A generalization of the propagation of singularities theorem on asymptotically anti-de Sitter spacetimes

TL;DR

This paper extends the propagation of singularities theorem to a broader class of boundary conditions on asymptotically anti-de Sitter spacetimes, using advanced microlocal analysis techniques.

Contribution

It generalizes previous results by incorporating pseudodifferential boundary conditions and employing $b$-calculus and twisted Sobolev spaces.

Findings

Proves propagation of singularities along generalized broken bicharacteristics.

Highlights the influence of pseudodifferential boundary operators on singularity propagation.

Extends the mathematical framework for analyzing wave equations in complex spacetime geometries.

Abstract

In a recent paper O. Gannot and M. Wrochna considered the Klein-Gordon equation on an asymptotically anti-de Sitter spacetime subject to Robin boundary conditions, proving in particular a propagation of singularity theorem. In this work we generalize their result considering a more general class of boundary conditions implemented on the conformal boundary via pseudodifferential operators of suitable order. Using techniques proper of $b$-calculus and of twisted Sobolev spaces, we prove also for the case in hand a propagation of singularity theorem along generalized broken bicharacteristics, highlighting the potential presence of a contribution due to the pseudodifferential operator encoding the boundary condition.