Finite codimension stability of some time-periodic hyperbolic equations (via compact resolvents)

TL;DR

This paper demonstrates that certain time-periodic hyperbolic equations are finite codimension stable due to the associated operator having a compact resolvent, linking abstract theory to numerical observations in general relativity.

Contribution

It introduces a class of hyperbolic equations with stability properties characterized by compact resolvents, connecting abstract operator theory to numerical relativity phenomena.

Findings

Identification of a class of hyperbolic equations with finite codimension stability

Establishment of the link between compact resolvent and stability in these equations

Application of abstract operator theory to phenomena observed in numerical relativity

Abstract

We identify a class of time-periodic linear symmetric hyperbolic equations that are finite codimension stable, because an associated operator has compact resolvent, sufficiently far to the right in the complex plane. This paper is an attempt to capture abstractly the observation in numerical general relativity that some discretely self-similar spacetimes, such as Choptuik's critical spacetime, are finite codimension stable.