Develop a rigorous theory of excitation factors in Kerr and prove universal bounds across (l,m).

Summary

Develop a rigorous theory of excitation factors in Kerr and prove universal bounds across (l,m).

Why this matters

Needed for universal ringdown amplitude claims.

Exact scope

Background / setting

asymptotically flat general relativity context; see family and coupling tags for matter model.

Equation type

PDE level: full-einstein.

Linearity

linearized

Regularity

Smooth / Sobolev hypotheses must be stated precisely in any final theorem; this provisional entry does not fix minimal regularity.

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

Theorem status follows literature as summarized in known results and references (not upgraded without verified solution pointers).

Problem statement

Develop a rigorous theory of excitation factors in Kerr and prove universal bounds across (l,m).

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: Context: Needed for universal ringdown amplitude claims.

What remains open

Develop a rigorous theory of excitation factors in Kerr and prove universal bounds across (l,m).

Mathematical prerequisites

Match hypotheses to primary sources cited on this page; state minimal regularity, gauge class, and parameter windows in any claimed theorem.

Completion criteria

Prove a theorem or give a rigorous counterexample that matches the scoped statement under explicitly listed hypotheses.

Implications if solved

Impact depends on the solved formulation; sharpen once the statement is pinned to a literature-compatible theorem.

Formal verification suitability

FV: low

Global PDE or phenomenological target; lemma-level formalization may be possible after scoping.

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-603 — Prove nonlinear stability of Kerr with quantitative scattering (asymptotic completeness near Kerr).
• K-660 — Establish a full nonlinear ringdown plus tail decomposition for near-Kerr vacuum spacetimes.
• K-683 — Prove a sharp characterization of the Kerr trapped set as a normally hyperbolic invariant manifold uniformly in a/M.
• 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

Source manifest: N-062 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.