Spectral stability and pseudospectrum of Kerr QNMs

Summary

Spectral stability and pseudospectrum of Kerr QNMs

Why this matters

QNMs are not enough by themselves if non-normal transient amplification dominates some time windows.

Exact scope

Background / setting

Asymptotically flat four-dimensional general relativity unless the statement specifies otherwise.

Equation type

PDE level: spectral-operator.

Linearity

Includes or emphasizes nonlinear dynamics.

Regularity

Smooth / Sobolev classes as in the problem statement; tighten when citing a specific theorem.

Parameter regime

Subextremal Kerr (or Kerr–de Sitter where tagged); spectral parameters $(l,m)$ and frequency $ω$ regimes as in cited microlocal frameworks.

Asymptotics

asymptotically-flat

Gauge / formulation

State gauge/fixing class compatible with cited stability or interior programs (e.g. generalized harmonic, double-null interior charts).

Status explanation

Entry imported without two independent bibliographic pointers in this repository; treat theorem-level claims as unverified here.

Problem statement

Quantify how Kerr resonances and pseudospectra move under small perturbations of the geometry or operator and identify transient-growth mechanisms associated with non-normality.

What is already known

• Microlocal/resolvent frameworks yield decay and mode stability for waves on exact Kerr and Kerr–de Sitter under stated spectral assumptions.
  Regime: Linear waves; fixed background.
  Standard input for QNM expansions and superradiance discussions.
• Nonlinear Kerr stability is proved in a small-$|a|/M$ vacuum window (Klainerman–Szeftel).
  Regime: Nonlinear vacuum, restricted parameters.
  Closest nonlinear analogue for exterior stability conjectures.
• Complete QNM expansion as a spectral representation (including branch cuts) for Kerr remains an open mathematical framework problem.
  Regime: Spectral theory on Kerr.
  Distinguishes partial mode stability from full expansion/completeness.

Progress summary: Partial progress exists in adjacent regimes;

What remains open

A complete answer must give either stability theorems or explicit instability mechanisms for resonance motion and pseudospectral growth under a stated perturbation class.

Mathematical prerequisites

Pseudospectral theory; non-selfadjoint semiclassics; resolvent norm growth; perturbation of resonances.

Completion criteria

A complete answer must give either stability theorems or explicit instability mechanisms for resonance motion and pseudospectral growth under a stated perturbation class.

Implications if solved

Would sharpen the mathematical meaning of mode stability in non-selfadjoint black-hole problems.

Formal verification suitability

FV: medium

Some subquestions may formalize before the full statement.

See Formal verification for how this database uses these labels.

References

• primary Global analysis of linear waves on Kerr–de Sitter space — Hintz, Vasy (2016)
  Linear wave decay and spectral gap on Kerr–de Sitter; standard microlocal input for $Lambda>0$ decay.
• survey Brief introduction to the nonlinear stability of Kerr — Klainerman, Szeftel (2022)
  Program overview, gauge structure, and relation between linear tools and nonlinear stability.

Related problems

Related by shared tags

Heuristic matches on family, cluster, equation level, asymptotics, and relevance.

• K-507 — Finite-data inverse resonance recovery of Kerr parameters
• K-506 — High-frequency Kerr quasinormal-mode laws with explicit remainder bounds
• K-624 — Prove a sharp characterization of near-extremal QNM clustering with explicit remainders.

Editorial / maintainer notes

Open: no complete theorem matching the statement is currently recorded on this site. : tighten if community consensus differs.