Surface concentration of transmission eigenfunctions

TL;DR

This paper investigates how transmission eigenfunctions in wave scattering problems tend to concentrate on the boundary surface of the domain, providing theoretical support for observed numerical phenomena.

Contribution

It establishes the geometric rigidity and boundary concentration of transmission eigenfunctions using generalized Weyl's law and ergodic properties of boundary operators.

Findings

Transmission eigenfunctions concentrate on the boundary surface.

Supports numerical observations of spectral phenomena.

Provides theoretical framework for boundary concentration.

Abstract

The transmission eigenvalue problem is a type of non-elliptic and non-selfadjoint spectral problem that arises in the wave scattering theory when invisibility/transparency occurs. The transmission eigenfunctions are the interior resonant modes inside the scattering medium. We are concerned with the geometric rigidity of the transmission eigenfunctions and show that they concentrate on the boundary surface of the underlying domain in two senses. This substantiates the recent numerical discovery in [10] on such an intriguing spectral phenomenon of the transmission resonance. Our argument is based on generalized Weyl's law and certain novel ergodic properties of the coupled boundary layer-potential operators which are employed to analyze the generalized transmission eigenfunctions.