Real analyticity of radiation patterns of resonant states on asymptotically hyperbolic manifolds

TL;DR

This paper proves that resonant states on asymptotically hyperbolic manifolds have analytic radiation patterns at infinity, and that solutions to Vasy operators with analytic coefficients are also analytic, confirming a conjecture by Zworski.

Contribution

It establishes the analyticity of radiation patterns for resonant states and solutions to Vasy operators on asymptotically hyperbolic manifolds, answering a conjecture by Zworski.

Findings

Resonant states have analytic radiation patterns at infinity.

Solutions to Vasy operators with analytic coefficients are analytic.

Confirms Zworski's conjecture on analyticity.

Abstract

We show that resonant states in scattering on asymptotically hyperbolic man-ifolds that are analytic near conformal infinity, have analytic radiation patterns at infinity. On even asymptotically hyperbolic manifolds we also show that smooth solutions of Vasy operators with analytic coefficients are also analytic. That answer a question of M.Zworski ([14] Conjecture 2). The proof is based on previous results of Baouendi-Goulaouic and Bolley-Camus-Hanouzet and for convenience of the reader we present an outline of the proof of the latter.