Localization of the interior transmission eigenvalues for a ball

We study the localization of the interior transmission eigenvalues (ITEs) in the case when the domain is the unit ball $\{x \in {\mathbb R}^d:\: |x| \leq 1\}, \: d\geq 2,$ and the coefficients $c_j(x), \: j =1,2,$ and the indices of refraction $n_j(x), \: j =1,2,$ are constants near the boundary $|x| = 1$. We prove that in this case the eigenvalue-free region obtained in [16] for strictly concave domains can be significantly improved. In particular, if $c_j(x), n_j(x), j = 1,2$ are constants for $|x| \leq 1$, we show that all (ITEs) lie in a strip $\{ \lambda \in {\mathbb C}:\:|{\rm Im}\: \lambda| \leq C\}$.


Introduction and statement of the result
Let Ω ⊂ R d , d ≥ 2, be a bounded, connected domain with a C ∞ smooth boundary Γ = ∂Ω. A complex number λ = 0 with Re λ ≥ 0 will be called interior transmission eigenvalue (ITE) if the following problem has a non-trivial solution:      ∇c 1 (x)∇ + λ 2 n 1 (x) u 1 = 0 in Ω, ∇c 2 (x)∇ + λ 2 n 2 (x) u 2 = 0 in Ω, u 1 = u 2 , c 1 ∂ ν u 1 = c 2 ∂ ν u 2 on Γ, (1.1) where ν denotes the Euclidean unit inner normal to Γ and c j (x), n j (x) ∈ C ∞ (Ω), j = 1, 2, are strictly positive real-valued functions. The (ITEs) were first studied by Kirsch [7] and by Colton and Monk [2] in the context of inverse scattering problems. It was shown that the real (ITEs) correspond to the frequencies for which the reconstruction algorithm in inverse scattering based on the so-called linear sampling methods breaks down. This subject attracted the attention of many researchers and the number of publications devoted to the (ITEs) considerably increased in the recent ten years. The reader may consult the survey [1] for a complete list of references and historical remarks.
It is well-known (e.g. see [15]) that there exists a closed non-symmetric operator, A, associated in a natural way to the problem (1.1), such that the possible (ITEs) can be considered as the eigenvalues of A. The analysis of the (ITEs) leads to the following three problems: (A) Prove the discreteness of the spectrum of A in C; (B) Find eigenvalue-free regions in C; (C) Establish a Weyl formula for the counting function of all (ITEs) N (r) = #{λ j is (ITE), |λ j | ≤ r}. 1 Note that the problem (A) is now relatively well studied (see [9], [14], [11], [4] and the references therein). In fact, the problem (A) is reduced to that one of showing that the resolvent of A is meromorphic with residues of finite rank. On the other hand, this is true (see [15]) if the inverse of the operator T (λ) introduced in Section 4 is meromorphic. The latter fact can be proved if the parametrix of the operator T (λ) constructed in the deep elliptic zone is invertible.
The problems (B) and (C) are more difficult, and they are of some interest for the numerical analysis of the (ITEs). In this direction it is interesting to find an optimal eigenvalue-free region and a Weyl formula with optimal remainder (see [12], [5], [13], [8] and the references therein). In a recent work [15] the authors showed that (B) and (C) are closely related each other, and a larger eigenvalue-free region leads to a Weyl asymptotics with a smaller remainder term. More precisely, we proved that the remainder in the Weyl formula is O ε (r d−κ+ε ), ∀ 0 < ε ≪ 1, where 0 < κ ≤ 1 is such that there are no (ITEs) in We conjecture that the optimal value of κ must be κ = 1.
The present paper is devoted to the problem (B). More precisely, we are interested in finding as small as possible neighborhoods of the real axis containing all (ITEs). The first result of this type was obtained in [6] assuming n 1 (x) > 1 inΩ and n 2 (x) ≡ 1, c 1 (x) ≡ c 2 (x) ≡ 1. For domains Ω with arbitrary geometry, it has been shown in [16] that under the condition (isotropic case) or the condition (anisotropic case) there are no (ITEs) in the region The localization of the (ITEs) has been recently studied in [17] in the case when the boundary Γ is strictly concave with respect to both Riemannian metrics d k=1 n j (x) c j (x) dx 2 k , j = 1, 2. Under the conditions (1.2) or (1.3) it has been proved in [17] that there are no (ITEs) in the region The approach in [16] and [17] is based on the construction of a semi-classical parametrix near the boundary for the problem where 0 < h ≪ 1 is a semi-classical parameter and z ∈ C with Re z = 1. For domains with arbitrary geometry the parametrix construction for (1.5) works for |Im z| ≥ h 1/2−ǫ , 0 < ǫ ≪ 1, while for strictly concave domains, by a more complicated construction, one can cover the region |Im z| ≥ h 1−ǫ . It is a challenging problem to construct a semi-classical parametrix for (1.5) when |Im z| ≥ Ch, C ≫ 1 being a constant. The purpose of the present paper is to improve the eigenvalue-free region (1.4) in the case when the domain is the unit ball in R d , d ≥ 2. Given a parameter 0 < δ ≪ 1, denote Ω(δ) = {x ∈ Ω : dist(x, Γ) ≤ δ}. Our main result is the following Theorem 1.1. Let Ω = {x ∈ R d : |x| ≤ 1}, d ≥ 2, and suppose that there is a constant 0 < δ 0 ≪ 1 such that the functions c j (x), n j (x), j = 1, 2, are constants in Ω(δ 0 ). Assume also either the condition (1.2) or the condition (1.3). Then, there is a constant C > 0 such that there are no (IT Es) in the region {λ ∈ C : Re λ > 1, |Im λ| ≥ C log (Re λ + 1)}. (1.6) If in addition the functions c j , n j , j = 1, 2, are constants everywhere in Ω, then there are no (IT Es) in a larger region of the form Remark 1. The eigenvalue-free region (1.6) is still valid if we add a compact cavity K ⊂ Ω and consider the equation (1.1) in Ω \ K with Dirichlet condition on ∂K. Indeed, the only fact we need is the coercivity of the corresponding Dirichlet realization (see the operator G D in Section 3), and this is used only in the proof of Lemma 3.4 below.
Remark 2. It is clear from the proof that the fact that the boundary Γ is a sphere is not essential. In other words, the eigenvalue-free regions (1.6) and (1.7) are still valid for any Riemannian manifold Ω = (0, 1) × Γ with metric g = dr 2 + r 2 σ, where r ∈ (0, 1), and (Γ, σ) is an arbitrary (d−1)-dimensional Riemannian manifold without boundary, the metric σ being independent of r.
In the isotropic case when c j ≡ 1, j = 1, 2 and n 1 = 1, n 2 = 1 is constant, the eigenvalue-free region (1.7) has been established in the one-dimensional case Ω = {x ∈ R : |x| ≤ 1} (see [14], [11]). Moreover, the case of the ball {x ∈ R d : |x| ≤ 1}, d = 2, 3, and radial refraction indices have been studied in [9], [3], [4], where spherical symmetric eigenfunctions depending only on the radial variable r = |x| has been considered. For example, the analysis of such eigenfunctions in R 3 leads to the following one-dimensional problem where n(r) is a strictly positive function. Among other things, it was shown in [4] that if n(1) = 1 and n ′ (1) or n ′′ (1) is non-zero, then there may exist infinitely many complex eigenvalues of the problem (1.8) lying outside any strip parallel to the real axis. This example shows that in the isotropic case the condition n(1) = 1 (resp. (1.2)) is important to have an eigenvalue-free region like (1.7). It also follows from the analysis in [9] (see Sections 3 and 4) that when n = Const and √ n is a rational number belonging to the interval (1, 2), then there exists a sequence of (ITEs), λ k = αk + β, k = 0, 1, 2, ..., with some constants α > 0 and β ∈ C, Im β = 0. This example shows that the eigenvalue-free region (1.7) is sharp and cannot be improved in general.
To study all (ITEs) and all eigenfunctions, however, one has to consider a family of infinitely many one-dimensional problems. Such an analysis is carried out in [11] in the isotropic case when the domain is the ball {x ∈ R d : |x| ≤ 1}, d ≥ 1, and In this case the (ITEs) are the zeros in C of the family of functions .., where J ν denotes the Bessel function of order ν. It has been proved in [11] that there are infinitely many real (ITEs) whose counting function has a Weyl asymptotics. When d = 1 a Weyl asymptotics for the counting function of all (ITEs) is also obtained.
To prove Theorem 1.1 we follow the same strategy as in [16], [17], which consists of deriving the eigenvalue-free region from some approximation properties of the interior Dirichlet-to-Neumann (DN) map. In our case we have to approximate the DN map where ν is the unit inner normal to Γ = ∂Ω and u solves the equation ∆ being the negative Euclidean Laplacian. Recall that the interior DN map is a meromorphic operator-valued function with poles lying on the positive real axis. Thus, the eigenvalue-free region turns out to be the region in which the DN map can be approximated by a simpler operator of the form f (∆ Γ ), where f is a complex-valued function and ∆ Γ denotes the negative Laplace-Beltrami operator on the boundary Γ equipped with the Riemannian metric induced by the Euclidean one. With such an approximation in hands, the problem of proving the eigenvaluefree region is transformed into the much simpler one of inverting complex-valued functions, which in turn is done using the conditions (1.2) or (1.3) (see Section 4). Therefore, a large portion of the present paper is devoted to the study of the interior DN map N 0 (λ) using the Bessel functions. Thus, instead of a parametrix we have an exact formula of the DN map (see Theorem 3.1 and its proof). Then we use the asymptotic expansions of the Bessel functions in terms of the Airy function to get the desired approximation (see Theorems 2.1 and 3.1). Of course, we cannot proceed in this way when the coefficients are supposed to be constants only in a neighborhood of the boundary. In this latter case we show that the DN map can be approximated by the DN map associated to the corresponding problem with constant coefficients everywhere and for which we have an explicit expression in terms of the Bessel functions (see Lemma 3.4). We expect that the eigenvalue-free regions (1.6) and (1.7) are still true for more general domains, but this is hard to prove because the available semi-classical parametrix constructions for the DN map lead to the existence of smaller regions (see [16], [17]). Theorem 2.1. For every 0 < δ ≪ 1, there are positive constants C δ , C ′ δ and δ 1 such that for There exist also positive constants C, C ′ , C 1 , C 2 and δ 1 independent of ν but depending on κ such that for Re λ ≥ C 1 , C 2 ≤ |Im λ| ≤ δ 1 Re λ, ν ≥ 0, we have the estimate Proof. We will consider several cases.
are the Hankel functions 1 having the asymptotic expansions (see (4.03) and (4.04), p.238 in [10]) provided |λ| and Im λ are taken large enough. We can write the function ψ ν as follows . By using the above inequalities, we get . Indeed, as above, one can easily see that η In this case we will use the asymptotic expansions of the Bessel functions in terms of the Airy function Ai(σ). Recall first that Ai(σ) has the expansion are the Hankel functions of first and second kind for |σ| ≫ 1, σ ∈ Λ ε := {σ ∈ C : |arg σ| ≤ π − ε}, 0 < ε ≪ 1, where β ℓ are real numbers and the fractional powers of σ take their principal values. The expansion (2.5) implies where β 0 = 1, β 1 = 1/4. The behavior of the function F in C \ Λ ε is more complicated and is given by the following

Some properties of the interior Dirichlet-to-Neumann map
Let Ω = {x ∈ R d : |x| ≤ 1}, Γ = ∂Ω, and let λ ∈ C with 1 ≪ |Im λ| ≪ Re λ. Given a function f ∈ H s+1 (Γ), let u solve the equation where ∆ is the negative Euclidean Laplacian. We define the interior Dirichlet-to-Neumann (DN) map ν being the unit inner normal to Γ. Let ∆ Γ be the negative Laplace-Beltrami operator on the boundary Γ equipped with the Riemannian metric induced by the Euclidean one. In what follows we will denote by H 1 sc (Γ) the Sobolev space equipped with the semi-classical norm f H 1 sc (Γ) = (I − |λ| −2 ∆ Γ ) 1/2 f L 2 (Γ) where I denotes the identity. For σ ≥ 0, set Theorem 3.1. For every 0 < δ ≪ 1, independent of λ, there are positive constants C δ , C δ and δ 1 = δ 1 (δ) such that for Re λ ≥ C δ , C δ ≤ |Im λ| ≤ δ 1 Re λ, we have the estimate Proof. We will express the DN map in terms of the Bessel functions. If r = |x| is the radial variable, we have Let {µ 2 j } be the eigenvalues of −∆ Γ repeated with their multiplicities and let {e j }, e j = 1, be the corresponding eigenfunctions, that is, −∆ Γ e j = µ 2 j e j . Denote by ·, · and · the scalar We will now study the DN map in a more general situation. Let c(x), n(x) ∈ C ∞ (Ω) be strictly positive functions and define the DN map associated to these functions by where u is the solution to the equation We suppose that there exist a constant 0 < δ 0 ≪ 1 and positive constants c and n such that c(x) = c, n(x) = n in Ω(δ 0 ). Set Theorem 3.3. For every 0 < δ ≪ 1, independent of λ, there are positive constants C δ , C δ and δ 1 = δ 1 (δ) such that for Re λ ≥ C δ , C δ log |λ| ≤ |Im λ| ≤ δ 1 Re λ, we have the estimate Proof. We will compare N (λ) with the DN map N (λ) defined by where u is the solution of the equation In other words, the estimate (3.2) holds true with N 0 and ρ 0 replaced by N and ρ, respectively. Therefore, one can easily see that Theorem 3.3 follows from Theorem 3.1 and the following Proof. Denote by G D and G D the Dirichlet self-adjoint realizations of the operators −n −1 ∇c∇ and − n −1 c∆ on the Hilbert spaces L 2 (Ω, n(x)dx) and L 2 (Ω, dx), respectively. Let χ 1 be a smooth function depending only on the radial variable such that χ 1 = 1 in Ω(δ 0 /3), χ 1 = 0 in Ω \ Ω(δ 0 /2). Let u 1 be the solution to (3.7) and u 2 the solution to (3.9), u 1 = u 2 = f on Γ. We have u 1 − χ 1 u 2 = 0 on Γ and Hence where γ denotes the restriction on Γ. By (3.11) we obtain where the Sobolev space H 2 (Ω) is equipped with the usual norm. We will use now the fact that the norm in H 1 sc (Γ) is bounded from above by the usual norm in H 1 (Γ). Thus, by the trace theorem and the coercivity of G D and G D we have