On Novel Geometric Structures of Laplacian Eigenfunctions in $\mathbb{R}^3$ and Applications to Inverse Problems

Two single-measurement uniqueness results for inverse scattering problems within polyhedral geometries

“…We derive two new unique identifiability results by a single measurement for the inverse scattering problem in the aforementioned two challenging setups. The main technical idea is to exploit certain quantitative geometric properties of the Laplacian eigenfunctions which were initiated in our recent works [12,13]. In this paper, we derive novel geometric properties that generalize and extend the related results in [13], which further enable us to establish the new unique identifiability results.…”

“…The main technical idea is to exploit certain quantitative geometric properties of the Laplacian eigenfunctions which were initiated in our recent works [12,13]. In this paper, we derive novel geometric properties that generalize and extend the related results in [13], which further enable us to establish the new unique identifiability results. It is pointed out that in addition to the shape of the obstacle or the grating profile, we can simultaneously recover the boundary impedance parameters.…”

“…Following the notations in [13], we let Π 1 , Π 2 be any two adjacent faces of a polyhedron Ω. Denote E(Π 1 , Π 2 , l) to be an edge corner associated with Π 1 and Π 2 . V({Π } n =1 , x 0 ) signifies the vertex corner formulated by Π 1 , Π 2 , • • • , Π n at the vertex x 0 ∈ ∂Ω.…”

“…It is clear that a vertex corner V({Π } n =1 , x 0 ) is composed of finite many edge corners, which intersect at x 0 , ; see [13, Definitions 1.2 and 1.3] for more details. Now, recall the definitions for rational and irrational obstacles based on the concept of the rational and irrational corners introduced in [13] as follows.…”

Invisibility enables super-visibility in electromagnetic imaging

“…Although the nodal lines of eigenfunctions appear in different locations for different eigenvalues (cf. [12,13]), the nodal lines always go through the corners. So, the corners of the domain will stand out if we superimpose the indicator function with multi-frequency farfield data.…”