On the existence of transmission eigenvalues in an inhomogeneous medium

Abstract

Abstract:We prove the existence of transmission eigenvalues corresponding to the inverse scattering problem for isotropic and anisotropic media for both scalar Helmholtz equation and Maxwell's equations. Considering a generalized abstract eigenvalue problem, we are able to extend the ideas of Päivärinta and Sylvester [15] to prove the existence of transmission eigenvalues for a larger class of interior transmission problems. Our analysis includes both the case of a medium with positive contrast and of a medium with negative contrast provided that this contrast is large enough.

Surface integral formulation of the interior transmission problem

“…We now can state a concluding theorem for this section, which is a classical result on ITP [9] related to the discreteness of transmission eigenvalues for contrasts that does not change sign and is now is straightforward corollary of Theorem 2.3 and Lemma 2.4 and the fact that the set of transmission eigenvalues is contained in the set of points where Z(k) is not invertible. Theorem 2.5.…”

“…The theory of inverse scattering for acoustic and electromagnetic waves, is an active area of research with significant developments in the past few years and more specifically the so-called interior transmission problem (ITP) and its transmission eigenvalues [3,11,12,19,22,8,4,6,9,15,17,7,21]. Although simply stated, the interior transmission problem is not covered by the standard theory of elliptic partial differential equations since as it stands it is neither elliptic nor self-adjoint.…”

A Spectral-Element Method for Transmission Eigenvalue Problems

“…The existence and computation of transmission eigenvalues have been studied recently by many researchers [6][7][8]13,14,18,19]. In particular, several finite-element methods are presented in [8].…”

Efficient Spectral Methods for Transmission Eigenvalues and Estimation of the Index of Refraction

“…On the other hand, it is shown in [8] that for τ 0 > 0 sufficiently small, we have λ 1 (τ 0 ,D,n(x))−τ 0 ≥ 0, which means that there is a transmission eigenvalue for (D,n * ) less than the first one, a contradiction to the assumption.…”

“…P N ∩ H 1 0 (I)). Hence, we have 8) whereū is the vector consisting of all unknown coefficients in (4.7). Note that in the AlgorithmN (see Section 2), we only need to compute the transmission eigenvalues for n being a constant.…”