solutions. Then we establish accurate characterisations of the asymptotic behaviours of the Green functions and use them to establish the ingoing and outgoing radiation conditions, which are analogues to the Sommerfeld radiation conditions in the Euclidean setting. Moreover, we prove a Rellich's type theorem which guarantees that the scattered field as well as its far-field pattern are uniquely defined. Within the framework, we consider the scattering from a source and a potential respectively. To our best knowledge, the theoretical framework is new to the literature and it paves the way for many subsequent developments for wave scattering on hyperbolic spaces.