Inside-outside duality with artificial backgrounds

We use the inside-outside duality approach proposed by Kirsch-Lechleiter to identify transmission eigenvalues (TEs) 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 TEs 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 TEs. This work is based on several of the pioneering works of our dearest colleague and friend Armin Lechleiter and is dedicated to his memory.