ANALYSIS OF RADIAL COMPLEX SCALING METHODS: SCALAR RESONANCE PROBLEMS

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 affine, smooth, and unbounded profiles. We report a swift technique to analyze the convergence of domain truncations and a more technical one for approximations through simultaneous truncation and discretization. Our established results include convergence rates of eigenvalues and eigenfunctions. The framework introduced is based on the following 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 multiplication operator, to smooth its symbol, and to apply the discrete commutator technique.

Automated summary

The study explores methods for solving scalar resonance problems in homogeneous exterior domains, introducing a new abstract framework for analyzing the convergence of domain truncations and discretizations with minimal requirements on scaling profiles. The established results include convergence rates of eigenvalues and eigenfunctions, based on interpreting domain truncations as Galerkin approximation and utilizing various theoretical concepts.