The Interior Transmission Eigenvalue Problem

Abstract

We consider the inverse problem of determining the spherically symmetric index of refraction n(r) from a knowledge of the corresponding transmission eigenvalues (which can be determined from field pattern of the scattered wave). We also show that for constant index of refraction n(r) = n, the smallest transmission eigenvalue suffices to determine n, complex eigenvalues exist for n sufficiently small and, for homogeneous media of general shape, determine a region in the complex plane where complex eigenvalues must lie.

On the use of T-coercivity to study the interior transmission eigenvalue problem

“…one can check that the first real transmission eigenvalue k 1 such that k 1 = 0 satisfies k 2 1 (A − Λ 1 (D)/n + ) where Λ 1 (D) is the first Dirichlet eigenvalue of − on D. This is one of the results of [5].…”

“…Our result applies to situations for which A − Id and n − 1 are positive or negative in a neighbourhood of the boundary and change sign inside the domain. That latter case was not covered by the usual techniques (see [3,5,8]) and has been an open problem for a long time. We also give estimates for the first eigenvalue in the case when A is smaller or greater than one all over the domain.…”

“…Nous montrons que ce dernier est bien posé au sens de Fredholm et que l'ensemble des valeurs propres de transmission est discret dénombrable. Notre résultat s'applique à des situations qui n'avaient pas encore pu être traitées par les techniques usuelles (voir [3,5,8]), comme par exemple le cas où l'indice de réfraction prend des valeurs supérieures et inférieures à 1 sur le domaine. Plus précisément, avec les notations du §1, nous prouvons les théorèmes suivants :…”

An inverse uniqueness on the index of refraction with a transition region

“…It is also another subject of research interest to study the existence or location of the eigenvalues. [2,5,[9][10][11][12][13][14][15][16][17] It is expected to find a Weyl type of asymptotics for the interior transmission eigenvalues. In that case, the distribution of the eigenvalues is directly connected to certain invariant characteristics of the scatterer.…”

“…For the non-symmetrically stratified medium, there are merely a few results available. [4,9,12,14] In this paper, we mainly follow the methods in [13,[24][25][26] to study the non-symmetrical scatterers as a series of one-dimensional problems. The new ingredient in this paper is the negative part of the index of refraction and its realization in indicator diagram.…”