1305.1723 (Semyon Dyatlov)

Asymptotics of linear waves and resonances with applications to black
holes    [PDF]

Semyon Dyatlov

We apply the results of arXiv:1301.5633 to describe asymptotic behavior of linear waves on stationary Lorentzian metrics with r-normally hyperbolic trapped sets, in particular Kerr and Kerr-de Sitter metrics with |a|> 1, then the energy norm of the solution is bounded by O(\lambda^{1/2} exp(-(\nu_min - \epsilon)t/2) + \lambda^(-\infty)), for t < C log\lambda, where \nu_min is a natural dynamical quantity. The key tool is a microlocal projector splitting the solution into a component with controlled rate of exponential decay and an O(\lambda exp(-(\nu_min -\epsilon)t) + \lambda^(-\infty)) remainder; this splitting can be viewed as an analog of resonance expansion. Moreover, for the Kerr-de Sitter case we study quasi-normal modes; under a dynamical pinching condition, a Weyl law in a band holds.

View original: http://arxiv.org/abs/1305.1723