Parametrix construction and numerical approximation of resonances of azimuthal harmonics of the charged Klein-Gordon operator on general cosmological slowly accelerating and rotating charged black hole type spacetimes

TL;DR

This paper establishes the Fredholm property of the charged Klein-Gordon operator's azimuthal harmonics on complex black hole spacetimes and introduces a numerical method to compute their resonances with error estimates.

Contribution

It provides a novel parametrix construction for these operators and a numerical scheme for resonance computation on general cosmological black hole backgrounds.

Findings

Proved the index 0 Fredholm property for the spectral family.

Developed a numerical scheme with explicit error bounds for resonance computation.

Applied the method to the De Sitter-Kerr-Newman family.

Abstract

We show the index 0 Fredholm property of the spectral family of the azimuthal harmonics of the charged Klein-Gordon operator on general cosmological slowly accelerating and rotating charged black hole type spacetimes, including the De Sitter-Kerr-Newman family, using a parametrix construction for abstract totally characteristic operators. We then present a numerical scheme to compute the meromorphic poles of the inverse of the spectral family and provide an explicit estimate of the numerical error.