Prove sharp semiclassical quantization of Kerr QNMs with explicit high-frequency error bounds.

Summary

Prove sharp semiclassical quantization of Kerr QNMs with explicit high-frequency error bounds.

Why this matters

Improves QNM asymptotics into fully controlled mathematics.

Exact scope

Background / setting

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

Equation type

PDE level: spectral-operator.

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

Prove sharp semiclassical quantization of Kerr QNMs with explicit high-frequency error bounds.

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: Improves QNM asymptotics into fully controlled mathematics.

What remains open

Prove sharp semiclassical quantization of Kerr QNMs with explicit high-frequency error bounds.

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-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.
• K-625 — Prove completeness/expansion of solutions in Kerr via QNMs plus branch-cut contributions (mathematical ringdown expansion).
• K-627 — Establish pseudospectrum bounds for Kerr wave operators and relate to transient growth near superradiance.
• K-661 — Determine whether QNM expansions are stable under small nonlinearities (nonlinear resonance theory).
• K-681 — Develop a mathematically rigorous definition of nonlinear QNMs as poles of a suitable nonlinear response functional.
• K-682 — Prove that Kerr QNMs are stable under small perturbations of the metric in a topology relevant to stability proofs.

Editorial / maintainer notes

Source manifest: N-036 (expansion_from_manifest.tsv). Numeric footnotes from the original table are not reproduced in this repository.