Analyticity of Quasinormal Modes in the Kerr and Kerr–de Sitter Spacetimes

Abstract

We prove that quasinormal modes (or resonant states) for linear wave equations in the subextremal Kerr and Kerr–de Sitter spacetimes are real analytic. The main novelty of this paper is the observation that the bicharacteristic flow associated to the linear wave equations for quasinormal modes with respect to a suitable Killing vector field has a stable radial point source/sink structure rather than merely a generalized normal source/sink structure. The analyticity then follows by a recent result in the microlocal analysis of radial points by Galkowski and Zworski. The results can then be recast with respect to the standard Killing vector field.

Stationarity and Fredholm Theory in Subextremal Kerr-de Sitter Spacetimes

“…we conclude that ker(P σ ) is non-trivial precisely on a discrete set A ⊂ C. Following the arguments in the proof of[7, Theorem 1.2] line by line, using Theorem 3.1 in place of the [8, Theorem 1.1], it follows that the elements in ker(P σ ) are real analytic if the coefficients of P are real analytic. ■Proof of Theorem 1.13.…”

“…The bicharacteristic flow at the event horizon is, however, slightly different. Though a similar computation was already done in the proof of [7,Theorem 1.7], let us explicitly compute the Hamiltonian vector field at N * {r = r e } in case r 0 = r e . The principal symbol p σ of P σ is given by In particular, P σ is invertible everywhere in Ω s apart form a discrete set.…”

“…For our application to wave equations, we rely on microlocal estimates near normally hyperbolic trapping in the sense of Wunsch and Zworski in [9], see also the improved results by Dyatlov in [1,2,3]. For more references on results related to this paper, we refer to the introductions in [6,7].…”