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

TL;DR

This paper studies the spectral behavior of surface-localized transmission eigenmodes in wave scattering, providing bounds and insights that enhance understanding of their geometric and inverse scattering properties.

Contribution

It derives sharp spectral density estimates for transmission eigenfunctions, revealing their boundary localization and connecting spectral properties with geometric rigidity.

Findings

Many eigenfunctions localize near the boundary

Established bounds on spectral density of eigenfunctions

Implications for inverse scattering theory

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.