# On vanishing near corners of transmission eigenfunctions

## TL;DR

This paper proves that transmission eigenfunctions vanish near corners with interior angles less than pi when the potential is nonzero there, revealing geometric information and impacting inverse scattering and invisibility studies.

## Contribution

It establishes the first quantitative result showing eigenfunctions vanish near corners with specific geometric conditions, linking spectral properties to domain geometry.

## Key findings

- Eigenfunctions vanish near corners with interior angle less than pi.
- Vanishing occurs when the potential does not vanish at the corner.
- Results have implications for inverse scattering and invisibility.

## Abstract

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$, $n\geq 2$, and $V\in L^\infty(\Omega)$ be a potential function. Consider the following transmission eigenvalue problem for nontrivial $v, w\in L^2(\Omega)$ and $k\in\mathbb{R}_+$, \[(\Delta+k^2)v= 0 \quad \text{in } \Omega,\] \[(\Delta+k^2(1+V))w= 0 \quad \text{in } \Omega,\] \[w-v \in H^2_0(\Omega), \quad \lVert v \rVert_{L^2(\Omega)}=1. \] We show that the transmission eigenfunctions $v$ and $w$ carry the geometric information of $\mathrm{supp}(V)$. Indeed, it is proved that $v$ and $w$ vanish near a corner point on $\partial \Omega$ in a generic situation where the corner possesses an interior angle less than $\pi$ and the potential function $V$ does not vanish at the corner point. This is the first quantitative result concerning the intrinsic property of transmission eigenfunctions and enriches the classical spectral theory for Dirichlet/Neumann Laplacian. We also discuss its implications to inverse scattering theory and invisibility.