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.


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 R 3 containing the origin with the Lipschitz boundary ; () ∈ C 2 (R 3 ); () > 0, for  ∈ ; () = 1, for  ∉ .Equation ( 1) is called the homogeneous interior transmission eigenvalue problem.We say  ∈ C is an interior transmission eigenvalue of (1) if there is a nontrivial pair of solutions (, V) such that , V ∈  2 (),  − V ∈  2 0 ().The last two conditions in (1) are the Sommerfeld radiations condition to ensure the uniqueness on the scattered waves.We assume that () ̸ = 1 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 farfields.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][4][5][6][7][8][9][10][11][12][13].To study the existence or location of the eigenvalues is a subject of high research interest [1,2,5,6,8,11,[14][15][16][17][18]].Weyl's type of asymptotics for the interior transmission eigenvalues is expected, even though problem (1) is defined in noncompact R 3 .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][20][21][22][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 L 2 -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 onedimensional 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  1 ,  2 be two unknown indices of refraction as assumed in (1).If they have the same set of eigenvalues, then  1 ≡  2 .

Preliminaries
We apply Rellich's expansion in scattering theory.Firstly we expand the solution (V, ) of (1) in two series of spherical harmonics by Rellich's lemma [8, page 32] in the far-fields: where  fl ||, 0 ≤  < ∞; x = (,) ∈ S in which the system is independent of  and x.The forward problem describes the distribution of the zeros of   (;  0 ), 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   (;  0 ).
Let  ∈ C be a possible eigenvalue of (7).Applying the analytic continuation of the Helmholtz equation and Rellich's lemma [8, page 32, 33, 222], the solutions parameterized by  solve outside the simple domain .
We note that representation (3) initially holds outside || ≥  0 , 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 || ≤  0 for some possible set of .Let x1 ∈ S 2 be a given scattered direction satisfying the following geometric condition: For x1 , we extend each Fourier coefficient   (; ) with  ∈ C determined by system (7) for all , toward the origin until it meets the boundary  at ( 1 , x1 ).Along the given x1 , we apply the differential operator Δ +  2  with to { , ()}, which accordingly can solve problem (1) replaced with the manmade radially symmetric index of refraction () = (x) = (x 1 ) for all x ∈ S 2 .More importantly, the interior transmission condition implies the following ODE: If there is merely one intersection point for [0,  0 ]× x1 with , then we set the initial conditions of   (; ) according to the following condition: The behavior of the Bessel function   () near  = 0 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,   (;  1 ) =   (;  0 ) by the uniqueness of ODE (13) along the line segment ( 0 , x1 ) to ( 1 , x1 ).
For  ≥ 0, we can take  , =  , = 1 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 x whenever entering the perturbation, so we denote the extended solution of (13) as ŷ (; ) and accordingly the functional determinant as D (;  1 ).Thus, ( 13) is relabeled as The eigenvalues of ( 20) are discussed in [14,26,27] by the singular Sturm-Liouville theory in [29][30][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  ∈ C that satisfies the following conditions: in which R is the intersection set along the scattered angle x defined by and we will discuss the well-posedness of ŷ (; ).For each fixed x ∈ S 2 , (21) provides an initial condition at r ∈ R.
Hence, the solution ŷ () of is constructed piecewise from infinity to the origin, at which We put it as a lemma.
By the assumption of (1), we deduce that R is a finite discrete set and r1 < r2 < ⋅ ⋅ ⋅ < rM for each fixed x ∈ S 2 , in the case that (, x), (, x), and (, x) are any three consecutive points along the incident direction x.Whenever (, x) is a tangent point at the boundary, we disregard it and consider the line segment from (, x) to (, x) as either completely inside or outside the perturbation.Without loss of generality, we assume that R 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 rM : which has unique solution inward up to rM −1 given that   (r) is a known function for  ∈ C. The behavior of the solution is understood by the singular Sturm-Liouville theory provided in Section 2. Because rM is the first intersection, the uniqueness of (25) holds up to rM for  ∈ C. We deduce from the unique analytic continuation that hold outside  for each fixed .In particular, D (; rM −1 ) = 0 holds by the construction of ŷ (; ).More importantly, the functional determinant D (; rM −1 ) = 0 is an algebraic condition that filters out a discrete set of eigenvalues from C for the problem The spectral theory of (28) is simply singular Sturm-Liouville theory [29][30][31].We have taken a similar approach in [14,26,27,32].Let  1 be one of its eigenvalues.Leaving the perturbation at rM −1 toward the origin,  1 defines another ODE system: in which D ( 1 ; rM −2 ) = 0 holds due to the construction of ŷ (; ) and the analytic continuation of the Helmholtz equation.The same  1 appears at rM −2 and is ready to define yet another new ODE: By the analytic continuation of of the Helmholtz equation, the same  1 satisfies (28), (29), and ( 30) and appears at rM −3 and then consecutively into each intersection interval by its construction.That is, system (28) extends to R + .There are several ways to produce an ODE flow from the origin to the infinity that satisfies D (; r0 ) = 0, D (; r1 ) = 0, . . .D (; rM ) = 0.
(32) More importantly, system (30) inside  also produces its group of eigenvalues that appear at rM −3 and consecutively into all other intersection intervals by repeating the argument after (28).That is, system (30) extends to R + .In this way, we consider the piecewise construction of eigenfunctions for all possible discrete  ∈ C that make sense of the following system: ( Each element of the zero set of D (; r ) 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 ŷ (; ) for each possible  the eigenvalue tunneling in the interior transmission problem.In this paper, we use the solution ŷ (; ) constructed as the procedure above.Such a construction can be set to initiate at r0 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 D (; r0 ) = 0, the function D (; r1 ) is an entire function of exponential type [3,4,14,26,27,32].Thus, the eigenvalues of (34) form a discrete set in C, accumulate into the eigenvalues of (33), and tunnel to the infinity.Conversely, once we find an eigenvalue of ( 33) for some  along some x, it solves (7) by the uniqueness of ODE up to || =  0 and then (10) by the analytic continuation of the Helmholtz equation.Whenever we collect all such eigenvalues from each incident x ∈ S 2 , 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.

Asymptotic Expansions and Cartwright-Levinson Theory
To study the functional determinants D (; ), we collect the following asymptotic behaviors of ŷ (; ) and ŷ  (; ).For each fixed x, we apply the Liouville transformation [7,8,[29][30][31]33]: where Here we recall that  is 1 outside : in which For simplicity of the notation, we drop all the superscripts about x 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 as with some fixed  0 ∉  such that  is at most a countable set [21][22][23]34].
Lemma 4. The functional determinant D (; ) is of order one and of type  + B(), r0 ≤  ≤ r1 .In particular, one has the following density identity: Proof.We begin with (8):

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We have outside the zeros of    (); similarly, outside the zeros of ŷ (; ).The term α() is bounded and bounded away from zero outside the zeros of    () and ŷ (; ) on real axis.The behaviors of the ẑ  (; ) and ẑ (; ) of ( 35) are well-known from the works [3,8,14,26,27,32,33,35].In particular, the following asymptotics hold.For  > 0 and R ≥ 0, there is a constant  such that Consequently, we can compute Lindelöf 's indicator function [14,22,23,26,27,34] for ŷ (; ) and then D (; ) for (41): In case that α () ≡ 0, instead of (48), we have  [3,33], we have  ≡ 1.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 [r 1 , r2 ], 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 ŷ (; )/ and   () satisfy the same differential equation and initial condition at r1 until r2 .Thus, there are only trivial eigenfunctions in [r Proof.This is only Lemma 4 and the discussion on (51).Now we refresh the idea of the eigenvalue tunneling.(55) Proof.Let the eigenvalue  solve the system of (54) for some  ≥ 0, 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 C [14,26,27].With  given by (54), we continue ODE system (54) with the mixed boundary condition: In general, we can take r as the reference point by proceeding with the previous argument.(61)