The Sommerfeld-Rellich Framework for Scattering on Hyperbolic Space: Far-Field Patterns and Inverse Problems

Abstract

This paper establishes a complete time-harmonic scattering theory for hyperbolic space, formulating it within the classical Sommerfeld-Rellich paradigm centered on far-field patterns--a foundational framework that has been absent despite the well-developed spectral and time-dependent theories for this geometry. We explicitly construct the ingoing and outgoing fundamental solutions for the Helmholtz operator and perform a precise asymptotic analysis at the conformal boundary to derive a {hyperbolic Sommerfeld radiation condition}. This condition, which is a local criterion at infinity, uniquely selects physically admissible outgoing solutions. We prove a {hyperbolic Rellich theorem} guaranteeing the uniqueness of the scattered field and its far-field pattern from asymptotic data. Within this rigorous framework, we solve the direct scattering problem for compact sources, penetrable media, and impenetrable obstacles, providing explicit representations for the corresponding {far-field patterns}. As a principal application and demonstration of the framework's utility, we initiate the study of inverse scattering on hyperbolic space, formulating both the inverse obstacle and inverse medium problems where the objective is to reconstruct the scatterer from measurements of its far-field pattern. Our work lays the groundwork for a far-field-based approach to scattering and inversion in hyperbolic geometry. Due to the local feature of the {hyperbolic Sommerfeld radiation condition}, our study can be readily extended to the broader asymptotically hyperbolic manifolds.