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Threshold asymptotics and decay for massive Maxwell on subextremal Reissner--Nordström

Bobby Eka Gunara

Abstract

We study the neutral massive Maxwell (Proca) equation on subextremal Reissner--Nordström exteriors. After spherical-harmonic decomposition, the odd sector is scalar, while the even sector remains a genuinely coupled $2\times2$ system. Our starting point is that this even system admits an exact asymptotic polarization splitting at spatial infinity. The three resulting channels carry effective angular momenta $\ell-1$, $\ell$, and $\ell+1$, and these are precisely the indices that govern the late-time thresholds. % For each fixed angular momentum we develop a threshold spectral theory for the cut-off resolvent. We prove meromorphic continuation across the massive branch cut, rule out upper-half-plane modes and threshold resonances, and obtain explicit small- and large-Coulomb expansions for the branch-cut jump. Inverting this jump yields polarization-resolved intermediate tails together with the universal very-late $t^{-5/6}$ branch-cut law. % At the full-field level, high-order angular regularity allows us to sum the modewise leading terms on compact radial sets and obtain a two-regime asymptotic expansion for the radiative branch-cut component of the Proca field, with explicit coefficient fields and quantitative remainders. We also analyze the quasibound resonance branches created by stable timelike trapping, prove residue and reconstruction bounds, and derive a fully self-contained dyadic packet estimate. As a result, the unsplit full Proca field obeys logarithmic compact-region decay, while the radiative branch-cut contribution retains explicit polynomial asymptotics and explicit leading coefficients.

Threshold asymptotics and decay for massive Maxwell on subextremal Reissner--Nordström

Abstract

We study the neutral massive Maxwell (Proca) equation on subextremal Reissner--Nordström exteriors. After spherical-harmonic decomposition, the odd sector is scalar, while the even sector remains a genuinely coupled system. Our starting point is that this even system admits an exact asymptotic polarization splitting at spatial infinity. The three resulting channels carry effective angular momenta , , and , and these are precisely the indices that govern the late-time thresholds. % For each fixed angular momentum we develop a threshold spectral theory for the cut-off resolvent. We prove meromorphic continuation across the massive branch cut, rule out upper-half-plane modes and threshold resonances, and obtain explicit small- and large-Coulomb expansions for the branch-cut jump. Inverting this jump yields polarization-resolved intermediate tails together with the universal very-late branch-cut law. % At the full-field level, high-order angular regularity allows us to sum the modewise leading terms on compact radial sets and obtain a two-regime asymptotic expansion for the radiative branch-cut component of the Proca field, with explicit coefficient fields and quantitative remainders. We also analyze the quasibound resonance branches created by stable timelike trapping, prove residue and reconstruction bounds, and derive a fully self-contained dyadic packet estimate. As a result, the unsplit full Proca field obeys logarithmic compact-region decay, while the radiative branch-cut contribution retains explicit polynomial asymptotics and explicit leading coefficients.
Paper Structure (48 sections, 67 theorems, 433 equations)

This paper contains 48 sections, 67 theorems, 433 equations.

Table of Contents

  1. Introduction and main results
  2. Background and scope
  3. Relation to the literature and the main innovation
  4. Notation and conventions
  5. Main results
  6. Self-contained summation and proof architecture
  7. Organization of the paper
  8. Proof dependency diagram
  9. Appendix guide
  10. Geometry, mode reduction, and polarization splitting
  11. Geometry and the field equation
  12. Reduced odd/even mode equations
  13. Exact asymptotic polarization diagonalization
  14. Operator-theoretic framework and limiting absorption
  15. Channel Hamiltonians and asymptotic operator structure
  16. ...and 33 more sections

Key Result

Theorem 1.1

Fix a subextremal Reissner--Nordström exterior eq:intro_subextremal and an angular momentum $\ell\ge1$. For each polarization $P\in\{-1,0,+1\}$ there exists a threshold index such that the following hold.

Theorems & Definitions (137)

  • Theorem 1.1: Fixed-mode spectral and threshold theorem
  • Theorem 1.2: Modewise branch-cut tails on subextremal Reissner--Nordström
  • Remark 1.1
  • Theorem 1.3: Uniform angular summability of branch-cut kernels
  • Theorem 1.4: Full-field pointwise branch-cut decay
  • Corollary 1.5: Polynomial time decay for the radiative Proca field
  • Theorem 1.6: Mode stability and threshold exclusion
  • Theorem 1.7: Quasibound branches and Bohr--Sommerfeld quantization
  • Theorem 1.8: Residue projectors and Agmon localization
  • Theorem 1.9: Self-contained summed quasibound decay
  • ...and 127 more