Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime

TL;DR

This paper provides a quantitative analysis of mode stability for the wave equation on Kerr spacetimes, offering simplified proofs and controlling energy fluxes to demonstrate decay properties across the entire sub-extremal range.

Contribution

It introduces a refined, quantitative approach to mode stability on Kerr backgrounds with simplified proofs and extends energy flux control to all bounded-frequency solutions.

Findings

Quantitative control of energy flux along horizon and null infinity

Establishment of integrated local energy decay in bounded-frequency regimes

Simplified proofs of mode stability for Kerr spacetimes

Abstract

We give a quantitative refinement and simple proofs of mode stability type statements for the wave equation on Kerr backgrounds in the full sub-extremal range (|a| < M). As an application, we are able to quantitatively control the energy flux along the horizon and null infinity and establish integrated local energy decay for solutions to the wave equation in any bounded-frequency regime.