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A Rellich-type theorem for the Helmholtz equation in a junction of stratified media

Sarah Al Humaikani, Anne-Sophie Bonnet-Ben Dhia, Sonia Fliss, Christophe Hazard

TL;DR

This work proves a Rellich-type uniqueness theorem for the Helmholtz equation in unbounded domains formed by junctions of three stratified half-planes. The authors develop a generalized Fourier framework tailored to stratified media, introducing generalized eigenfunctions and a corresponding half-plane representation to decompose $L^2$ solutions into modal components. By establishing analyticity properties of the boundary traces with respect to a complex spectral parameter and exploiting unique continuation, they show that no nontrivial $L^2$ solutions exist when the angular gaps between branches are at least $\pi/2$, implying absence of trapped modes at the junction. The analysis extends from two-layered to general stratified media, carefully handling potential $L^2$ eigenfunctions and meromorphic continuations of scattering data. The results provide a robust obstruction to guided/trapped waves in such junctions and connect to broader Rellich-type uniqueness and spectral theory in exterior-like stratified domains.

Abstract

We prove that there are no non-zero square-integrable solutions to a two-dimensional Helmholtz equation in some unbounded inhomogeneous domains which represent junctions of stratified media. More precisely, we consider domains that are unions of three half-planes, where each half-plane is stratified in the direction orthogonal to its boundary. As for the well-known Rellich uniqueness theorem for a homogeneous exterior domain, our result does not require any boundary condition. Our proof is based on half-plane representations of the solution which are derived through a generalization of the Fourier transform adapted to stratified media. A byproduct of our result is the absence of trapped modes at the junction of open waveguides as soon as the angles between branches are greater than $π$/2.

A Rellich-type theorem for the Helmholtz equation in a junction of stratified media

TL;DR

This work proves a Rellich-type uniqueness theorem for the Helmholtz equation in unbounded domains formed by junctions of three stratified half-planes. The authors develop a generalized Fourier framework tailored to stratified media, introducing generalized eigenfunctions and a corresponding half-plane representation to decompose solutions into modal components. By establishing analyticity properties of the boundary traces with respect to a complex spectral parameter and exploiting unique continuation, they show that no nontrivial solutions exist when the angular gaps between branches are at least , implying absence of trapped modes at the junction. The analysis extends from two-layered to general stratified media, carefully handling potential eigenfunctions and meromorphic continuations of scattering data. The results provide a robust obstruction to guided/trapped waves in such junctions and connect to broader Rellich-type uniqueness and spectral theory in exterior-like stratified domains.

Abstract

We prove that there are no non-zero square-integrable solutions to a two-dimensional Helmholtz equation in some unbounded inhomogeneous domains which represent junctions of stratified media. More precisely, we consider domains that are unions of three half-planes, where each half-plane is stratified in the direction orthogonal to its boundary. As for the well-known Rellich uniqueness theorem for a homogeneous exterior domain, our result does not require any boundary condition. Our proof is based on half-plane representations of the solution which are derived through a generalization of the Fourier transform adapted to stratified media. A byproduct of our result is the absence of trapped modes at the junction of open waveguides as soon as the angles between branches are greater than /2.

Paper Structure

This paper contains 15 sections, 8 theorems, 113 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Motivation and main result
  3. Sketch of the proof for a homogeneous medium
  4. Fourier half-plane representation for a two-layered medium
  5. Generalized eigenfunctions
  6. Generalized Fourier transform
  7. Deriving the half-plane representation
  8. Junction of two-layered media
  9. Right-angle configuration
  10. General angle configuration
  11. Extension to the general stratification case
  12. Generalized Fourier transform
  13. Half-space representation
  14. Proof of \ref{['Theorem: main']}
  15. Conclusion

Key Result

Theorem 1.1

\newlabelTheorem: main0 Considering $\Omega\subset \mathbb{R}^2$ and a function $k\in L^\infty(\Omega)$ as described above, if $u\in L^2(\Omega)$ satisfies the Helmholtz equation in the distributional sense, then $u=0$ in $\Omega$.

Figures (6)

  • Figure 1: The two considered configurations. (a): right-angle, (b): general angle. The light gray represents various constant values of $k$, whereas the dark gray stands for possible stratifications.
  • Figure 1: Behavior of the generalized eigenfunctions in the case where $\textsc{k}_{-} > \textsc{k}_{+}$.
  • Figure 1: Junctions of two-layered media.
  • Figure 2: Examples of situations where \ref{['Theorem: main']} can be deduced from existing results.
  • Figure 2: Relevant domain of analyticity of $\widehat{\varphi}_\textsc{n}^+$ (gray area) for $\textsc{k}_{\textsc{n},+}=10$, $\textsc{k}_{\textsc{n},-}=9$, $\theta_\textsc{e}=-5\pi/6$, $\theta_\textsc{w}=2\pi/3$.
  • ...and 1 more figures

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • Proof 1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 2.5
  • Remark 2.6
  • Proof 2
  • ...and 10 more