Inverse Uniqueness in Interior Transmission Problem and Its Eigenvalue Tunneling in Simple Domain
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.
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  and Colton and Monk . 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  for the reconstruction of the interior transmission eigenvalues and  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  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 .
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 . That is,
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.
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.
To sketch an idea of how we construct a discrete set of for Lemma 2, we start with the first segment into the perturbation and discuss the well-posedness of the initial value problem starting at :which has unique solution inward up to given that is a known function for . The behavior of the solution is understood by the singular Sturm-Liouville theory provided in Section 2. Because is the first intersection, the uniqueness of (25) holds up to for . We deduce from the unique analytic continuation that hold outside for each fixed . In particular, holds by the construction of . More importantly, the functional determinant is an algebraic condition that filters out a discrete set of eigenvalues from for the problemThe spectral theory of (28) is simply singular Sturm-Liouville theory [29–31]. We have taken a similar approach in [14, 26, 27, 32]. Let be one of its eigenvalues. Leaving the perturbation at toward the origin, defines another ODE system:in which holds due to the construction of and the analytic continuation of the Helmholtz equation. The same appears at and is ready to define yet another new ODE:By the analytic continuation of of the Helmholtz equation, the same satisfies (28), (29), and (30) and appears at and then consecutively into each intersection interval by its construction. That is, system (28) extends to .
There are several ways to produce an ODE flow from the origin to the infinity that satisfiesMore importantly, system (30) inside also produces its group of eigenvalues that appear at and consecutively into all other intersection intervals by repeating the argument after (28). That is, system (30) extends to . In this way, we consider the piecewise construction of eigenfunctions for all possible discrete that make sense of the following system:Each element of the zero set of defines an initial value condition for the ODE and an algebraic condition to filter out a discrete spectrum of (33). There is the uniqueness and the existence to the solution of (33) defined by the piecewise construction as shown above, and we call the extended solution for each possible the eigenvalue tunneling in the interior transmission problem. In this paper, we use the solution constructed as the procedure above. Such a construction can be set to initiate at and tunnels to the infinity.
We have already discussed the simple case in the radially symmetric and starlike domains [14, 26, 27]:For example, with the initial condition , the function is an entire function of exponential type [3, 4, 14, 26, 27, 32]. Thus, the eigenvalues of (34) form a discrete set in , accumulate into the eigenvalues of (33), and tunnel to the infinity.
Conversely, once we find an eigenvalue of (33) for some along some , it solves (7) by the uniqueness of ODE up to and then (10) by the analytic continuation of the Helmholtz equation. Whenever we collect all such eigenvalues from each incident , they constitute the interior transmission eigenvalues of (1). The geometric characteristics of the perturbation are connected by rays of ODE system to the far-fields.
3. Asymptotic Expansions and Cartwright-Levinson Theory
To study the functional determinants , we collect the following asymptotic behaviors of and . For each fixed , we apply the Liouville transformation [7, 8, 29–31, 33]:whereHere we recall that is outside :in whichFor simplicity of the notation, we drop all the superscripts about whenever the context is clear.
Definition 3. Let be an integral function of order and let denote the number of the zeros of inside the angle and . One defines the density function aswith some fixed such that is at most a countable set [21–23, 34].
Lemma 4. The functional determinant is of order one and of type , . In particular, one has the following density identity:
Proof. We begin with (8):in whichWe have outside the zeros of ; similarly, outside the zeros of . The term is bounded and bounded away from zero outside the zeros of and on real axis. The behaviors of the and of (35) are well-known from the works [3, 8, 14, 26, 27, 32, 33, 35]. In particular, the following asymptotics hold. For and , there is a constant such that and that with the bounded error term Consequently, we can compute Lindelöf’s indicator function [14, 22, 23, 26, 27, 34] for and then for (41):in which , and we apply an inequality of Lindelöf’s indicator function in [22, page 51].
In case that , instead of (48), we haveReferring Cartwright theory to [22, page 251], we conclude that is of Cartwright’s class, and the lemma is thus proven.
Let us discuss the special case that .
Lemma 5. The meromorphic function if and only if along the incident angle .
Proof. We begin withas a meromorphic function in . Referring to [29–31, 33], has zeros asymptotically distributed near the zeros of , near the zero set of . A similar property holds for and . Hence, whenever has a zero, has a zero; has one whenever has one. The two perturbations, and , have the same set of Neumann and Dirichlet eigenvalues. By the inverse spectral uniqueness of Sturm-Liouville problem [3, 33], we have . The sufficient condition is obvious. This proves the lemma.
Thus, Lemma 4 merely describes the eigenvalue density of problem (34). To describe the density for (33), we may apply the translation invariant properties of interior transmission eigenvalues. Alternatively, we may consider the problem from the point of view of uniqueness theorem of ODE as in Section 1. In , except the previous eigenvalues of (34), we consider the new eigenvalue density of the problem produced on the second interval; that is,The density is zero, because and satisfy the same differential equation and initial condition at until . Thus, there are only trivial eigenfunctions in . One can prove inductively the density in each intersection interval, so we state the following theorem.
Theorem 6. Let denote the number of the eigenvalues of (33) inside the angle and in . One defines the density function asThenin which is in , zero otherwise, and is the Lebesgue measure.
Now we refresh the idea of the eigenvalue tunneling.
Proposition 7. If solves the following ODE for some ,then is the eigenvalue to the following system for the same also:
Proof. Let the eigenvalue solve the system of (54) for some , in which the first two equations there give an entire function in and the third condition implies that the eigenvalues of (54) form a discrete set in [14, 26, 27]. With given by (54), we continue ODE system (54) with the mixed boundary condition:The second equation above with actually gives a new initial value problem at ; that is,Similarly, initiates at again and then tunnels consecutively into the infinity with the same according to our construction in Section 1. The solution is a known function throughout . Hence, the given satisfiesThis proves the lemma.
Definition 8. One calls the eigenvalues of (55) that contributed by the eigenvalues generated in interval , where , .
In general, we can take as the reference point by proceeding with the previous argument.
Proposition 9. If satisfies the following ODE for some with a nontrivial solution,then is the eigenvalue to the following system:
4. A Proof of Theorem 1
Let be an eigenvalue of (54). Then it solves (7) at , and is defined outside through the analytic continuation of the Helmholtz equation. System (7) is radially symmetrically defined. Thus, with this same , we can solve the ODE solution of (55) along all other directions in by the piecewise construction in Lemma 2. Accordingly, the Fourier coefficients are defined for all angle , and is then extended into with the given . That is, the ODE spectrum along each scattered angle makes a part of the PDE spectrum. The spectrum of (54) is a subset of (1).
If , , are two indices of refraction with solution and determinant and given with identical set of interior transmission eigenvalues, then for all and all the following Sturm-Liouville problems in form of (54) hold with identical set of eigenvalues: This is one-dimensional inverse Sturm-Liouville problem [14, 26, 27, 29–31, 33]. The argument in [14, 26, 27] shows that and have identical solutions and , which we briefly explain as follows. Lemma 4 suggests and have zero density of On the other hand, the equation has only density of or , and then we deduce from the maximal zero set density from Cartwright theory [22, page 251] that In particular, they have the same set of Dirichlet eigenvalues.
Similar argument shows that Particularly, they have the same set of Neumann eigenvalues. Hence, inverse uniqueness of the Bessel operator [30, Theorems 1.2 and 1.3] is applied to (62) and proves that in for all . In this case, we have for for all .
Now we see that trivially in . We use a similar argument on the interval :If and have the same set of interior transmission eigenvalues, then they have the same spectrum for (67). From previous discussion, we conclude that for for all . The argument holds for all intersection points inductively. This proves Theorem 1.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author wants to thank Professor Chao-Mei Tu at NTNU for proofreading an earlier version of this paper.
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