The pseudospectrum for the Kerr black hole with spin $s=0$ case

TL;DR

This paper studies the pseudospectrum of Kerr black holes with zero spin, revealing how spectral instability varies with different norms and the imaginary part of frequencies, using a hyperboloidal framework and Chebyshev collocation method.

Contribution

It introduces a novel approach to analyze the pseudospectrum of Kerr black holes with spin zero, employing a hyperboloidal framework and convergence analysis with various norms.

Findings

Pseudospectrum convergence improves with higher derivative norms.

Increasing imaginary parts of frequencies can impair pseudospectrum convergence.

The hyperboloidal framework effectively transforms the QNM problem into a two-dimensional eigenvalue problem.

Abstract

We investigate the pseudospectrum of the Kerr black hole, which indicates the instability of the spectrum of quasinormal modes (QNMs) of the Kerr black hole. Methodologically, we use the hyperboloidal framework to cast the QNM problem into a two-dimensional eigenvalue problem associated with a non-self-adjoint operator, and then the spectrum and pseudospectrum are solved by imposing the two-dimensional Chebyshev collocation method. The (energy) norm is constructed by using the conserved current method for the spin $s=0$ case. For the finite rank approximation of the operator, we discuss the convergence of pseudospectra using various norms, each involving different orders of derivatives. The convergence of the pseudospectrum improves as the order of the derivatives increases. We find that an increase in the imaginary part of complex frequency can deteriorate the convergence of the pseudospectrum under the condition of the same norms.