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Spectral properties of surface-localized transmission eigenmodes and applications to inverse scattering problems

Yan Jiang, Hongyu Liu, Kai Zhang, Haoran Zheng

Abstract

This paper investigates a distinctive spectral pattern exhibited by transmission eigenfunctions in wave scattering theory. Building upon the discovery in [7, 8] that these eigenfunctions localize near the domain boundary, we derive sharp spectral density estimates--establishing both lower and upper bounds--to demonstrate that a significant proportion of transmission eigenfunctions manifest this surface-localizing behavior. Our analysis elucidates the connection between the geometric rigidity of eigenfunctions and their spectral properties. Though primarily explored within a radially symmetric framework, this study provides rigorous theoretical insights, advances new perspectives in this emerging field, and offers meaningful implications for inverse scattering theory.

Spectral properties of surface-localized transmission eigenmodes and applications to inverse scattering problems

Abstract

This paper investigates a distinctive spectral pattern exhibited by transmission eigenfunctions in wave scattering theory. Building upon the discovery in [7, 8] that these eigenfunctions localize near the domain boundary, we derive sharp spectral density estimates--establishing both lower and upper bounds--to demonstrate that a significant proportion of transmission eigenfunctions manifest this surface-localizing behavior. Our analysis elucidates the connection between the geometric rigidity of eigenfunctions and their spectral properties. Though primarily explored within a radially symmetric framework, this study provides rigorous theoretical insights, advances new perspectives in this emerging field, and offers meaningful implications for inverse scattering theory.

Paper Structure

This paper contains 11 sections, 14 theorems, 194 equations, 5 figures.

Table of Contents

  1. Physical origin and connection to existing studies
  2. Mathematical setup and statement of the main results
  3. Proof of main results: 2D case
  4. A uniform lower bound estimate in 2D
  5. Further comments on the lower bound estimate
  6. Case 1. $k \in I_1$.
  7. Case 3. $k \in I_3$.
  8. Upper bound of $\rho_{\tilde{\varepsilon}, \delta}$ in 2D
  9. Proof of main results: 3D case
  10. A uniform lower bound estimate in 3D
  11. Upper bound of $\rho_{\tilde{\varepsilon},\delta}$ in 3D

Key Result

Theorem 2.3

For given $0<\varepsilon<\frac{1}{2}$, $0<\delta<1$, and refractive index $0<\mathbf{n}<1$, we have Furthermore, the following dimension-specific bounds hold: where $P^{(2)}$ and $P^{(3)}$ are specified in eq:P2P3. Crucially, these lower bounds depend solely on $\mathbf{n}$ and are independent of $\varepsilon$ and $\delta$.

Figures (5)

  • Figure 1: The graph of transmission eigenfunctions $u$(left) and $v$(right) to system \ref{['eq:trans1']} associated with $\mathbf{n}=10$ and $k=101.84852668$.
  • Figure 2: The graph of the lower bound for $\rho_{\varepsilon, \delta}$ in 2D: $B_{L}^{(2)}(\mathbf{n})$.
  • Figure 3: The graph of the lower bound for $\rho_{\varepsilon, \delta}$ and upper bound for $\rho_{\tilde{\varepsilon}, \delta}$ in 2D.
  • Figure 4: The graph of the lower bound for $\rho_{\varepsilon, \delta}$ in 3D: $B_{L}^{(3)}(\mathbf{n})$.
  • Figure 5: The graph of the lower bound for $\rho_{\varepsilon, \delta}$ and upper bound for $\rho_{\tilde{\varepsilon}, \delta}$ in 3D.

Theorems & Definitions (35)

  • Definition 2.1
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3: Lower bound
  • Theorem 2.4: Upper bound
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Remark 3.1
  • Remark 3.2
  • ...and 25 more