Advances in Mathematical Physics

Volume 2016, Article ID 2438253, 9 pages

http://dx.doi.org/10.1155/2016/2438253

## Inverse Uniqueness in Interior Transmission Problem and Its Eigenvalue Tunneling in Simple Domain

Department of Mathematics, National Chung Cheng University, 168 University Road, Min-Hsiung, Chia-Yi County 621, Taiwan

Received 7 October 2015; Accepted 6 December 2015

Academic Editor: Ivan Avramidi

Copyright © 2016 Lung-Hui Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We study inverse uniqueness with a knowledge of spectral data of an interior transmission problem in a penetrable simple domain. We expand the solution in a series of one-dimensional problems in the far-fields. We define an ODE by restricting the PDE along a fixed scattered direction. Accordingly, we obtain a Sturm-Liouville problem for each scattered direction. There exists the correspondence between the ODE spectrum and the PDE spectrum. We deduce the inverse uniqueness on the index of refraction from the discussion on the uniqueness anglewise of the Strum-Liouville problem.

#### 1. Introduction

In this paper, we study the inverse spectral problem in the following homogeneous interior transmission problem:where is the unit outer normal; is a simple domain in containing the origin with the Lipschitz boundary ; ; , for ; , for . Equation (1) is called the homogeneous interior transmission eigenvalue problem. We say is an interior transmission eigenvalue of (1) if there is a nontrivial pair of solutions such that , . The last two conditions in (1) are the Sommerfeld radiations condition to ensure the uniqueness on the scattered waves. We assume that near from its interior, which minimizes the support of . To ensure the uniqueness of the scattered solution, we impose the Sommerfeld radiation condition:

Problem (1) occurs naturally when one considers the scattering of the plane waves by certain inhomogeneity defined by an index of refraction inside the domain . The inverse problem is to determine the index of refraction by the measurement of the scattered waves in the far-fields. The inverse scattering problem plays a role in various disciplines of science and technology such as sonar and radar, geophysical sciences, medical imaging, remote sensing, and nondestructive testing in instrument manufacturing. For the origin of interior transmission eigenvalue problem, we refer to Kirsch [1] and Colton and Monk [2]. For theoretical study and historic literature, we refer to [1, 3–13]. To study the existence or location of the eigenvalues is a subject of high research interest [1, 2, 5, 6, 8, 11, 14–18]. Weyl’s type of asymptotics for the interior transmission eigenvalues is expected, even though problem (1) is defined in noncompact . In that case, the distribution of the eigenvalues is directly connected to certain invariant characteristics on the scatterer. In this regard, we apply the methods from entire function theory [19–23] to study the distributional laws of the eigenvalues. We also refer to [24] for the reconstruction of the interior transmission eigenvalues and [25] for a numerical description on the distribution of the eigenvalues. It is remarkable that an example on the nonuniqueness of the index of refraction is constructed in [4, Section 6] for the class of radially symmetric indices of refraction with a jump discontinuity of . Finding the optimal regularity assumption on the index of refraction to attain the uniqueness or the nonuniqueness remains an open problem. The breakthrough is made from the point of view of inverse Sturm-Liouville theory [18] that inverse -uniqueness on the radially symmetric index of refraction is obtained if certain extra local information [18, Theorem 1] is provided.

For the nonsymmetrically stratified medium, there are not too many known results [6, 7, 16, 17]. In this paper, we mainly follow the complex analysis methods [3, 14, 18, 26, 27] to study the nonsymmetrical scatterers as a series of one-dimensional problems along the rays scattering from the origin. The analysis along each ray possibly has multiple intersection points with , so we expect certain tunneling effect in a penetrable domain. In this paper, the new perspective is the asymptotic analysis inside and outside the perturbation. We give a global uniqueness on the index of refraction in simple domain by stating the following result.

Theorem 1. *Let , be two unknown indices of refraction as assumed in (1). If they have the same set of eigenvalues, then .*

#### 2. Preliminaries

We apply Rellich’s expansion in scattering theory. Firstly we expand the solution of (1) in two series of spherical harmonics by Rellich’s lemma [8, page 32] in the far-fields:where , ; ; is the spherical Bessel function of first kind of order . The summations converge uniformly and absolutely on suitable compact subsets in , for some . We note that expansion (3) holds for the Helmholtz equation outside the simple domain [8, page 31, Lemma 2.11]. Particularly, the spherical harmonicsis a complete orthonormal system in . Here,where the Legendre polynomials , , form a complete orthogonal system in . By the orthogonality of the spherical harmonics, the functions in the formsatisfy the first two equations in (1) independently [8, page 227] for .

Now we look for , , and , if any satisfies the following independent system for and :In terms of elementary linear algebra, the existence of the coefficients and is equivalent to finding the zeros of the following functional determinant:in which the system is independent of and . The forward problem describes the distribution of the zeros of , while the inverse problem specifies the index of refraction by the topology of the zero set. In [14, 18, 26, 27], we have discussed the methods to find the zeros of .

Let be a possible eigenvalue of (7). Applying the analytic continuation of the Helmholtz equation and Rellich’s lemma [8, pages 32, 33, and 222], the solutions parameterized by solveoutside the simple domain .

We note that representation (3) initially holds outside , and the core of many inverse problems is to extend the solution into the perturbation. For our case, we want to extend representation (3) into for some possible set of . Let be a given scattered direction satisfying the following geometric condition:For , we extend each Fourier coefficient with determined by system (7) for all , toward the origin until it meets the boundary at . Along the given , we apply the differential operator withto , which accordingly can solve problem (1) replaced with the manmade radially symmetric index of refraction for all . More importantly, the interior transmission condition implies the following ODE:

*If there is merely one intersection point* for with , then we set the initial conditions of according to the following condition:The behavior of the Bessel function near is found in [28, page 437]. We refer initial condition (14) to [18]. That is,

We observe that the uniqueness of the ODE (7) is valid up to the boundary :In particular, by the uniqueness of ODE (13) along the line segment to .

For , we can take in (16). From the point of view of the Helmholtz equation, both satisfy the Sommerfeld radiation condition whenever solves (13) and (14). By the uniqueness implied by the Sommerfeld radiation condition, we can choose that Using a similar argument, we deduce that

In general, the solution depends on the scattered direction whenever entering the perturbation, so we denote the extended solution of (13) as and accordingly the functional determinant as . Thus, (13) is relabeled asThe eigenvalues of (20) are discussed in [14, 26, 27] by the singular Sturm-Liouville theory in [29–31].

However,* we are working on a simple domain * in this paper. Hence, we modify the solution extension into in previous discussion. Instead of (16), we now ask for any that satisfies the following conditions:in which is the intersection set along the scattered angle defined byand we will discuss the well-posedness of . For each fixed , (21) provides an initial condition at . Hence, the solution ofis constructed piecewise from infinity to the origin, at whichWe put it as a lemma.

Lemma 2. *For , there is unique solution that satisfies (21), (22), (23), and (24) for the fixed .*

By the assumption of (1), we deduce that is a finite discrete set and for each fixed , in the case that , , and are any three consecutive points along the incident direction . Whenever is a tangent point at the boundary, we disregard it and consider the line segment from to as either completely inside or outside the perturbation. Without loss of generality, we assume that contains no tangent point. See Figure 1.