Quasimodes and a Lower Bound on the Uniform Energy Decay Rate for Kerr-AdS Spacetimes

TL;DR

This paper constructs quasimodes for the Klein-Gordon equation on Kerr-AdS spacetimes, demonstrating that solutions cannot decay faster than logarithmically, thus establishing a lower bound on decay rates.

Contribution

The authors develop a semi-classical eigenvalue problem approach to construct quasimodes with exponentially small errors, providing a lower bound on energy decay rates in Kerr-AdS backgrounds.

Findings

Solutions decay at most logarithmically

Constructed quasimodes with exponentially small errors

Established a lower bound on decay rate

Abstract

We construct quasimodes for the Klein-Gordon equation on the black hole exterior of Kerr-Anti-de Sitter (Kerr-AdS) spacetimes. Such quasi-modes are associated with time-periodic approximate solutions of the Klein Gordon equation and provide natural candidates to probe the decay of solutions on these backgrounds. They are constructed as the solutions of a semi-classical non-linear eigenvalue problem arising after separation of variables, with the (inverse of the) angular momentum playing the role of the semi-classical parameter. Our construction results in exponentially small errors in the semi-classical parameter. This implies that general solutions to the Klein Gordon equation on Kerr-AdS cannot decay faster than logarithmically. The latter result completes previous work by the authors, where a logarithmic decay rate was established as an upper bound.