On the Fourier analysis of the Einstein-Klein-Gordon system: Growth and Decay of the Fourier constants
Résumé
We consider the $(1+3)-$dimensional Einstein equations with negative cosmological constant coupled to a spherically-symmetric, massless scalar field and study perturbations around the Anti-de-Sitter solution. We derive the resonant systems, pick out vanishing secular terms and discuss issues related to small divisors. Most importantly, we rigorously establish (sharp, in most of the cases) asymptotic behaviour for all the interaction coefficients. The latter is based on uniform estimates for the eigenfunctions associated to the linearized operator and their first order derivatives as well as on oscillating integrals.