A uniqueness theorem on the eigenvalues of spherically symmetric interior transmission problem in absorbing medium

Abstract

We study the asymptotic distribution of the eigenvalues of interior transmission problem in absorbing medium. We apply Cartwright's theory and the technique from entire function theory to find a Weyl's type of density theorem in absorbing medium. Given a sufficient quantity of transmission eigenvalues, we obtain limited uniqueness on the refraction index as a uniqueness problem in entire function theory.

On Certain Translation Invariant Properties of Interior Transmission Spectra and Their Doppler’s Effect

“…For ≥ −1/2, there are much more generalized results from [28,29]. In particular, the solution of (15) is an entire function of order one and of finite type [4,9,10,14,[27][28][29][30] that has a Sturm-Liouville type of spectral analysis. Now we fulfill the boundary conditions in (1).…”

“…Previously [9,10,26], we have shown that if the interior transmission eigenvalues are given identical for all possible incident angles with 1 (0) = 2 (0) = 1, then we can prove the inverse uniqueness. Here we are given a potpourri of spectral data.…”

“…in which we recall from [9] and [10, Lemma 2.5] that ( )/ ( ) = (1) and̂( ; )/̂( ; ) = ( ) outside the zeros in the denominators. Without loss of generality, we drop the lower order term (1/ )( ( )/ ( )) in the bracket in (36) and the term 1 − (1/ )( ( )/ ( ))(̂( ; )/̂( ; )) is bounded and bounded away from zero near the zeros of ( ) and̂( ; ).…”

“…For the nonsymmetrically stratified medium, there are not too many known results [3,8,23]. In this paper, we mainly follow the idea in [9,[24][25][26] to study the nonsymmetrical scatterers as a series of one-dimensional problems and consider the analytic continuation theorems of the Helmholtz equation. This paper aims to examine some spectral invariants associated to (1) and analyze the functional correspondence between the variation of the spectral density function and perturbation of the index of refraction.…”

“…For a theoretical study and historic literature, we refer to [3][4][5][6]. It is also another subject of research interest to study the existence or location of the eigenvalues [1,4,[7][8][9][10][11][12][13][14]. It is expected to find a Weyl's type of asymptotics for the interior transmission eigenvalues.…”