径向复标度法分析：标量共振问题
Analysis of radial complex scaling methods: scalar resonance problems
摘要：
考虑径向复尺度/完全匹配层方法求解齐次外部区域中的标量共振问题。我们引入了一个新的抽象框架来分析域截断和离散化的收敛性。我们的理论需要相当小的假设上的缩放配置文件，包括仿射，光滑和无界的配置文件。我们报告了一个快速的技术来分析域截断的收敛性和一个更技术性的近似通过不合理的截断和离散。我们适应后者的技术，也涵盖所谓的确切的方法，不需要域截断。我们建立的结果包括特征值和特征函数的收敛速度。所介绍的框架是基于这样的思想：将区域截断解释为Galerkin逼近，将全纯弗雷德霍姆算子本征值逼近论应用于线性本征值问题，采用弱T-凸性和T-相容逼近的概念，构造一个合适的T-算子作为乘法算子，光滑其符号并应用离散交换子技术。
英文摘要：
We consider radial complex scaling/perfectly matched layer methods for scalar resonance problems in homogeneous exterior domains. We introduce a new abstract framework to analyze the convergence of domain truncations and discretizations. Our theory requires rather minimal assumptions on the scaling profile and includes affin, smooth and also unbounded profiles. We report a swift technique to analyze the convergence of domain truncations and a more technical one for approximations through simultaneaous truncation and discretization. We adapt the latter technique to cover also so-called exact methods which do not require a domain truncation. Our established results include convergence rates of eigenvalues and eigenfunctions. The introduced framework is based on the ideas to interpret the domain truncation as Galerkin approximation, to apply theory on holomorphic Fredholm operator eigenvalue approximation theory to a linear eigenvalue problem, to employ the notion of weak T-coercivity and T-compatible approximations, to construct a suitable T-operator as multiplicatin operator, to smooth its symbol and to apply the discrete commutator technique.
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