The inside–outside duality for scattering problems by inhomogeneous media

Abstract

This paper investigates the relationship between interior transmission eigenvalues k0 > 0 and the accumulation point 1 of the eigenvalues of the scattering operator when k approaches k0. As is well known, the spectrum of is discrete, the eigenvalues μn(k) lie on the unit circle in and converge to 1 from one side depending on the sign of the contrast. Under certain (implicit) conditions on the contrast it is shown that interior transmission eigenvalues k0 can be characterized by the fact that one eigenvalue of converges to 1 from the opposite side if k tends to k0 from below. The proof uses the Cayley transform, Courant’s maximum–minimum principle, and the factorization of the far field operator. For constant contrasts that are positive and large enough or negative and small enough, we show that the conditions necessary to prove this characterization are satisfied at least for the smallest transmission eigenvalue.