Inside-outside duality with artificial backgrounds

TL;DR

This paper advances the understanding of transmission eigenvalues by using artificial backgrounds within the inside-outside duality framework, providing necessary and sufficient spectral conditions and convergence results.

Contribution

It introduces a novel approach with artificial backgrounds, especially zero-index ones, to characterize transmission eigenvalues and proves convergence of associated invisible fields.

Findings

Necessary and sufficient spectral conditions for transmission eigenvalues.

Convergence of the invisible generalized incident field.

Extension of the inside-outside duality method with artificial backgrounds.

Abstract

We use the inside-outside duality approach proposed by Kirsch-Lechleiter to identify transmission eigenvalues associated with artificial backgrounds. We prove that for well chosen artificial backgrounds, in particular for the ones with zero index of refraction at the inclusion location, one obtains a necessary and sufficient condition characterizing transmission eigenvalues via the spectrum of the modified far field operator. We also complement the existing literature with a convergence result for the invisible generalized incident field associated with the transmission eigenvalues. This work is based on several of the pioneering works of our dearest colleague and friend Armin Lechleiter and is dedicated to his memory.