An inversion formula for the X-ray normal operator over closed hyperbolic surfaces

TL;DR

This work addresses recovering a function from the X-ray transform on closed hyperbolic surfaces by deriving an explicit inversion for Guillarmou's normal operator. The authors reduce the problem to an attenuated inversion on the Poincaré disk, formalize the limit via meromorphic resolvent theory and Anosov dynamics, and then transport the result to closed manifolds through metric normalization and covering arguments. The main achievement is the explicit inversion formula $\Delta S_K \Pi_0 f = -8\pi^2 f$ for mean-zero $f$, together with an explicit construction of invariant distributions with prescribed pushforward, which strengthens connections to inverse problems and spectral rigidity in hyperbolic geometry. The approach blends geometric analysis on the Poincaré disk with microlocal and representation-theoretic tools to extend Helgason-type inversion formulas to the Anosov setting, and sets the stage for potential extensions to variable curvature and approximate inversions in more general geometries.

Abstract

We construct an explicit inversion formula for Guillarmou's normal operator on closed surfaces of constant negative curvature. This normal operator can be defined as a weak limit for an "attenuated normal operator", and we prove this inversion formula by first constructing an additional inversion formula for this attenuated normal operator on both the Poincaré disk and closed surfaces of constant negative curvature. A consequence of the inversion formula is the explicit construction of invariant distributions with prescribed pushforward over closed hyperbolic manifolds.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• proof
• proof : Proof of Theorem \ref{['thm:D-att-inv']}
• Lemma 5
• proof
• Lemma 6
• proof