Applications of elliptic operator theory to the isotropic interior transmission eigenvalue problem

Abstract

The paper concerns the isotropic interior transmission eigenvalue (ITE) problem. This problem is not elliptic, but we show that, using the Dirichlet-to-Neumann map, it can be reduced to an elliptic one. This leads to the discreteness of the spectrum as well as to certain results on possible location of the transmission eigenvalues. If the index of refraction n(x) is real, we get a result on the existence of infinitely many positive ITEs and the Weyl type lower bound on its counting function. All the results are obtained under the assumption that n(x) − 1 does not vanish at the boundary of the obstacle or it vanishes identically, but its normal derivative does not vanish at the boundary. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle. Some results on the discreteness and localization of the spectrum are obtained for complex valued n(x).

Localization of the interior transmission eigenvalues for a ball

“…The problems (B) and (C) are more difficult, and they are of some interest for the numerical analysis of the (ITEs). In this direction it is interesting to find an optimal eigenvalue-free region and a Weyl formula with optimal remainder (see [12], [5], [13], [8] and the references therein). In a recent work [15] the authors showed that (B) and (C) are closely related each other, and a larger eigenvalue-free region leads to a Weyl asymptotics with a smaller remainder term.…”

High-frequency approximation of the interior Dirichlet-to-Neumann map and applications to the transmission eigenvalues

“…In the isotropic case the eigenvalue-free region (2.5) has been also obtained in [14] when the dimension is one. In the general case of arbitrary domains transmission eigenvalue-free regions have been previously proved in [5], [6] and [12] (isotropic case), [15] and [16] (both cases). For example, it has been proved in [15] that, under the conditions (2.2) and (2.4), there are no transmission eigenvalues in λ ∈ C : Re λ > 1, |Im λ| ≥ C ε (Re λ) 1 2 +ε , C ε > 0, for every 0 < ε ≪ 1.…”

Self-adjoint indefinite Laplacians

“…The direct substitution of the values of (25) and (29) into (23) and (24) shows that one always has v ∈ H 1 (B re,R ), while the condition v ∈ L 2 (B r i ,re ) appears to be equivalent to (11), so it holds for any h for a ≥ r 2 e /r i as before, otherwise a very strong regularity of h is required.…”