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Eigenvalues arising in scattering theory have been envisioned as a potential source of target signatures in nondestructive testing of materials, whereby perturbations of the eigenvalues computed for a penetrable medium would be used to…

Analysis of PDEs · Mathematics 2021-04-06 Samuel Cogar

In this paper, we provide an analytical study of the transmission eigenvalue problem in the context of biharmonic scattering with a penetrable obstacle. We will assume that the underlying physical model is given by an infinite elastic…

Analysis of PDEs · Mathematics 2025-10-10 Rafael Ceja Ayala , Isaac Harris , Andreas Kleefeld

In this paper, we consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which…

Numerical Analysis · Mathematics 2016-08-24 Fang Zeng , Jiguang Sun , Liwei Xu

In this paper, we investigate the interior transmission eigenvalue problem for an inhomogeneous media with conductive boundary conditions. We prove the discreteness and existence of the transmission eigenvalues. We also investigate the…

Analysis of PDEs · Mathematics 2016-10-31 Oleksandr Bondarenko , Isaac Harris , Andreas Kleefeld

The transmission eigenvalues corresponding to the half-line Schr\"odinger equation with the general selfadjoint boundary condition is analyzed when the potential is real valued, integrable, and compactly supported. It is shown that a…

Spectral Theory · Mathematics 2016-10-06 Tuncay Aktosun , Vassilis G. Papanicolaou

The discrete Schr\"odinger equation with the Dirichlet boundary condition is considered on a half-line lattice when the potential is real valued and compactly supported. The inverse problem of recovery of the potential from the so-called…

Spectral Theory · Mathematics 2018-05-22 Tuncay Aktosun , Vassilis G. Papanicolaou

The interior transmission eigenvalue problem (ITP) plays a central role in inverse scattering theory and in the spectral analysis of inhomogeneous media. Despite its smooth dependence on the refractive index at the PDE level, the…

Numerical Analysis · Mathematics 2026-05-20 Davide Pradovera , Alessandro Borghi , Lukas Pieronek , Andreas Kleefeld

In this paper, we discuss a virtual element approximation for the modified transmission eigenvalue problem in inverse scattering for natural materials. In this case, due to the positive artificial diffusivity parameter in the considered…

Numerical Analysis · Mathematics 2025-10-30 Liangkun Xu , Shixi Wang , Hai Bi

The goal of this paper is to develop numerical methods computing a few smallest elasticity transmission eigenvalues, which are of practical importance in inverse scattering theory. The problem is challenging since it is nonlinear,…

Numerical Analysis · Mathematics 2018-02-13 Xia Ji , Peijun Li , Jiguang Sun

This paper investigates the inverse scattering problem for the magnetic Schr\"odinger equation. We first establish the well-posedness of the direct problem through a variational approach under physically meaningful assumptions on the…

Analysis of PDEs · Mathematics 2025-10-10 Chaohua Duan , Zhen Xue

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…

Analysis of PDEs · Mathematics 2023-04-24 Yat Tin Chow , Youjun Deng , Hongyu Liu , Mahesh Sunkula

The transmission eigenvalue problem arising from the inverse scattering theory is of great importance in the theory of qualitative methods and in the practical applications. In this paper, we study the transmission eigenvalue problem for…

Numerical Analysis · Mathematics 2022-12-23 Shixi Wang , Hai Bi , Yidu Yang

Scattering resonances arise in wave phenomena and play an important role in many applications. While extensive theoretical studies have been conducted, effective numerical computation remains limited, and most existing methods suffer from…

Numerical Analysis · Mathematics 2026-04-17 Bo Gong , Jiguang Sun

We discuss the use of differential sampling method to image local perturbations in anisotropic periodic layers, extending earlier works on the isotropic case. We study in particular the new interior transmission problem that is associated…

Numerical Analysis · Mathematics 2018-11-09 Thi-Phong Nguyen

In this paper we survey some recent results concerning scattering and non-scattering in the context of the linear Helmholtz equation and inhomogeneities of nontrivial contrast. We examine isotropic as well as anisotropic media. Part of the…

Analysis of PDEs · Mathematics 2026-02-09 Fioralba Cakoni , Michael S. Vogelius

We consider inverse obstacle and transmission scattering problems where the source of the incident waves is located on a smooth closed surface that is a boundary of a domain located outside of the obstacle/inhomogeneity of the media. The…

Mathematical Physics · Physics 2017-10-12 Evgeny Lakshtanov , Boris Vainberg

This paper is concerned with the inverse electromagnetic scattering problem for anisotropic media. We use the interior resonant modes to develop an inverse scattering scheme for imaging the scatterer. The whole procedure consists of three…

Analysis of PDEs · Mathematics 2022-09-12 Youzi He , Hongjie Li , Hongyu Liu , Xianchao Wang

In this paper, we provide an analytical study of the transmission eigenvalue problem with two conductivity parameters. We will assume that the underlying physical model is given by the scattering of a plane wave for an isotropic scatterer.…

Analysis of PDEs · Mathematics 2022-09-16 Rafael Ceja Ayala , Isaac Harris , Andreas Kleefeld , Nikolaos Pallikarakis

In this paper, we consider an interior transmission eigenvalue (ITE) problem on some compact $C^{\infty }$-Riemannian manifolds with a common smooth boundary. In particular, these manifolds may have different topologies, but we impose some…

Spectral Theory · Mathematics 2019-05-27 Hisashi Morioka , Naotaka Shoji

We study an inverse uniqueness with a knowledge of spectral data in the interior transmission problem defined by an index of refraction in a simple domain. We expand the solution in such a domain into a series of one dimensional problems.…

Analysis of PDEs · Mathematics 2015-08-10 Lung-Hui Chen