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

Lu Chen, Hongyu Liu

TL;DR

The paper develops a complete time-harmonic scattering theory on hyperbolic space by transferring the Sommerfeld-Rellich-Colton–Kress (SRCK) framework to a curved setting. It explicitly constructs outgoing/ingoing Green's functions for the Helmholtz operator $-\\ riangle_{\\bbB^n}-\frac{(n-1)^2}{4}-\mu^2$ and derives a hyperbolic Sommerfeld radiation condition at conformal infinity, together with a Rellich-type uniqueness theorem. Direct scattering problems for compact sources, penetrable media, and impenetrable obstacles are solved with precise far-field representations, and two inverse problems are formulated: obstacle reconstruction and potential reconstruction from far-field data. The work establishes hyperbolic analogues of SRCK components, density theorems, and foundational inverse results, with natural extensions to asymptotically hyperbolic manifolds relevant to geometric inverse problems and AdS/CFT contexts.

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 f{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 provides the essential theoretical underpinning for a far-field-based approach to scattering and inversion in hyperbolic geometry and lays the groundwork for extensions to asymptotically hyperbolic manifolds.

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

TL;DR

The paper develops a complete time-harmonic scattering theory on hyperbolic space by transferring the Sommerfeld-Rellich-Colton–Kress (SRCK) framework to a curved setting. It explicitly constructs outgoing/ingoing Green's functions for the Helmholtz operator and derives a hyperbolic Sommerfeld radiation condition at conformal infinity, together with a Rellich-type uniqueness theorem. Direct scattering problems for compact sources, penetrable media, and impenetrable obstacles are solved with precise far-field representations, and two inverse problems are formulated: obstacle reconstruction and potential reconstruction from far-field data. The work establishes hyperbolic analogues of SRCK components, density theorems, and foundational inverse results, with natural extensions to asymptotically hyperbolic manifolds relevant to geometric inverse problems and AdS/CFT contexts.

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 f{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 provides the essential theoretical underpinning for a far-field-based approach to scattering and inversion in hyperbolic geometry and lays the groundwork for extensions to asymptotically hyperbolic manifolds.
Paper Structure (32 sections, 14 theorems, 316 equations)

This paper contains 32 sections, 14 theorems, 316 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. The time-harmonic scattering theory in Euclidean space
  4. Direct scattering problems
  5. Inverse scattering problems
  6. Fourier analysis on hyperbolic space
  7. Geometric models of hyperbolic space
  8. Hyperboloid model
  9. Poincaré ball model
  10. Half-space model
  11. Foliations by totally geodesic hypersurfaces
  12. Fourier analysis on the Poincaré ball $\mathbb{B}^n$
  13. Mobius transformations on $\mathbb{B}^n$
  14. Fourier transform on $\mathbb{B}^n$
  15. Green's formulas on bounded hyperbolic domains
  16. ...and 17 more sections

Key Result

Lemma 3.1

For any $u\in C_c^\infty(\mathbb{B}^n)$, Equality holds only for $u\equiv 0$.

Theorems & Definitions (33)

  • Lemma 3.1: Poincaré inequality on $\mathbb{B}^n$
  • proof
  • Remark 3.2
  • Lemma 3.3: Green's identities on a hyperbolic ball
  • proof
  • Remark 4.1
  • Theorem 4.1: Fundamental solutions of the Helmholtz operator
  • proof
  • Remark 4.2
  • Theorem 4.2
  • ...and 23 more