The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation

Abstract

The recovery of a spherically-symmetric wave speed v is considered in a bounded spherical region of radius b from the set of the corresponding transmission eigenvalues for which the corresponding eigenfunctions are also spherically symmetric. If the integral of 1/v on the interval [0, b] is less than b, assuming that there exists at least one v corresponding to the data, it is shown that v is uniquely determined by the data consisting of such transmission eigenvalues and their "multiplicities," where the "multiplicity" is defined as the multiplicity of the transmission eigenvalue as a zero of a key quantity.When that integral is equal to b, the unique recovery is obtained when the data contains one additional piece of information. Some similar results are presented for the unique determination of the potential from the transmission eigenvalues with "multiplicities" for a related Schrödinger equation. Mathematics Subject Classification (2010): 34B07 34B24 47E05Short title: Inverse problem for transmission eigenvalues

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

“…Alternatively, we consider the other theoretical framework. [14] Because (1.2) is spherically symmetric, we define the transformation:…”

“…Secondly, assuming a real refraction index n 1 , the uniqueness result [14] holds for the spectral invariants under the assumption However, the information we need to decipher appears on the second term in the exponent in the expansion (1.23) and (1.24). Now we need the vocabulary from entire function theory to describe the distribution of the zeros of the functional determinant D(k).…”

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

“…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.…”

“…In this paper, we follow the approaches in [13,24,26] to prove the following theorem: the new ingredient is the asymptotic expansion of the solution of (1.13) and its connection to Cartwright-Levinson theory.…”

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

“…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.…”