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

Authors: Lung-Hui Chen
Journal: Advances in Mathematical Physics
 (2017)

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## Abstract

We study the translation invariant properties of the eigenvalues of scattering transmission problem. We examine the functional derivative of the eigenvalue density functionΔ(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 ∂rΔx^=(1+nrx^)/π.


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## Body

## 1. Introduction

In this paper, we investigate an inverse spectral problem in the following homogeneous interior transmission problem:(1)Δw+k2nxw=0,inD;Δv+k2v=0,inD;w=v,on∂D;∂w∂ν=∂v∂ν,on∂D,where ν is the unit outer normal; D is a starlike domain in R3 containing the origin with boundary ∂D; n(x)∈C2(R3); n(x)>0, for x∈D; n(x)=1, for x∉D; n(0)=1. Equation (1) is called the homogeneous interior transmission eigenvalue problem. We also assume the boundary is given and defined by (2)R=Rθ,φ∈C1S2;R+,where S2 is the unit sphere with the spherical coordinate x^≔(θ,φ). Problem (1) occurs naturally when one considers the scattering of the plane waves hinging on certain inhomogeneity inside the domain D 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:(3)Δw+k2nxw=0,inD;Δv+k2v=0,inD;w-v=hl2krYlmx^,on∂D;∂w∂ν-∂v∂ν=∂∂νhl2krYlmx^,on∂D,in which hl(2)(kr) is the spherical Hankel function of order l and Ylm is the spherical harmonics of order l,m. We say k∈C is an interior transmission eigenvalue of (1) if there is a nontrivial pair of solution w,v∈C2(D)∩C1(D-) to the boundary value problem (1). To ensure the uniqueness of the scattered solution outside D, we impose the Sommerfeld radiation condition; that is, (4)limr→∞⁡r∂w∂r-ikw=0;(5)limr→∞⁡r∂v∂r-ikv=0.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 [3–6]. It is also another subject of research interest to study the existence or location of the eigenvalues [1, 4, 7–14]. 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 [15–20] 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, 24–26] 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.


## 2. Preliminaries and Main Result

Firstly, we expand the solution(v,w) 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]. (6)vx;k=∑l=0∞∑m=-lm=lal,mjlkrYlmx^;wx;k=1r∑l=0∞∑m=-lm=lbl,mylrYlmx^,where (7)r≔x,R0≤r<∞,x^=θ,φ∈S2,and jl is the spherical Bessel function of the first kind of order l. The summations converge uniformly and absolutely on compact subsets forsufficiently large|x|≥R0≫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(8)Ylmθ,φ≔2l+14πl-m!l+m!Plmcos⁡θeimφ,m=-l,…,l;l=0,1,2,…,is a complete orthonormal system in L2(S2), in which (9)Pnmt≔1-t2m/2dmPntdtm,m=0,1,…,n,where the Legendre polynomials Pn, n=0,1,…, form a complete orthogonal system in L2[-1,1]. We refer this to [4, p. 25]. By the orthogonality of the spherical harmonics, the functions (10)vl,mx≔al,mjlkrYlmx^;wl,mx≔bl,mylrrYlmx^satisfy (1) independently for |x|≥R0. It is well-known that (11)ΔR3=1r2∂∂rr2∂∂r+1r2sin⁡φ∂∂φsin⁡φ∂∂φ+1r2sin2⁡φ∂2∂θ2,so we apply the Laplacian to the first equation of (1) for |x|≫R0, where n(x)=1, and we observe that (12)∑l=0∞∑m=-lm=lbl,mΔR3ylrrYlmx^=∑l=0∞∑m=-lm=lbl,m1r2∂∂rr2∂∂rylrrYlmx^+∑l=0∞∑m=-lm=lbl,mylrr1r2ΔS2Ylmx^=-k2nx∑l=0∞∑m=-lm=lbl,mylrrYlmx^.Accordingly, the Fourier coefficients yl(r) trivially satisfy the following equation: (13)yl′′+k2nx^r-ll+1r2yl=0,R0≤r<∞.Most importantly, we extend the Fourier coefficients yl(r) along some fixed incident direction x^ into |x|≤R0 by considering the following scheme. Let us define (14)n^r≔nrx^.For the fixed x^∈S2, we consider the radially symmetric index of refraction in C2(R3) by rotating n^(r) around the origin. Due to the radial-symmetry, the solution yl of (13) has an extension into |x|≤R0 depending on the incident direction x^ by solving the following ODE: (15)y^l′′+k2n^r-ll+1r2y^l=0,0<r<R0;limr→0⁡y^lrr-jlkr=0.We denote the solution to (15) as y^l(r;k). We renormalize the initial condition as follows: (16)limr→0⁡y^lrr-jlkr=0is independent of the incident direction x^. We refer the initial condition to [4, 14].Now we consider the following Liouville transformation for each fixedx^∈S2: (17)z^liξ^i;k≔nirx^1/4y^lr;k,whereξ^i≔∫0Rx^niρx^1/2dρ,i=1,2.Let us define (18)B^i≔∫0Rx^niρx^1/2dρ,i=1,2.Then, we deduce from (15) and (17) that (19)z^li′′+k2-p^iξ^iz^li=0,0≤ξ^i≤B^i,i=1,2,in which (20)p^iξ^i≔n^i′′r4n^ir2-516n^i′r2n^ir3+ll+1r2n^ir.Here ξ^i=B^i if and only if r=R(x^) for the fixed x^. Moreover, (21)q^iξ^i≔n^i′′r4n^ir2-516n^i′r2n^ir3+ll+1r2n^ir-ll+1ξ^i2.Thus, we deduce from (19), (20), and (21) that (22)z^li′′+k2-q^iξ^i-ll+1ξ^i2z^li=0,0≤ξ^i≤B^i.For the simplicity of the notation, we drop the superscripts on the variables. When l=0, the asymptotic estimates for the solution z0(r) are classic and can be found in [27] (23)z0ξ;k=sin⁡kξk-cos⁡kξ2k2Qξ+sin⁡kξ4k3pξ+p0-12Q2ξ+Oexp⁡Ikξk4,where Q(ξ)≔∫0ξp(s)ds, and the error term can be improved [27, p. 17] by (24)cos⁡kξ8k4p′ξ-p′0-pξ+p0Qξ-∫0ξp2sds+16Q3ξ+oexp⁡Ikξk4.Similarly,(25)z0′ξ;k=cos⁡kξ+sin⁡kξ2kQξ+cos⁡kξ4k2pξ-p0-12Q2ξ+Oexp⁡Ikξk3,in which the boundary behavior of p(ξ) plays a role in determining the inverse spectral uniqueness on the scatterer, and thus C2-assumption on the index of refraction is convenient.Forl≥-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, 27–30] 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 -l≤m≤l,l=0,1,2,…, (26)a^l,mjlkrr=R-b^l,my^lrrr=R=0;a^l,mjlkr′r=R-b^l,my^lrr′r=R=0,in which we note that R=R(x^). The eigenvalues are possibly the intersection points of two complex-valued functions, and the constants a^l,m and b^l,m depend on x^ as well. Let us define (27)D^lk≔det⁡jlkrr=R-y^lrrr=Rjlkr′r=R-y^lrr′r=R,in which D^l(k) is independent of index m. The existence of the nontrivial (a^l,m,b^l,m) of (26) is equivalent to finding thezeros of (28)D^lk=0.The following fundamental lemma connects thezeros of D^l(k)=0 and theeigenvalues of (1).Lemma 1.
Letk be an eigenvalue of (1) if and only if it satisfies (28) for all l and some x^∈S2.Proof.
Letv(x;k) and w(x;k) be a nontrivial pair of solutions of (1). Then, (6) holds and uniquely defines the coefficients y^l(r;k) and all l≥0 for r≥R0 and extends into |x|≤R0 by solving (15) with some fixed x^. The solution, (29)wl,mx≔bl,mylrrYlmx^,∀l≥0,each satisfies the Helmholtz equation independently and meets with vl,m(x) in |x|≫R0. By the analytic continuation of the Helmholtz equation [1], they extend to meet on ∂D. That is, (28) is satisfied at x^∈S2. Hence, (28) holds on x^ for all l≥0.
For the sufficient condition, if (28) is valid on the boundary for some x~, then y~l(r) define a Fourier coefficient by ODE (15) by some eigenvalue k0, which defines a solution of the Helmholtz equation (30)wl,mx;k0≔bl,my~lr;k0rYlmx^,∀l≥0,for |x|≥R0. By the analytic extension of the Helmholtz equation, wl,m(x;k0) is defined outside D and meets vl,m(x;k0) on ∂D. That is, (28) holds for all other x^∈S2 with k0. Hence, (31)a~l,mjlk0rr=R=b~l,my^lr;k0rr=R;a~l,mjlk0r′r=R=b~l,my^lr;k0r′r=R,x^≠x~,which we consider as aninitial condition for that Bessel function jl(k0r), are known functions. Combining the initial condition with (32)y^l′′+k2n^r-ll+1r2y^l=0,we find that the solution w(x;k0) defines an eigenfunction to (1). We refer more detailed construction to [11].Forl=0, D^l(k) is easier to compute [24, (2.10)]: (33)D^0k=sin⁡kRky^0′R;k-cos⁡kRy^0R;k,in which all four elements have zeros distributed asymptotically like sine and cosine functions.Moreover,D^0(k) has the following Hadamard representation: (34)D^0k=γ^kd^∏j1-kk^j,where γ^ is a constant [24, (2.14)]. From [9, 10], the zeros spread along the real axis by Cartwright theory [19, Ch. 6] and [18, 20]. In general l≥-1/2, we state the following lemma.Lemma 2.
For the fixedx^∈S2, the determinant D^l(k) has only finitely many different zeros to the ones of kjl′(k)y^l(R;k) or jl(kR)yl′(R;k) in each parallel adjacent strips to the imaginary axis.Proof.
We infer from (27) that (35)D^lk=-jlkRy^l′R;kR+jlkRy^lR;kR2+kjl′kRy^lR;kR;(36)D^lk=kjl′kRy^lR;kR1-1kjlkRjl′kRy^l′R;ky^lR;k+1kRjlkRjl′kR,in which we recall from [9] and [10, Lemma 2.5] that jl(kR)/jl′(kR)=O(1) and y^l′(R;k)/y^l(R;k)=O(k) outside the zeros in the denominators. Without loss of generality, we drop the lower order term (1/k)(jlkR/jl′kR) in the bracket in (36) and the term 1-(1/k)(jlkR/jl′kR)(y^l′R;k/y^lR;k) is bounded and bounded away from zero near the zeros of jl′(kR) and y^l(R;k). The asymptotic expansion of y^l(R;k) for large k is given in (23) and in [28, 29]. In particular, D^l(k) is bounded near the real axis and then in the Cartwright class [19, p. 251].
More importantly, we drop the lower order term in (35) and then consider Rouché’s Theorem for the identity (37)D^lk;R-kjl′kRy^lR;kR=jlkRyl′R;kR.The inequality (38)jlkRyl′R;kR<kjl′kRy^lR;kRholds if and only if (39)y^l′R;ky^lR;k<kjl′kRjlkR.We recall that (40)jlkRjl′kR=sin⁡kcos⁡k1+O1k;y^l′R;ky^lR;k=kcos⁡kB^sin⁡kB^1+O1k,outside their poles, where (41)B^≔∫0Rx^nx^ρdρ.Hence, (39) holds if and only if (42)cot⁡kB^<cot⁡kR,for  largek,which are two asymptotically periodic meromorphic functions along the real axis. Referring the graph of cot⁡k to [31], there are vertical lines in C passing through the real axis at which cot⁡k=0 and then strictly increases to 1 or at which cot⁡k=∞ and strictly decreases to 1, as Ik goes to infinity. For large k, there are infinitely many vertical lines of this property on which the following inequality holds:(43)D^lk;R-kjl′kRy^lR;kR<kjl′kRy^lR;kR.
Similarly, we may repeat the argument from (37) to deduce that there are infinitely many vertical lines in C on which the following equality holds: (44)D^lk;R-jlkRyl′R;kR<jlkRyl′R;kR.
Therefore, we apply Rouché’s theorem to (43) and (44) on the boundary of a family of suitable vertical strips along the real axis and conclude that D^l(k) have only finite many of different zeros to the ones of kjl′(k)y^l(R;k) or jl(kR)yl′(R;k) in the strips.One step further, we can describe the zero distribution ofD^l(k) more precisely by applying Wilder’s theorem [32, 33] to the term kjl′(k)y^l(R;k) or jl(kR)yl′(R;k) in (35) and obtain the following density distribution theorem for (33): let R11, R12, and R13 be three adjacent strips containing all but a finite exceptions of the zeros along the positive real axis, and let N(R1i(α,s,K)),i=1,2,3, be the zero counting function in the strip starting with Ik=α, of length s and of width K. Then there exist some K>0 and large α such that (45)B^≔∫0Rx^nx^ρdρ;(46)NR11α,s,K~sB^π,Rx^≥B^;sRx^π,Rx^<B^;(47)NR12α,s,K~sRx^-B^π;(48)NR13α,s,K~sB^π,Rx^≥B^;sRx^π,Rx^<B^.A similar argument holds for jl(kR)yl′(R;k) that yields identical asymptotics to (46), (47), and (48). We refer the details to [10, Theorem 4.2].From the point of view of the inhomogeneous system (3), (49)a^l,mjlkrr=R-b^l,my^lrrr=R=hl2krYlmx^r=R;a^l,mddrjlkrr=R-b^l,mddry^lrrr=R=ddrhl2krYlmx^r=R,to which we can solve for a^l,m, b^l,m by Cramer’s rule: (50)a^l,mk=det⁡y^lRhl2krYlmx^r=Ry^lr/r′r=Rd/drhl2krYlmx^r=RD^lk;b^l,mk=det⁡hl2krYlmx^r=R-jlkRd/drhl2krYlmx^r=R-jl′krr=RD^lk,where a^l,m(k), b^l,m(k) are understood as two meromorphic functions. An interior transmission eigenvalue means that the inhomogeneous system (3) can not be solved explicitly in C2(D)∩C1(D¯) whenever D^l(k) has a zero in the denominator. We observe that (w(r;k),v(r;k)) is not a solution in the classic sense if there is any l and any x^ such that D^l(k) has a zero. Whenever that happens, we have an interior transmission eigenvalue.Definition 3.
We defineZ(x^) to be the union of the zero set of entire function D^0(k) at x^ and -x^.We need the following vocabulary to describe the asymptotic behavior of the eigenvalues.Definition 4.
Letf(z) be an integral function of order ρ, and N(f,α,β,r) denote the number of the zeros of f(z) inside the angle [α,β] and |z|≤r; we define the zero density function (51)Δfα,β≔limr→∞⁡Nf,α,β,rrρ,Δfβ≔Δfα0,β,with some fixed α0∉E such that E is at most a countable set [18–20].
Following the results in [9–11, 26], we have the density asymptotics: (52)ΔD^0-ϵ,ϵ=ΔD^0π-ϵ,π+ϵ=Rx^+∫0Rx^nx^ρdρπ;(53)ΔD^0ϵ,π-ϵ=ΔD^0π+ϵ,2π-ϵ=0.For a simpler notation, we use (54)Δx^≔ΔZx^-ϵ,ϵto denote the density of zero set of Z(x^).
One can show the following.Lemma 5.
Let us denote(55)B^≔∫-R-x^Rx^nx^ρdρ,and the following identity holds: (56)Δx^=Rx^+R-x^+B^π.Accordingly, the density function is invariant to the rigid translations of D.Proof.
We observe that(57)∫0Rx^nx^ρdρ+∫0R-x^n-x^ρdρ=B^.By applying (52) in (54) to x^ and -x^, respectively, (56) is proven. The invariant property comes from the translation transformation along r of integral B^. The invariant property on rotation is trivial.We state the inverse spectral uniqueness in this paper.Theorem 6.
Letn1,n2 be two unknown indices of refraction parameterized in (1) in domains D1 and D2 with eigenvalue sets Z1(x^) and Z2(x^), respectively, as defined in Definition 3. D1 and D2 are assumed to be identical up to a translation in (r,x^)∈R+×S2. If Z1(x^)=Z2(x^) for infinitely many pairs of (x^,-x^)∈S2×S2, then n1≡n2 up to a translation in (r,x^)∈R+×S2.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 n1(0)=n2(0)=1, then we can prove the inverse uniqueness. Here we are given a potpourri of spectral data.


## 3. Analysis along an Angle

The following results hold up to a rigid translation inR+×S2.Lemma 7.
LetΔi(x^) be the density function of ni defined as in (54), i=1,2. Up to a translation in (r,x^), there exists at least one accumulation point of pair of common incident directions {x^0,-x^0} and {x^0′,-x^0′}∈S2 from n1 and n2, respectively, such that Δ1(x^)=Δ2(x^′) holds for x^ near x^0 and x^′ near x^0′, respectively, in S2.Proof.
S
2 is compact. By applying the assumption in Theorem 6, there are the accumulation points x^0 and x^0′ in S2, respectively, such that Δ1(x^)=Δ2(x^′) holds near x^0 and x^0′, respectively, in S2. This proves the lemma.Lemma 8.
LetRi(x^) be the boundary defining function of domain Di,i=1,2. Up to a translation in (r,x^), there exists one pair of incident directions {x^0,-x^0} and {x^0′,-x^0′} in S2, respectively, such that R1(±x^)=R2(±x^) in the neighborhood of x^0 and x^0′, respectively.Proof.
By Theorem6 assumption, we can assume the eigenvalues of n1 and n2 coincide for some ±x^0 and ±x^0′, respectively, in S2. 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, R1(±x^)=R2(±x^) for x^ near x^0 and x^0′, respectively. This proves the lemma.Proposition 9.
Let us assumen1(0)=n2(0)=1. There exists one fixed pair of incident directions {x^0,-x^0} and {x^0′,-x^0′} in S2, respectively, such that n1(±rx^)≡n2(±rx^′) for x^ near x^0 and x^′ near x^0′, respectively.Proof.
Applying Lemmas7 and 8, this is the previous uniqueness result from [9–11, 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 onS2 that there are finitely many of such neighborhoods on S2 described in Lemmas 7 and 8.Lemma 10.
The density functionΔ(x^) has the derivatives on ∂D: (58)∂θΔx^=2π∂θRx^;(59)∂φΔx^=2π∂φRx^,x^∈S2.In particular, ∂θΔ(x^) and ∂φΔ(x^) are invariant to the rigid translation of (n,D).Proof.
Applying the chain rule tox^ in (56) with -x^ fixed and setting n′(x^R(x^))=0, (60)∂θΔx^=1π∂θRx^1+nRx^;∂φΔx^=1π∂φRx^1+nRx^.Observing the boundary conditions of (1), n(R(x^))=1,∀x^∈S2. Thus, we deduce (58) and (59). We note that ∂θΔ(x^) and ∂φΔ(x^) depend only on the boundary ∂D. This proves the lemma.Proposition 11.
Under the assumption of Theorem6, (61)Δ1≡Δ2,up  to  a  translation.Proof.
We deduce from Lemma8, Proposition 9, and Lemma 10 that (62)∂θΔ1x^=∂θΔ2x^;(63)∂φΔ1x^=∂φΔ2x^;(64)R1±x^=R2±x^;(65)n1±rx^=n2±rx^,near some x^0 and x^0′, respectively, such that R1(±x^)=R2(±x^) and n1(rx^)=n2(rx^) up to a translation as well. For each index ni,i=1,2, we collect all local coordinates such that (62) and (63) hold. As a result of the Weierstrass-Bolzano property on S2, there are mostly finitely many choices of neighborhoods of x^0 and x^0′ in which (62) and (63) hold, respectively, on ∂D1 and ∂D2, say, the collection of the neighborhoods (66)O1≔ξ1,ξ2,…,ξm;O2≔η1,η2,…,ηn.Because the boundary Di,i=1,2, is given, all of the other neighborhoods satisfying (62) and (63) in each ξj, j=1,…,m, or ηj,j=1,…,n are known. Fix one of ξj0 and ηj0, 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+×S2→R+×S2 such that (67)ΨR1≡R2.In this case, up to the chosen Ψ, we deduce that (68)∂θΔ1≡∂θΔ2;∂φΔ1≡∂φΔ2.
Applying (68) under Lemmas 10 and 7, we consider the initial value problems up to the translation Ψ and deduce (69)∂θΔ1Ψx^=∂θΔ2x^,x^∈S2;∂φΔ1Ψx^=∂φΔ2x^,x^∈S2;Δ1Ψx^0=Δ2x^0′.By the uniqueness of ODE of (58), (59), and (69), we obtain (70)Δ1≡Δ2,up  to  the  translationΨ.Surely we have∂D1≡∂D2 up to a translation by assumption, but Proposition 11 holds with some local information on the index of refraction.Corollary 12.
Let us assumeni and Ri,i=,1,2, are analytical functions with the assumption of Theorem 6 without assuming that ∂D1≡∂D2 up to a translation. If we assume the results of Lemmas 7 and 8, then we can conclude that (71)Δ1≡Δ2,up  to  a  translation.Proof.
Ifni and Ri,i=1,2, are analytical functions, then (60) implies that Δi(x^),i=,1,2, are analytical in S2. Then (62) and (63) imply an analytic extension to S2. This and ODE (69) prove the corollary.


## 4. Proof of Theorem6 via the Doppler’s Effect

We want to show that, for a fixedx^, (72)∂rΔx^=1+nrx^π,-R-x^<r<Rx^.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 formn(rx^)χ[-R(-x^),r] where χ[-R(-x^),r] is the cut-off function defined on the interval [-R(-x^),r]. We fix -R(-x^) and start the perturbation at r=R(x^). Outside [-R(-x^),r], we still take the index of refraction to be 1. We can mollify the cusp of n(r)χ[-R(-x^),r] near r in an arbitrarily small neighborhood by standard measure theory. Without loss of generality, we assume n(r)χ[-R(-x^),r],i=1,2, is differentiable, and the perturbation is carried out under the uniform norm. We recall from [9] the following theorem.Theorem 13.
Let{kj} be the zeros of D^0(k). Then, for any σ>0, there exists (73)η=sup⁡n′-n,depending on σ, such that there exists an interior transmission eigenvalues of the index of refraction n′, in σ-neighborhood of each kj, whenever η 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 x^0 and x^0′, respectively, for any common interior eigenvalue k of index ni,i=1,2, the following identities hold up to the translation Ψ: (74)y^l1R1x^0;k=y^l2R2x^0′;k;y^l1′R1x^0;k=y^l2′R2x^0′;k.That is, the common interior transmission eigenvalues are the intersection points of two entire functions that move continuously to the perturbation to the index of refraction on either side of (74) according to Theorem 13.Following [10, (2.36)] and [9], the zero density of entire functions (75)y^l1R1x^0;k-y^l2R2x^0′;kor (76)y^l1′R1x^0;k-y^l2′R2x^0′;kis (77)max⁡∫0R1x^0n1x^0ρdρπ,∫0R2x^0′n2x^0′ρdρπ,while the density for the common zero set is (78)ΔD^0i-ϵ,ϵ=Rix^0+∫0Rix^0nix^0ρdρπ,i=1,2,according to the assumption of Theorem 6 and (52). This contradicts with (77), and so Cartwright’s theory [9, 10, 18–20, 26] suggests that (79)y^l1R1x^0;k≡y^l2R2x^0′;k;y^l1′R1x^0;k≡y^l2′R2x^0′;k.We refer the details of (58) up to (63) to [10] or [19, p. 251] for a Cartwright theory. In particular, n1 and n2 have identical sets of norming constants and Dirichlet/Neumann eigenvalues for the Sturm-Liouville problem (15). Hence, [29, Theorem 1.2] implies n1=n2 along x^0 and x^0′, respectively.Let us consider the perturbation on both sides of (79) in the following form: (80)nirx^χRi-x^,r⟼nirx^χRi-x^,r′,r′<r<Rix^.We denote the new solutions again by y^li,i=1,2. For each r′, Theorem 13 and (52) imply that the new density function of the common set of interior transmission eigenvalues equals to (81)r′x^0+∫0r′x^0n1x^0ρdρπorr′x^0′+∫0r′x^0′n2x^0′ρdρπ.Without loss of generality, we take x^0=x^0′ due to (67).Wheneverr′ is close to r, this overdetermined density (81) again, by Cartwright’s theory, implies the identities for the new solutions. (82)y^l1r′;k≡y^l2r′;k;(83)y^l1′r′;k≡y^l2′r′;k.Moreover, we deduce from (82) and (83) again that the perturbed index of refraction nirx^0χ-Ri-x^0,r′,i=1,2, have an identical set of interior transmission eigenvalues [9, 10]. From (81) and Theorem 13, (84)r′x^0+∫0r′x^0n1x^0ρdρπ=r′x^0+∫0r′x^0n2x^0ρdρπ.The perturbation applies in -Ri-x^0<r′<Rix^0,i=1,2.We repeat the same argument to the other incident directionsx^∈S2. Thus, we apply Proposition 11 to consider the limit quotient for all x^∈S2: (85)limr′→r⁡r′x^-rx^+∫rx^r′x^nix^ρdρπr′x^-rx^,i=1,2.Thus, we apply the Lebesgue Differentiation Theorem to deduce that (86)∂rΔix^=1+nirx^π,r>-Rix^,∀x^∈S2.From Proposition 11, we conclude that (87)n1≡n2up  to  the  translationΨ.



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*Source: 3838507-2017-05-21.xml*