Analysis of radial complex scaling methods: scalar resonance problems

TL;DR

This paper develops a new abstract framework for analyzing the convergence of radial complex scaling methods used in scalar resonance problems, accommodating various profiles and including exact methods without domain truncation.

Contribution

It introduces a minimal-assumption, comprehensive framework for convergence analysis of radial complex scaling methods, covering a wide range of profiles and exact techniques.

Findings

Established convergence rates for eigenvalues and eigenfunctions.

Unified analysis approach for domain truncation and discretization.

Extended framework to include exact methods without domain truncation.

Abstract

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.