The wave equation on asymptotically de Sitter-like spaces

TL;DR

This paper studies the asymptotic behavior of solutions to the Klein-Gordon equation on Lorentzian manifolds resembling de Sitter space at infinity, defining a scattering operator linked to the bicharacteristic flow.

Contribution

It characterizes the asymptotics of solutions and introduces a scattering operator as a Fourier integral operator in de Sitter-like spaces.

Findings

Solutions exhibit specific asymptotic behavior at infinity.

The scattering operator is a Fourier integral operator.

Bicharacteristic flow determines the scattering dynamics.

Abstract

In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds $(X^\circ,g)$ which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y+ and Y-, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to plus infinity, and to the other manifold as the parameter goes to minus infinity, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y+ to Y-.