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

We study the translation invariant properties of the eigenvalues of scattering transmission problem. We examine the functional derivative of the eigenvalue density function Δ(x) to the defining index of refraction n(x). By the limit behaviors in frequency sphere, we prove some results on the inverse uniqueness of index of refraction. In physics, Doppler’s effect connects the variation of the frequency/eigenvalue and the motion velocity/variation of position variable. In this paper, we proved the functional derivative ∂rΔ(x) = (1 + √n(r(x)))/π.


Introduction
In this paper, we investigate an inverse spectral problem in the following homogeneous interior transmission problem: Δ +  2  ()  = 0, in ; ΔV +  2 V = 0, in ;  = V, on ; where ] is the unit outer normal;  is a starlike domain in R 3 containing the origin with boundary ; () ∈ C 2 (R 3 ); () > 0, for  ∈ ; () = 1, for  ∉ ; (0) = 1.Equation ( 1) is called the homogeneous interior transmission eigenvalue problem.We also assume the boundary is given and defined by  =  (, ) ∈ C 1 (S 2 ; R + ) , where S 2 is the unit sphere with the spherical coordinate x fl (, ).Problem (1) occurs naturally when one considers the scattering of the plane waves hinging on certain inhomogeneity inside the domain  that is defined by an index of refraction in many models.In the reduced scattering model (1), the inverse problem is to determine the index of refraction by the measurement of the scattered wave fields collected in the far fields.The inverse scattering problem plays a role in various disciplines of science and technology such as the applications in sonar and radar, geophysical sciences, astrophysics, medical imaging, remote sensing, and nondestructive testing in instrument manufacturing.By an inhomogeneous interior transmission problem, we mean the following system: Δ +  2  ()  = 0, in ; ΔV +  2 V = 0, in ;  − V = ℎ (2)   ()    ( x) , on ; ] − V ] =  ] [ℎ (2)   ()    ( x)] , on , in which ℎ (2)   () is the spherical Hankel function of order  and    is the spherical harmonics of order , .We say  ∈ C is an interior transmission eigenvalue of (1) if there is a nontrivial pair of solution , V ∈ C 2 () ∩ C 1 () to the boundary value problem (1).To ensure the uniqueness of The interior transmission eigenvalues play a role in the inverse scattering theory both in numerical computation and in theoretical scattering theory.For the origin of interior transmission eigenvalue problem, we refer to Colton and Monk [1] and Kirsch [2].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.In this case, the distribution of transmission eigenvalues is directly related to certain spectral invariants of the scatterer.In this regard, we apply the methods from entire function theory [15][16][17][18][19][20] to study the distributional laws of the eigenvalues.We also refer to [21,22] for the reconstruction of the interior transmission eigenvalues.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.

Preliminaries and Main Result
Firstly, we expand the solution (V, ) of (1) in two series of spherical harmonics by Rellich's lemma.This is a Fourier type of expansion theory, and we refer to [4, p. 32, p. 227] and [14, p. 353]. where and   is the spherical Bessel function of the first kind of order .The summations converge uniformly and absolutely on compact subsets for sufficiently large || ≥  0 ≫ 0. We refer the method to [4, p. 32, Lemma 2.11] which holds for a bounded perturbation.
Here, we note that the spherical harmonics is a complete orthonormal system in L 2 (S 2 ), in which where the Legendre polynomials   ,  = 0, 1, . .., form a complete orthogonal system in so we apply the Laplacian to the first equation of (1) for || ≫  0 , where () = 1, and we observe that Accordingly, the Fourier coefficients   () trivially satisfy the following equation: Most importantly, we extend the Fourier coefficients   () along some fixed incident direction x into || ≤  0 by considering the following scheme.Let us define n () fl  ( x) .
For the fixed x ∈ S 2 , we consider the radially symmetric index of refraction in C 2 (R 3 ) by rotating n() around the origin.Due to the radial-symmetry, the solution   of ( 13) has an extension into || ≤  0 depending on the incident direction x by solving the following ODE: We denote the solution to (15) as ŷ (; ).We renormalize the initial condition as follows: is independent of the incident direction x.We refer the initial condition to [4,14].Now we consider the following Liouville transformation for each fixed x ∈ S 2 : where ξ fl Let us define Then, we deduce from ( 15) and ( 17) that in which Here ξ = B if and only if  = (x) for the fixed x.Moreover, Thus, we deduce from ( 19), (20), and ( 21) that For the simplicity of the notation, we drop the superscripts on the variables.When  = 0, the asymptotic estimates for the solution  0 () are classic and can be found in [27] where () fl ∫  0 (), and the error term can be improved [27, p. 17] by Similarly, in which the boundary behavior of () plays a role in determining the inverse spectral uniqueness on the scatterer, and thus C 2 -assumption on the index of refraction is convenient.
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).The nontrivial solutions (10) of the homogeneous system are required to satisfy the boundary condition along the fixed x as follows: for in which we note that  = (x).The eigenvalues are possibly the intersection points of two complex-valued functions, and the constants â, and b, depend on x as well.Let us define in which D () is independent of index .The existence of the nontrivial (â , , b, ) of ( 26) is equivalent to finding the zeros of The following fundamental lemma connects the zeros of D () = 0 and the eigenvalues of (1).
Lemma 1.Let  be an eigenvalue of (1) if and only if it satisfies (28) for all  and some x ∈ S 2 .
Proof.Let V(; ) and (; ) be a nontrivial pair of solutions of (1).Then, (6) holds and uniquely defines the coefficients ŷ (; ) and all  ≥ 0 for  ≥  0 and extends into || ≤  0 by solving (15) with some fixed x.The solution, each satisfies the Helmholtz equation independently and meets with V , () in || ≫  0 .By the analytic continuation of the Helmholtz equation [1], they extend to meet on .That is, ( 28) is satisfied at x ∈ S 2 .Hence, (28) holds on x for all  ≥ 0.
For the sufficient condition, if ( 28) is valid on the boundary for some x, then ỹ () define a Fourier coefficient by ODE (15) we find that the solution (;  0 ) defines an eigenfunction to (1).We refer more detailed construction to [11].
For  = 0, D () is easier to compute [24, (2.10)]: in which all four elements have zeros distributed asymptotically like sine and cosine functions.
One step further, we can describe the zero distribution of D () more precisely by applying Wilder's theorem [32,33] to the term    () ŷ (; ) or   ()   (; ) in (35) and obtain the following density distribution theorem for (33): let  11 ,  12 , and  13 be three adjacent strips containing all but a finite exceptions of the zeros along the positive real axis, and let ( 1 (, , )),  = 1, 2, 3, be the zero counting function in the strip starting with I = , of length  and of width .Then there exist some  > 0 and large  such that where â, (), b, () are understood as two meromorphic functions.An interior transmission eigenvalue means that the inhomogeneous system (3) can not be solved explicitly in C 2 () ∩ C 1 () whenever D () has a zero in the denominator.We observe that ((; ), V(; )) is not a solution in the classic sense if there is any  and any x such that D () has a zero.Whenever that happens, we have an interior transmission eigenvalue.
Definition 3. We define Z( x) to be the union of the zero set of entire function D0 () at x and − x.
We need the following vocabulary to describe the asymptotic behavior of the eigenvalues.Definition 4. Let () be an integral function of order , and (, , , ) denote the number of the zeros of () inside the angle [, ] and || ≤ ; we define the zero density function with some fixed  0 ∉  such that  is at most a countable set [18][19][20].Following the results in [9][10][11]26], we have the density asymptotics: For a simpler notation, we use to denote the density of zero set of Z( x).

Advances in Mathematical Physics
One can show the following.

Lemma 5. us denote
and the following identity holds: Accordingly, the density function is invariant to the rigid translations of .
Proof.We observe that By applying (52) in (54) to x and − x, respectively, (56) is proven.The invariant property comes from the translation transformation along  of integral B. The invariant property on rotation is trivial.
We state the inverse spectral uniqueness in this paper.
Theorem 6.Let  1 ,  2 be two unknown indices of refraction parameterized in (1) in domains  1 and  2 with eigenvalue sets Z 1 ( x) and Z 2 ( x), respectively, as defined in Definition 3.  1 and  2 are assumed to be identical up to a translation in This is one local assumption implying a global uniqueness on the index of refraction.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.

Analysis along an Angle
The following results hold up to a rigid translation in R + ×S 2 .Lemma 7. Let Δ  ( x) be the density function of   defined as in (54),  = 1, 2. Up to a translation in (, x), there exists at least one accumulation point of pair of common incident directions { x0 , −x 0 } and { x 0 , −x  0 } ∈ S 2 from  1 and  2 , respectively, such that Δ 1 ( x) = Δ 2 ( x ) holds for x near x0 and x near x 0 , respectively, in S 2 .
Proof.S 2 is compact.By applying the assumption in Theorem 6, there are the accumulation points x0 and x 0 in S 2 , respectively, such that Δ 1 ( x) = Δ 2 ( x ) holds near x0 and x 0 , respectively, in S 2 .This proves the lemma.Lemma 8. Let   ( x) be the boundary defining function of domain   ,  = 1, 2. Up to a translation in (, x), there exists one pair of incident directions { x0 , −x 0 } and { x 0 , −x  0 } in S 2 , respectively, such that  1 (± x) =  2 (± x) in the neighborhood of x0 and x 0 , respectively.
Proof.By Theorem 6 assumption, we can assume the eigenvalues of  1 and  2 coincide for some ± x0 and ± x 0 , respectively, in S 2 .Considering some algebraic additions of ( 46) and (47), or (47) and (48), the boundary defining function at x appears as a part of eigenvalue asymptotics above.That is,  1 (± x) =  2 (± x) for x near x0 and x 0 , respectively.This proves the lemma.Proof.Applying Lemmas 7 and 8, this is the previous uniqueness result from [9][10][11]26].If two indices of refraction have an identical set of interior transmission eigenvalues along one of their incident directions, then they must be identical on that ray.
We deduce from the Weierstrass-Bolzano property on S 2 that there are finitely many of such neighborhoods on S 2 described in Lemmas 7 and 8.
In particular,   Δ( x) and   Δ( x) are invariant to the rigid translation of (, ).
Proposition 11.Under the assumption of Theorem 6, Proof.We deduce from 8, Proposition 9, and Lemma 10 that near some x0 and x 0 , respectively, such that  1 (± x) =  2 (± x) and  1 ( x) =  2 ( x) up to a translation as well.For each index   ,  = 1, 2, we collect all local coordinates such that (62) and (63) hold.As a result of the Weierstrass-Bolzano property on S 2 , there are mostly finitely many choices of neighborhoods of x0 and x 0 in which ( 62) and ( 63) hold, respectively, on  1 and  2 , say, the collection of the neighborhoods Because the boundary   ,  = 1, 2, is given, all of the other neighborhoods satisfying (62) and (63) in each   ,  = 1, . . ., , or   ,  = 1, . . .,  are known.Fix one of   0 and   0 , the one is locally isomorphic to all other neighborhoods by applying (62), ( 63), (64), and (65) up to some translations.Accordingly, we choose a translation Ψ : R + × S 2 → R + × S 2 such that Ψ ( 1 ) ≡  2 . (67) In this case, up to the chosen Ψ, we deduce that Applying (68) under Lemmas 10 and 7, we consider the initial value problems up to the translation Ψ and deduce By the uniqueness of ODE of (58), (59), and (69), we obtain

Proof of Theorem 6 via the Doppler's Effect
We want to show that, for a fixed x,   Δ ( x) = 1 + √ ( ( x))  , −(− x) <  <  ( x) .(72) We have the shifts of the frequency density to the space variable on the left hand side while the index of refraction is on the left hand side.Let us justify this differentiation by a perturbation theory on the index of refraction.We have seen the perturbation theory for finitely many interior transmission eigenvalues from [7,9].As observed in [9,10], the perturbation theory for finite eigenvalues implies a theory for all eigenvalues when considered with the asymptotic almost periodic structure (46), (47), and (48).
Let us consider a perturbation on the index of refraction in the form ( x) [−(− x),] where  [−(− x),] is the cutoff function defined on the interval [−(− x), ].We fix −(− x) and start the perturbation at  = (x).Outside [−(− x), ], we still take the index of refraction to be 1.We can mollify the cusp of () [−(− x),] near  in an arbitrarily small neighborhood by standard measure theory.Without loss of generality, we assume () [−(− x),] ,  = 1, 2, is differentiable, and the perturbation is carried out under the uniform norm.We recall from [9] the following theorem.
depending on , such that there exists an interior transmission eigenvalues of the index of refraction   , in -neighborhood of each   , whenever  is small enough.
The perturbation theory for finitely many eigenvalues is firstly considered in [8].Previously in [9,10], we have shown from Sommerfeld's radiation condition (4), (5), and (26) that, along x0 and x 0 , respectively, for any common interior eigenvalue  of index   ,  = 1, 2, the following identities hold up to the translation Ψ: That is, the common interior transmission eigenvalues are the intersection points of two entire functions that move

2
Advances in Mathematical Physicsthe scattered solution outside , we impose the Sommerfeld radiation condition; that is,

Lemma 2 .
For the fixed x ∈ S 2 , the determinant D () has only finitely many different zeros to the ones of    () ŷ (; ) or   ()   (; ) in each parallel adjacent strips to the imaginary axis.
2, up to the translation Ψ. (70) Surely we have  1 ≡  2 up to a translation by assumption, but Proposition 11 holds with some local information on the index of refraction.Let us assume   and   ,  =, 1, 2, are analytical functions with the assumption of Theorem 6 without assuming that  1 ≡  2 up to a translation.If we assume the results of Lemmas 7 and 8, then we can conclude thatΔ 1 ≡ Δ 2 , up to a translation.Proof.If   and   ,  = 1, 2, are analytical functions, then (60) implies that Δ  ( x),  =, 1, 2, are analytical in S 2 .Then (62) and (63) imply an analytic extension to S 2 .This and ODE (69) prove the corollary.