Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finiteness of Nonscattering Wavenumbers for Herglotz Incident Waves

Jingni Xiao

TL;DR

This work studies the finiteness of nonscattering wavenumbers for 2D inhomogeneous media under Herglotz incident waves. It develops a boundary-integral identity and reduces the nonscattering condition to oscillatory boundary integrals whose large-$k$ behavior is controlled via stationary-phase analysis. The authors prove finiteness for ellipses with a focus at the origin (including the $q>1$ regime) and for admissible $C^2$ star-shaped domains with $q\in(0,1)$ under broad geometric conditions on the boundary via a Vandermonde-system argument. A geometric rigidity principle emerges: infinite nonscattering sequences require exact radial symmetry and are destroyed by admissible perturbations of the boundary. The techniques combine boundary integral identities, explicit stationary-point analysis, and algebraic Vandermonde arguments to link spectral finiteness to geometric admissibility.

Abstract

This paper continues the study initiated in [30] on nonscattering phenomena for inhomogeneous media. We investigate star-shaped domains in $\mathbb{R}^2$ and establish finiteness results for nonscattering wavenumbers associated with Herglotz incident waves of fixed density. First, for ellipses with constant medium coefficient $q\in(0,1)\cup(1,\infty)$, we prove that there exist at most finitely many nonscattering wavenumbers. This generalizes and strengthens the corresponding results in [30], in particular removing additional geometric restrictions in the case $q>1$. Second, for admissible $C^2$ star-shaped domains with $q\in(0,1)$, we establish analogous finiteness results under broader geometric assumptions on the radius function. Our results reveal that infinite sequences of nonscattering wavenumbers are tied to exact radial symmetry and cannot persist under admissible geometric perturbations.

Finiteness of Nonscattering Wavenumbers for Herglotz Incident Waves

TL;DR

This work studies the finiteness of nonscattering wavenumbers for 2D inhomogeneous media under Herglotz incident waves. It develops a boundary-integral identity and reduces the nonscattering condition to oscillatory boundary integrals whose large- behavior is controlled via stationary-phase analysis. The authors prove finiteness for ellipses with a focus at the origin (including the regime) and for admissible star-shaped domains with under broad geometric conditions on the boundary via a Vandermonde-system argument. A geometric rigidity principle emerges: infinite nonscattering sequences require exact radial symmetry and are destroyed by admissible perturbations of the boundary. The techniques combine boundary integral identities, explicit stationary-point analysis, and algebraic Vandermonde arguments to link spectral finiteness to geometric admissibility.

Abstract

This paper continues the study initiated in [30] on nonscattering phenomena for inhomogeneous media. We investigate star-shaped domains in and establish finiteness results for nonscattering wavenumbers associated with Herglotz incident waves of fixed density. First, for ellipses with constant medium coefficient , we prove that there exist at most finitely many nonscattering wavenumbers. This generalizes and strengthens the corresponding results in [30], in particular removing additional geometric restrictions in the case . Second, for admissible star-shaped domains with , we establish analogous finiteness results under broader geometric assumptions on the radius function. Our results reveal that infinite sequences of nonscattering wavenumbers are tied to exact radial symmetry and cannot persist under admissible geometric perturbations.
Paper Structure (9 sections, 17 theorems, 161 equations, 2 figures)

This paper contains 9 sections, 17 theorems, 161 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main Results
  3. Preliminaries
  4. Ellipses
  5. Star-Shaped Domain
  6. Stationary Points
  7. Proof of the Main Results
  8. Proof of Theorem \ref{['thm:StarAnly']}
  9. Proof of Theorem \ref{['thm:main1Star']}

Key Result

Theorem 1

Let $\Omega$ be an ellipse whose one focus is located at the origin. Let $\phi\in C^1(\mathbb{S}^1)$ be nontrivial. Assume that either Then for the medium $(q;\Omega)$, the set of nonscattering wavenumbers with respect to the incident wave $u^i=\mathbf{H}[k,\phi]$ is at most finite.

Figures (2)

  • Figure 1: Graphs of $h$, $h'$, and $h"$, when $q=0.1$ ($q\le1/4$).
  • Figure 2: Graphs of $h$, $h'$, and $h"$, when $q=0.8$ ($1>q>1/4$).

Theorems & Definitions (33)

  • Theorem 1
  • Remark 1.1
  • Definition 1
  • Remark 1.2
  • Theorem 2
  • Remark 1.3
  • Remark 1.4
  • Theorem 3
  • Remark 1.5
  • Lemma 1.1
  • ...and 23 more