Counting function for interior transmission eigenvalues

In this paper we give results on the counting function associated with the interior transmission eigenvalues. For a complex refraction index we estimate of the counting function by Ctn. In the case where the refraction index is positive we give an equivalent of the counting function.

1. Introduction. In this paper we give an estimate to the counting function associated with the interior transmission eigenvalues. We recall the problem. Let Ω be a smooth bounded domain in R n . Let n(x) be a smooth function defined in Ω, called the refraction index. We say that k = 0 is a interior transmission eigenvalue if there exists (w, v) = (0, 0) such that        ∆w + k 2 n(x)w = 0 in Ω, ∆v + k 2 v = 0 in Ω, w = v on ∂Ω, where ∂ ν is the exterior normal derivative to ∂Ω. We consider here the function n(x) complex valued. In physical models, we have n(x) = n 1 (x)+in 2 (x)/k where n j are real valued. Taking u = w − v andṽ = k 2 v, we obtain the following equivalent system if k = 0,    ∆ + k 2 (1 + m) u + mv = 0 in Ω, (∆ + k 2 )v = 0 in Ω, u = ∂ ν u = 0 on ∂Ω, where, for simplicity, we have replacedṽ by v and n by 1 + m. When k ∈ R, this problem is related with scattering problem. We can find a precise result in Colton and Kress [8,Theorem 8.9] first proved by Colton, Kirsch and Päivärinta [7] and in a survey by Cakoni and Haddar [5].
As the problem si not self-adjoint even for n(x) real valued, usual tools used in self-adjoint cases cannot be applied, in particular, even for operator with compact resolvent, the existence of k is not always true.
A lot of results was obtained this last years using several methods. When n(x) is real, Päivärinta and Sylvester [20] proved that there exist interior transmission eigenvalues; Cakoni, Gintides, and Haddar [4] proved that the set of k 2 j is infinite and discrete. For n(x) complex valued Sylvester [23] proved that this set is discrete 168 LUC ROBBIANO finite or infinite. In [22] we proved that there exist a infinite number of complex eigenvalues and the associated generalized eigenspaces span a dense space in L 2 (Ω)⊕ L 2 (Ω).
Lakshtanov and Vainberg [15,16,17] studied the counting function for problems with different boundary conditions. For Problem (1), in [18] they obtain a lower estimate of the counting function for real interior transmission eigenvalues.
For counting function in [22] we gave some non optimal estimate. This estimate was improved by Dimassi and Petkov [9]. Their estimate have the same size than the one found below in Theorem 3.2 except for a factor 3 √ 3. For constant m there is a recent result given by Pham and Stefanov [21] where they give an equivalent of counting function in this case.
The main results of this paper are Theorem 3.1 and Theorem 3.2. In Theorem 3.1 we prove than the counting function satisfies an estimate in Ct n and we prove that if we denote by λ j the eigenvalues of the problem when |z| goes to ∞ and z in a line outside a domain related with the range of n(x).
In the case where n(x) > 0 is real valued or if n 1 (x) > 0 in the case n(x) = n 1 (x) + n 2 (x)/k this estimate allows, applying a tauberian theorem to give an equivalent of the counting function. We find that N (t) ∼ αt n where the precise value of α is given in Theorem 3.2. These results are also proven by Faierman [10] in a preprint. The methods used are very close that the one used here but he assumes that n(x) = 1 every where. This condition excludes the case of cavity.
Here we assume only that n(x) = 1 in a neighborhood of the boundary.

Notations and background.
Let Ω be a C ∞ bounded domain in R n . Let n(x) ∈ C ∞ (Ω) be complex valued. We set m(x) = n(x) − 1. We consider also the case where n(x) = n 1 (x) + in 2 (x)/k where n j (x) are real valued and k the spectral parameter. This case is different of the previous one but can be treated similarly. We assume that for all x ∈ Ω, n(x) = 0, or n 1 (x) = 0 or equivalently m(x) = −1. We assume that there exists a neighborhood W of ∂Ω such that for x ∈ W , n(x) = 1 or n 1 (x) = 1 or equivalently m(x) = 0. Actually if n(x) = 1 for all x ∈ ∂Ω, such a neighborhood W exists.
We denote by C e the cone in C defined by In the case where n(x) = n 1 (x) Here we give some notations useful for the statement of the results. We use the notations and the results proven in [22], except some change of sign.
Let z ∈ C, we denote by B z (u, v) = (f, g) the mapping defined from H 2 0 (Ω)⊕{v ∈ L 2 (Ω), ∆v ∈ L 2 (Ω)} to L 2 (Ω) ⊕ L 2 (Ω) by In the case where n = n 1 + in 2 /k we must change the definition of B z . We define m 1 (x) = n 1 (x) − 1 and m 2 (x) = n 2 (x). The mappingB k (u, v) = (f, g) is given by Remark that the principal symbol ofB k is the same than the one of B z if we set Under an assumption on z, B z is invertible for some z.
Theorem 2.2. There exists k ∈ C such thatB k is bijective from In particular, applying the Riesz theory, the spectrum is finite or is a discrete countable set. If λ = 0 is in the spectrum, λ is an eigenvalue associated with a finite dimensional generalized eigenspace.
Theorem 2.4. There exists k ∈ C such that the resolventR k from H 2 (Ω) ⊕ L 2 (Ω) to itself is compact.
In particular, we can apply the Riesz theory, the spectrum is finite or a discrete countable set. If λ = 0 is in the spectrum, λ is an eigenvalue associated with a finite dimensional generalized eigenspace. Remark 1. Actually if z 0 ∈ C e ∪ [0, ∞) for all λ > 0 large enough we can take z = λz 0 in Theorems 2.1 and 2.3.
If k 2 0 ∈ C e ∪ [0, ∞) for all λ > 0 large enough we can take k = λk 0 in the Theorems 2.2 and 2.4. Here we estimate the resolvent in the exterior of a conic neighborhood of C e ∪ [0, ∞). In particular if n 1 (x) > 0, the eigenvalues k 2 are in all small conic neighborhood of (0, +∞), except for a finite number of eigenvalues. In general for a non self-adjoint problem, we cannot claim that the spectrum is non empty. In the following theorem, with a stronger assumption on C e , we can prove that the spectrum is non empty.
Theorem 2.5. Assume that C e is contained in a sector with angle less than θ with θ < 2π/p where 4p > n and θ < π/2. Then there exists z such that the spectrum of R z is infinite and the space spanned by the generalized eigenspaces is dense in H 2 0 (Ω) ⊕ {v ∈ L 2 (Ω), ∆v ∈ L 2 (Ω)}.

LUC ROBBIANO
Theorem 2.6. There exists k such that the spectrum ofR k is infinite and the space spanned by the generalized eigenspaces (i.e. ker(R k − λ) j for j sufficiently large and λ an eigenvalue) is dense in the space Remark 2. These results are based on the theory given in Agmon [1] and using the spectral results on Hilbert-Schmidt operators. In this theory we deduce that the spectrum is infinite from the proof that the generalized eigenspaces form a dense subspace in the closure of the range of R z [resp.R k ]. In [22] we proved that R p z [resp.R p k ] is a Hilbert-Schmidt operator if 4p > n. We can deduce the spectral Let z j be the elements of the spectrum of R z [resp.R k ] and E j the generalized associated eigenspace. We denote by 3. Results. We denote by ω j for j = 1, · · · , p, the roots of z p = 1.
Moreover let µ ∈ C such that |µ| = 1 and we assume that ω j µ ∈ C e ∪ (0, +∞) for j = 1, · · · , p. We denote by a( We fix z 0 such that the resolvent R z0 exist and let µ j such that 1/µ j are the eigenvalues of R z0 counted with multiplicity. Then we have when z = rµ and r goes to ∞. Remark 3. The first part of the theorem improve [22,Theorem 7] where we found the estimate N (t) ≤ Ct n+4 .

Remark 4.
By a more precise study in a neighborhood of the boundary we can obtain a result with a smaller remainder in (6) but this estimate does not allow to prove a result better on the counting function. Malliavin [19] was proved a tauberian theorem with a sharp remainder but this requires an estimate on j∈N 1 λ p j −z p in a complex domain except in a parabola neighborhood of (0, ∞). Here the estimate are proved in a complex domain except in a conic neighborhood of (0, ∞). It is maybe possible to improve this result following Hitrik, Krupchyk, Ola and Päivärinta [13] where they prove that the eigenvalues are in a parabolic neighborhood of (0, ∞).

Proof of Theorem 3.1.
The proof is based on Lemmas 4.1 and 4.2 below. We introduce some notations.
We set S = R z0 and T = S p where p satisfies the assumption of Theorem 3.1.
As T λ is a matrix of operators we denote We denote by the kernel of T z p .
Lemma 4.2. With the notation of Theorem 3.1 we have when |z| goes to ∞.

4.1.
Proof of Lemma 4.1. We recall that ω j for j = 1, · · · , p, are the roots of z p = 1, we have The Thus we have If we denote by We recall that S = R z0 and if the resolvent R z0+z exists. In what follows, z 0 is fixed thus |z| ∼ |z + z 0 | for large |z|.
We introduce some notation for Sobolev spaces.
We denote the semi-classical H s norm by w 2 We recall that we denote D = −ih∂, and the quantity |α|≤s D α w 2 L 2 (Ω) , if s is an integer, is equivalent to w 2 H s sc (Ω) uniformly with respect to h. When h = 1 we denote the space by H s (Ω).
We apply the results of [22]. The estimates below are given by [22, theorem 10] for k ≥ 1. The estimate on S 12 and S 22 for k = 0 are also given by [22, theorem 10]. gives the estimate on S 11 . The relation between z and h is zh 2 = µ, in particular h 2 |z| = 1.
We denote by We remark that We deduce from (11) that Λ z S ωj z Λ −1 z : L 2 (Ω)⊕L 2 (Ω) → H (Ω) with an operator norm less than C|z| −1 . As we deduce that with an operator norm less than C|z| −p . We can prove that N (t) ≤ Ct n . First we weaken (14) to consider Λ z T z p Λ −1 z as a map between L 2 (Ω) ⊕ L 2 (Ω) → H (Ω), with an operator norm less than C|z| −p . As We can apply the theorem 13.5 in Agmon [1], that is, if m > n/2 we have where |||T ||| is the Hilbert-Schmidt norm and T m is the operator norm of the map . We apply this estimate with m = 2p − 2 > n/2 and we have We can follow the proof of Theorem 7 in [22]. If we denote by µ j complex numbers such that µ −1 j are eigenvalue of S counted with multiplicity, then 1 µ p j −z p are the eigenvalues of T z p and thus the eigenvalues of Λ z T z p Λ −1 z . We obtain Let µ ∈ C such that |µ| = 1, and ω j µ ∈ C e ∪ (0, ∞) for all j = 1, · · · , p. We take Then we obtain N (t) ≤ Ct n . Now we prove Formula (6). Estimate (14) implies that To estimate the norm of T 22 , we shall use that S 12 is a mapping from L 2 (Ω) to H 4 sc (Ω). Actually if we take g ∈ L 2 (Ω), we have Λ z S ωj z Λ −1 z (0, g) ∈ H (Ω) ≤ C|z| −1 g L 2 (Ω) . We can repeat the previous argument for the p − 1 other factors Λ z S ωj z Λ −1 z and we obtain that (Ω)⊕H 2p sc (Ω) ≤ C|z| −p g L 2 (Ω) . In particular this means that T 22 g H 2p sc (Ω) ≤ C|z| −p g L 2 (Ω) , and By (15), we have We can apply the theorem 13.9, Agmon [1]. If 2p > n we have for j = 1, 2, is well defined in L 2 (Ω) and Remark 5. The adjoint of B z is given by an analogous formula than (4). Indeed we find that the adjoint B * z (p, q) = (g 1 , g 2 ) is given by − ∆((1 +m) −1 p) −zp = g 1 in Ω − ∆q −zq +m(1 +m) −1 p = g 2 in Ω q |∂Ω = ∂ ν q |∂Ω = 0 on ∂Ω.
Using the relation between R z and S z (see (10)), we deduce that the adjoint of S z satisfies the same estimate than S z given in (11). By (13), the adjoint of T z satisfies the same properties than T z given in (16), (17) and (18).
For j = 1, 2,(19) implies, from (16), (17) and (18) We recall that λ −1 j are the eigenvalues of S counted with multiplicity the eigenvalues of T are λ −p j . The indices are such that |λ j | ≤ |λ j+1 |. As N (t) ≤ Ct n , this implies that |λ j | ≥ Cj 2/n where C > 0. In particular 1/|λ j | p converges if 2p > n. By Theorem 12.17 in Agmon [1], there exists a constant c ∈ C such that We recall that the trace is defined in the theorem 12.20 in Agmon [1] for an operator Q = Q 1 Q 2 where Q 1 and Q 2 are Hilbert-Schmidt operators. Moreover if K is the kernel of Q, K(x, x) is definite for almost all x, we have Ω |K(x, x)|dx < ∞ and We remark as by assumption µ p j and z p are not in the same cone, we have |µ p j − z p | ∼ |µ p j | + |z p |. Then This implies that j∈N 1 (µ p j −z p )µ p j → 0 when |z| → +∞.
In [22] we proved that |||T z p ||| ≤ C|z| 1−p+n/4 as 1 − p + n/4 < 0 thus |||T z p ||| goes to 0 as |z| goes to +∞. We have | Tr(T T z p )| ≤ |||T ||||||T z p ||| goes to 0 as |z| goes to +∞. Then c = 0 in (21). We obtain that (22) the trace of z p T T z p is given by the integral of its kernel and, as the integral of the kernel of T z p exists by (20), the integral of trace of kernel of z p T T z p − T z p does not depend of z, we obtain By (20), for j = 1, 2, Ω K jj (x, x)dx goes to 0 when |z| goes to 0 and as we have

Proof of Lemma 4.2.
We recall some facts on pseudo-differential operators. Let a(x, ξ) be in C ∞ (R n × R n ) we say that a is a symbol of order m if for all α, β ∈ N n , there exist C α,β > 0, such that where ξ 2 = 1 + |ξ| 2 . In particular a polynomial in ξ of order m with coefficients in C ∞ (R n ) with bounded derivatives of all orders, is a symbol of order m.
With a symbol we can associate an semi-classical operator by the following formula where b(x, ξ) and c(x, ξ) are symbols of order m, we call b(x, ξ) the principal symbol of a(x, ξ) which is definite modulo h. This formula makes sense for u ∈ S (R n ) and we can extend it to u ∈ H s for all s. For a, a symbol of order m, there exists C > 0 such that for all u ∈ H s , In the following, when we use pseudo-differential operator we have always cut-off functions supported in Ω in each side of the operator. We do not have to consider the action of pseudo-differential operator on H s (Ω) space as in [22].
We begin with a description on S z in all compact set in Ω.

Lemma 4.3.
Let µ ∈ C such that |µ| = 1 and we assume that µ ∈ C e ∪ (0, +∞) for j = 1, · · · , p. Let z ∈ C such that z/µ ∈ (0, ∞) is large enough. Let θ andθ be functions in C ∞ 0 (Ω) such that θ(x) = 1 if x in the support ofθ, and θ(x) = 1 if x in a compact subset of Ω. Then we havẽ where W = Λ z KΛ −1 z and W * are bounded on H s (Ω) and the principal symbol of B is given on the support ofθ by Proof. We apply the result proved in [22]. Let us recall the notations and the main results. We multiply Equations (4) by h 2 , we denote by µ = h 2 z where µ belongs to a bounded domain of C, a = 1/(1 + m) and V = m/(1 + m). We change (f, g) in (−f, −g).
We recall the assumption made on m, we have m(x) = −1 for all x ∈ Ω and m(x) = 0 for x in a neighborhood of ∂Ω.
We apply φ 2 on the equation on v in (25), we have Applying the parametrix φ 1Q , we have With this equation and (27) we obtain whereK −1 is an operator of order −1 . Let where K 11 = Kφ 2 , K 12 =K −1 and K 22 = K −1 φ 2 . In particular K jk is bounded on H s sc (Ω) for all s ≥ 0. Indeed, all the operators contain cut-off thus K jk φ 3 u is compactly supported in Ω if u ∈ H s sc (Ω). We recall that S = R z0 and by (10), S z = R z0+z , if the resolvent R z0+z exists. In what follows, z 0 is fixed and we have the relation z = −µ/h 2 − z 0 . In particular |z| −1/2 ∼ h for large |z|. With these relations we have, following (28) and (29), S z (f, g) = (u, v), and Thus we can write where W = Λ z KΛ −1 z is bounded on H s (Ω) and the principal symbol of B is given on the support of φ 0 by Asθφ 0 =θ and φ 3 θ = φ 3 , (30) gives (24). As K is a semi-classical pseudodifferential operator, W * is also bounded on H s (Ω) Remark 6. Formula (24) does not give a description on the operator S z in Ω. It gives only S z in all compact in Ω. In the proof below we need also estimates on S z up the boundary given in (11) to absorb the error terms.
where ϕ 1 R p ϕ 0 satisfies the following property if we denote by (Ω) where the norm operator are uniformly bounded with respect h, and the principal Moreover the adjoint of R jq p satisfies (32) where the norm operator are uniformly bounded with respect h.
Proof. We argue by induction on k and for that we must introduce a sequence of cut-off functions. Let χ k andχ k be cut-off functions such thatχ kχk+1 =χ k+1 , χ kχk =χ k ,χ k χ k+1 = χ k+1 . We can assume thatχ 0 = 1 on the support of ϕ 0 and andχ p = 1 on the support of ϕ 1 . We can apply Formula (24) whereθ is replaced byχ k and θ by χ k . We havẽ where W and W * are bounded on H s (Ω).
We prove by recurrence the following formulã where the semi-classical principal symbol of B k is given by where Q −2k−2 (x, ξ) is a symbol of order −2k − 2. The operators R jq k and their adjoints satisfy Estimates (32) with p = k.
If Formula (35) is true for k we havẽ By (34) for k + 1 and (35) for k, we have The term |z| −1−kχ k+1 Bχ k+1χk B kχ0 gives the first right hand side term of (35), where B k+1 = Bχ k+1χk B k and the principal symbol is given by Formula (36) on the support ofχ k+1 .
The three other terms have the form of R k+1 and satisfy Estimates (32). Indeed, the power of |z| is obtained as the operator norm of Λ z S z Λ −1 z is bounded by |z| −1 . To prove the mapping between the H s sc , we denote by A q a generic operator of order q mapping H s to H s−q . We check that The properties on adjoints follow from the recurrence assumptions on R k , the properties on W * and Remark 5. By (34) for k + 1 and as χ k+1 (1 −χ k ) = 0, we have By (12) and (11) the operator norm of this term is |z| −k−1/2 . The proof that L 2 satisfies Estimates (32) for p = k + 1 is obtained by (37). The properties on the adjoint of L 2 follow from Remark 5 and the properties on W * . From (36) for k = p, and as z p − µ p = p j=1 (z − ω j µ), we obtain that p j=1 (a|ξ| 2 − ω j µ) −1 = (a p |ξ| 2p − µ p ) −1 .
We shall prove and Obviously (41) and (42) indeed |ξ| 2p and a p |ξ| 2p are not in the same cone as µ p by assumption. Thus the principal term in (41) given by the principal term from (39), can be estimate by Cδ and the error terms from (39) can be estimate by C δ |z| −1/2 . To prove (42), using (20) we obtain which is Estimate (42).
We have for j large enough such that |ν j | ≤ εδ j , where C depends only on p.