A note on transmission eigenvalues in electromagnetic scattering theory

Fioralba Cakoni; Shixu Meng; Jingni Xiao

doi:10.3934/ipi.2021025

Article Contents

2021, Volume 15, Issue 5: 999-1014. Doi: 10.3934/ipi.2021025

This issue Previous Article Near-field imaging for an obstacle above rough surfaces with limited aperture data Next Article Two-dimensional inverse scattering for quasi-linear biharmonic operator

A note on transmission eigenvalues in electromagnetic scattering theory

1.
Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA
2.
Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

^* Corresponding author: jingni.xiao@rutgers.edu
^* Corresponding author: jingni.xiao@rutgers.edu

Received: October 2020

Revised: January 2021

Early access: March 2021

Published: October 2021

The author F. Cakoni is supported in part by the AFOSR Grant FA9550-20-1-0024 and NSF Grant DMS-1813492.

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Abstract

This short note was motivated by our efforts to investigate whether there exists a half plane free of transmission eigenvalues for Maxwell's equations. This question is related to solvability of the time domain interior transmission problem which plays a fundamental role in the justification of linear sampling and factorization methods with time dependent data. Our original goal was to adapt semiclassical analysis techniques developed in [21,23] to prove that for some combination of electromagnetic parameters, the transmission eigenvalues lie in a strip around the real axis. Unfortunately we failed. To try to understand why, we looked at the particular example of spherically symmetric media, which provided us with some insight on why we couldn't prove the above result. Hence this paper reports our findings on the location of all transmission eigenvalues and the existence of complex transmission eigenvalues for Maxwell's equations for spherically stratified media. We hope that these results can provide reasonable conjectures for general electromagnetic media.

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Mathematics Subject Classification: Primary: 35Q61, 35P25, 35P20; Secondary: 35R30, 78A46.

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