Scattering for the Wave Equation on de Sitter Space in All Even Spatial Dimensions

TL;DR

This paper develops a comprehensive scattering theory for the linear wave equation on even-dimensional de Sitter space, establishing existence, uniqueness, and asymptotic completeness of scattering states, and constructing an isomorphism between asymptotic data at past and future infinity.

Contribution

It introduces a novel scattering framework for the wave equation on de Sitter space in all even dimensions, including the construction of a scattering map as a Banach space isomorphism.

Findings

Proves existence and uniqueness of scattering states.

Constructs a scattering map as a Banach space isomorphism.

Establishes asymptotic completeness for the wave equation on de Sitter space.

Abstract

For any $n\geq4$ even, we establish a complete scattering theory for the linear wave equation on the $(n+1)$-dimensional de Sitter space. We prove the existence and uniqueness of scattering states, and asymptotic completeness. Moreover, we construct the scattering map taking asymptotic data at past infinity $\mathscr{I}^-$ to asymptotic data at future infinity $\mathscr{I}^+$. Identifying $\mathscr{I}^-$ and $\mathscr{I}^+$ with $S^n,$ we prove that the scattering map is a Banach space isomorphism on $H^{s+n}(S^n)\times H^{s}(S^n),$ for any $s\geq1.$ The main analysis is carried out at the level of the model equation obtained by differentiating the linear wave equation $\frac{n}{2}$ times in the time variable. The main result of the paper follows from proving a scattering theory for this equation. In particular, for the model equation we construct a scattering isomorphism from asymptotic data in $H^{s+\frac{1}{2}}(S^n)\times H^s(S^n)\times H^s(S^n)$ to Cauchy initial data in $H^{s+\frac{1}{2}}(S^n)\times H^{s+\frac{1}{2}}(S^n)\times H^{s-\frac{1}{2}}(S^n)$.