The essential self-adjointness of the wave operator on radiative spacetimes

TL;DR

This paper establishes the essential self-adjointness of the wave operator on a broad class of radiative spacetimes, including perturbations of Minkowski space, using advanced microlocal analysis techniques.

Contribution

It extends previous results by proving self-adjointness for spacetimes with gravitational radiation perturbations, employing modern microlocal methods.

Findings

Proves essential self-adjointness of the wave operator on radiative spacetimes.

Shows all tempered distributions satisfying certain wave equations are Schwartz.

Utilizes advanced microlocal analysis techniques for the proof.

Abstract

We prove the essential self-adjointness of the d'Alembertian $\square_g$, allowing a larger class of spacetimes than previously considered, including those that arise from perturbing Minkowski spacetime by gravitational radiation. We emphasize the fact, proven by Taira in closely related settings, that all tempered distributions $u$ satisfying $\square_g u = \lambda u +f$ for $\lambda\in \mathbb{C}\backslash \mathbb{R}$ and $f$ Schwartz are Schwartz. The proof is fully microlocal and relatively quick given the ``de,sc-'' machinery recently developed by the third author.