Location of eigenvalues for the wave equation with dissipative boundary conditions

We examine the location of the eigenvalues of the generator $G$ of a semi-group $V(t) = e^{tG},\: t \geq 0,$ related to the wave equation in an unbounded domain $\Omega \subset {\mathbb R}^d$ with dissipative boundary condition $\partial_{\nu}u - \gamma(x) \partial_t u = 0$ on $\Gamma = \partial \Omega.$ We study two cases: $(A): \: 0<\gamma(x)<1,\: \forall x \in \Gamma$ and $(B):\: 1<\gamma(x), \: \forall x \in \Gamma.$ We prove that for every $0<\epsilon \ll 1,$ the eigenvalues of $G$ in the case $(A)$ lie in the region $\Lambda_{\epsilon} = \{z \in {\mathbb C}:\: |\Re z | \leq C_{\epsilon} (|\Im z|^{\frac{1}{2} + \epsilon} + 1), \: \Re z<0\},$ while in the case $(B)$ for every $0<\epsilon \ll 1$ and every $N \in {\mathbb N}$ the eigenvalues lie in $\Lambda_{\epsilon} \cup {\mathcal R}_N,$ where ${\mathcal R}_N = \{z \in {\mathbb C}:\: |\Im z| \leq C_N (|\Re z| + 1)^{-N},\: \Re z<0\}.$


Introduction
Let K ⊂ R d , d ≥ 2, be a bounded non-empty domain and let Ω = R d \K be connected. We suppose that the boundary Γ of Ω is C ∞ . Consider the boundary problem with initial data (f 1 , f 2 ) ∈ H 1 (Ω) × L 2 (Ω) = H. Here ν is the unit outward normal to Γ pointing into Ω and γ(x) ≥ 0 is a C ∞ function on Γ. The solution of the problem (1.1) is given by V (t)f = e tG f, t ≥ 0, where V (t) is a semi-group in H whose generator G = 0 1 ∆ 0 has a domain D(G) which is the closure in the graph norm of functions (f 1 , f 2 ) ∈ C ∞ (0) (R n ) × C ∞ (0) (R n ) satisfying the boundary condition ∂ ν f 1 − γf 2 = 0 on Γ. For d odd it is well known [5] that the spectrum of G in Re z < 0 is formed by isolated eigenvalues with finite multiplicity. Notice that if Gf = λf with f = (f 1 , f 2 ) = 0, Re λ < 0 and ∂ ν f 1 − γf 2 = 0 on Γ, we get and u(t, x) = V (t)f = e λt f (x) is a solution of (1.1) with exponentially decreasing global energy. Such solutions are called asymptotically disappearing and they perturb the inverse scattering problems. Recently it was proved [2] that if we have at least one eigenvalue λ of G with Re λ < 0, then the wave operators W ± are not complete, that is Ran W − = Ran W + . Hence we cannot define the scattering 2000 Mathematics Subject Classification. Primary 35P20, Secondary 47A40, 35L05. The author was partially supported by the ANR project Nosevol BS01019 01.
operator S related to the Cauchy problem for the wave equations and (1.1) by the product W −1 + W − . When the global energy is conserved in time and the unperturbed and perturbed problems are associated to unitary groups the corresponding scattering operator S(z) : L 2 (S d−1 ) → L 2 (S d−1 ) satisfies the identity S −1 (z) = S * (z), z ∈ C if S(z) is invertible at z. Since S(z) and S * (z) are analytic in the "physical" half plane {z ∈ C : Im z < 0} (see [4]) the above relation implies that S(z) is invertible for Im z > 0. For dissipative boundary problems the above relation in general is not true and S(z 0 ) may have a non trivial kernel for some z 0 , Im z 0 > 0. For odd dimensions d Lax and Phillips [5] proved that this implies that iz 0 becomes an eigenvalue of G. Thus the analysis of the location of the eigenvalues of G is important for the location of the points where the kernel of S(z) is not trivial.
In the scattering theory of Lax-Phillips [5] the energy space can be presented as a direct sum H = D − a ⊕ K a ⊕ D + a , a > 0 and we have the relations V (t)D + a ⊂ D + a , V (t)(K a ) ⊂ K a ⊕ D a + , V (t)D − a ⊂ H, t ≥ 0. R. Phillips defined a system as non controllable if there exists a state f ∈ K a such that V (t)f ⊥ D + a , t ≥ 0. This means that there exists states in the "black box" K a which remain undetected by the scattering process. Majda [6] proved that if we have such state f , then u(t, x) = V (t)f is a disappearing solution, that is there exists T > 0 depending on f such that u(t, x) vanishes for all t ≥ T > 0. On the other hand, if 0 < γ(x) < 1, ∀x ∈ Γ or if 1 < γ(x), ∀x ∈ Γ, and the boundary is analytic there are no disappearing solutions (see [6]). Thus if γ(x) = 1, ∀x ∈ Γ, it is natural to search asymptotically disappearing solutions. From the results in [6] (see also [2]) it follows that if f ⊥ D − a , then f cannot be an eigenfunction of G, hence there are no outgoing eigenfunctions of G. However, we may have incoming asymptotically disappearing solutions as the following simple example shows.
In [7] Majda examined the location of the eigenvalues of G and he proved that if 0 ≤ γ(x) < 1, ∀x ∈ Γ, the eigenvalues of G lie in the region In the case γ(x) = 1, ∀x ∈ Γ one conjectures that for some obstacles there are no eigenvalues of G.
The purpose of this paper is to improve the above results for the location of eigenvalues. We consider two cases: Our main result is the following Theorem 1.1. In the case (A) for every , 0 < 1, the eigenvalues of G lie in the region Λ = {z ∈ C : | Re z| ≤ C (| Im z| 1 2 + + 1), Re z < 0}. In the case (B) for every , 0 < 1, and every N ∈ N the eigenvalues of G lie in the region Λ ∪ R N , where For strictly convex obstacles K we prove a better result in the case (B). Theorem 1.2. Assume that K is strictly convex. In the case (B) there exists R 0 > 0 such that for every N ∈ N the eigenvalues of G lie in the region {z ∈ C : The eigenvalues of G are symmetric with respect to the real axis, so it is sufficient to examine the location of the eigenvalues whose imaginary part is nonnegative. Introduce in {z ∈ C : Im z ≥ 0} the sets We set λ = i √ z h and we use the branch 0 ≤ arg z < 2π with Im √ z > 0 if Im z > 0. From (1.2) we deduce that the eigenfunctions u of G satisfy the problem The proofs of Theorems 1.1 and 1.2 are based on a semi-classical analysis of the equation is the Dirichlet-to-Neumann map, D ν = −i∂ ν and u is the solution of the problem In the paper we use the semi-classical Sobolev space H s h (Γ), s ∈ R, with norm hD s u L 2 (Γ) , where hD = (1 + (hD x ) 2 ) 1/2 . The purpose is to prove that if z ∈ Z 1 ∪Z 2 ∪Z 3 lies in some regions and h is small enough from (1.4) we get f = 0 which is not possible for an eigenfunction u. In this direction our strategy is close to that for the analysis of eigenvalues-free regions for the interior transmission eigenvalues in [13] and [14]. We apply some results for the interior Dirichlet-to-Neumann map established in [13] and [14] for bounded domains which after modifications and some constructions remain true for the exterior Dirichlet-to-Neumann map N (z, h) defined above.
For the reader convenience we recall two properties of semi-classical pseudodifferential operators Op h (a) (see Section 7 of [3] and Proposition 2.1 of [13]). Assume that a ∈ T * (X) satisfies the bounds Next for 0 ≤ δ < 1/2 we have a calculus and if a ∈ S l1,m1 We refer to [3] for more details concerning the calculus. The left hand side of (2.2) can be estimated also in some cases when one of the symbols a or b is in a class S l,m δ with 0 ≤ δ < 1. For the precise statements then reader should consult Proposition 2.2 in [13] and Proposition 4.2 in [14].
Let (x , ξ ) be the coordinates on T * (Γ). Denote by r 0 (x , ξ ) the principal symbol of the Laplace-Beltrami operator −∆ Γ on Γ equipped with the Riemannian metric induced by the Euclidean metric in R d .
The same result remains true for unbounded domains Ω with N int (z, h) replaced by N (z, h) applying some modifications to the proof in [13] based on the construction of a semi-classical parametrix close to the boundary. For reader convenience we will recall some facts and arguments from [13] which will be necessary for our exposition. Consider normal geodesic coordinates (x 1 , x ) in a neighborhood of a fixed point x 0 ∈ Γ, where locally the boundary Γ is given by x 1 = 0. Let ψ(x ) ∈ C ∞ 0 (Γ) be a cut-off function with support in a small neighborhood of x 0 ∈ Γ and ψ(x ) = 1 in another neighborhood of x 0 . Then −h 2 ∆ − z in these coordinates has the form In Section 3 in [13] for δ 1 > 0 small enough one constructs a semi-classical parametrix The phase ϕ(x, y , ξ , z) is complex-valued and 1 being a large integer. The phase ϕ and the amplitude a are determined so that For z ∈ Z 1,0 and any integer s ≥ 0, there exist l s , N s > 0 so that for N ≥ N s we have the estimate (see Proposition 3.7 in [13]) (2.5) For z ∈ Z 2 ∪ Z 3 the above estimate holds with | Im z| replaced by 1. Next introduce the operator Let G D be the self-adjoint realization of the operator −∆ on L 2 (Ω) with Dirichlet boundary condition on Γ. Since the spectrum of G D is the positive real axis, for z ∈ Z 1 we have the estimate while for z ∈ Z 2 ∪ Z 3 the above estimate holds with | Im z| replaced by 1. For k = 0 this estimate is trivial, while for k ≥ 1 it follows from the coercive estimates for the Dirichlet problem in unbounded domains Then This implies as in [13] the following with constants C N , s d > 0, independent of f, h and z, and s d independent of N . If z ∈ Z 2 ∪ Z 3 , then (2.6) holds with | Im z| replaced by 1.
Choose a partition of unity J j=1 ψ j (x) = 1 on Γ and set T (z, h) = J j=1 T ψj (z, h). Notice that the principal symbol of T (z, h) is ρ. By using Proposition 2.2 and repeating without any change the argument in Section 3 in [13], one concludes that the statement of Theorem 2.1 remains true replacing

Eigenvalues-free regions in the case (A)
In this section we suppose that 0 Indeed, if f = 0 on Γ, then u ∈ H 2 (Ω) will be eigenfunction of the Dirichlet problem in Ω and this is impossible. From (1.3) one obtains the equation (1.4).
According to Theorem 2.
where for z ∈ Z 2 ∪Z 3 the above estimate holds with | Im z| replaced by 1. Introduce the symbol We will show that c(x , ξ , z) is elliptic in a suitable class. Write Case I. z ∈ Z 1 . The symbol c is elliptic for |ξ | large enough and it remains to examine its behavior for |ξ | ≤ C 0 . For these values of ξ we have |ρ + γ √ z| ≤ C 1 . First consider the set Consequently, the symbol c is elliptic and Hence, for bounded |ξ | we have |c| ≥ C 3 h δ , C 3 > 0, while for large |ξ | we have |c| ∼ |ξ |. Introduce the function On the other hand, Thus for bounded |ξ | and (x , ξ ) / ∈ F, we deduce Combining this with the estimates (3.4), one concludes that | Im Case III. z ∈ Z 3 . In this case Im z = 1 and one has This implies that c ∈ S 0,1 0 is elliptic and c −1 ∈ S 0,−1 0 .

The analysis above shows that the operator Op
and, applying (3.1), we deduce On the other hand, for |α 1 | + |β 1 | ≥ 1, |α 2 | + |β 2 | ≥ 1 and |ξ | ≤ C 0 according to Consider the operator Op h (c −1 )Op h (c) − I. Following Section 7 in [3], the symbol of this operator is given by On the other hand, the estimates (3.9), (3.10) yield Thus, applying once more (2.1), one gets A combination of the above estimates implies (3.11) This shows that in the case (A) for every 0 < 1 the eigenvalues of G must lie in the region Λ defined in Theorem 1.1.

Eigenvalues-free region in the case (B)
In this section we deal with the case (B). The analysis of Section 3 works only for z ∈ Z 1 ∪ Z 3 . Indeed for z ∈ Z 1 we have The symbol g introduced in the previous section satisfies the estimates (3.6) and c ∈ S 0,1 δ , c −1 ∈ S 0,−1 δ . For z ∈ Z 3 we apply the same argument. Thus for z ∈ Z 1 ∪Z 3 we obtain that the eigenvalues must lie in Λ . For z ∈ Z 2 the argument exploited in the case (A) breaks down since for Re z = −1, Im z = 0 the symbol is not elliptic and it may vanish for some (x 0 , ξ 0 ).
In the following we suppose that z ∈ Z 2 . Then Proposition 2.3 yields a better approximation (4.1) If f = 0 is the trace of an eigenfunction of G, from the equality (1.4) on Γ we obtain Next, we write with z t = −1 + it Im z ∈ Z 2 , 0 < t < 1. The next Lemma is an analog of Lemma 3.9 in [13].
with a constant C > 0 independent of z, h and f . Moreover, Proof. The proof of (4.3) is the same as in [13] since for z ∈ Z 2 we get To establish (4.4), we apply Green's formula in the unbounded domain Ω. By using the notation of Section 3, setũ = J j=1ũ ψ . Then −ih∂ νũ | Γ = T (z, h)f and for R 1 the functionũ vanishes for |x| ≥ R. Thus one obtains Multiplying the above equality by h and taking the real part, we deduce Therefore, . It is easy to see that ũ L 2 (Ω) ≤ Ch −s d f L 2 (Ω) and combining this with (2.5) in the case z ∈ Z 2 , we obtain (4.4).
To prove that s is elliptic, it is sufficient to show that is elliptic. Consider the function Clearly, On the other hand, it is clear that 1 + y − y 2 > 0 for 0 ≤ y < 1+ This implies F (r 0 ) > 0 for r 0 ≥ 0 and s is elliptic. Consequently, and for small h and f L 2 (Γ) = 0, Im z = 0, we deduce from (4.5) and (4.6) Going back to λ = i √ z h , we see that Re √ z = y 1/2 sin ϕ/2, Im √ z = y 1/2 cos ϕ/2 and 0 ≤ sin ϕ ≤ B N h N . This implies for h small enough the estimate Thus for z ∈ Z 2 and every N ∈ N the eigenvalues of G lie in R N and this completes the proof of Theorem 1.1

Eigenvalue-free region for strictly convex obstacles in the case (B)
In this section we study the eigenvalues-free regions when K is a strictly convex obstacle. Let 0 < 1/2 be a small number. Introduce the regioñ where φ is the function introduced in Section 2. Notice that on the support of 1−χ 1 we have |1 − r 0 (x , ξ )| ≥ h /2 . By a modification of the construction in [13] (see also [10]) we can construct a semi-classical parametrixũ ψ having the form (2.4).
and for | Im z| ≤ h we have the estimate Thus the problem is to get an estimate of N (z, h)Op h (χ 0 /2 ) L 2 (Γ)→L 2 (Γ) . We will prove the following Set for simplicity of notation µ = Im z. We will follow closely the construction of a semi-classical parametrix in Sections 5, 6 in [14]. The only difference is that we deal with an unbounded domain and the local form of P slightly changes. For the convenience of the reader we are going to recall the result in [11]. Let Ω δ = {x ∈ Ω : dist(x, Γ) < δ}. Since K is strictly convex, in local normal geodesic coordinates (x, ξ) ∈ T * (Ω δ ), considered in Section 2, the principal symbol of P becomes Here locally in the interior of K we have x 1 > 0, while in the exterior of K we have x 1 < 0. Following [14], denote by R the set of functions a ∈ C ∞ (T * (Ω δ )) satisfying with all derivatives the estimates It was shown in Theorem 3.1 in [11] that there exists an exact symplectic map χ : T * (Ω δ ) → T * (Ω δ ) so that χ(x, ξ) = (y(x, ξ), η(x, ξ)) satisfies Let U ⊂ T * (Ω δ ) be a small neighborhood of K. By using a h-Fourier integral operator on Ω δ associated to the canonical relation one transforms P into an operator P 0 which in the new coordinates denoted again by (x, ξ) has the form . By a simple change of variable t = −x 1 , we pass to the situation when the exterior of K is presented by t > 0. Next one applies a new symplectic transformation of the tangential variables ( 1 (x , ξ ) (see Section 2 in [14]). Therefore the operator P 0 is transformed intõ where q(x # , ξ # ) > 0, q ∈ S 0 0 in a neighborhood of ξ # d = 0 and The only difference with [14] is the sign (−) in front of t in the form ofP 0 . For simplicity of the notations we denote the coordinates (x # , ξ # ) by (y, η) and consider the operator P 0 = D 2 t − t + D y d − iµq(y, D y ) + hq(y, D y ; µ, h) with 0 < C 1 ≤ q(y, η) ≤ C 2 , q ∈ S 0 0 ,q ∈ S 0 0 . Notice that we have the term −iµq(y, η) with µ > 0, while in [14] the model operator involves iµq(y, η) since the sign of µ is not important for the argument in Sections 5, 6 of [14].
First we will treat the situation examined in Section 6 in [14] when µ > 0 and η d satisfy the conditions Let ρ be the solution of the equation with Im ρ > 0. With a minor modifications of the argument in Section 6 in [14] we may construct a parametrixũ 2 = Op h (A(t))f , where A(t) = φ t δ 1 |ρ| 2 a(t, y, η; µ, h)e iϕ(t,y,η:µ) h .
Here δ 1 > 0 is small enough and φ is the function introduced in Section 2. We take ϕ and a in the form where M 1 and ϕ k and a k,ν do not depend on t. Let Φ 2 (η 1 ) ∈ C ∞ 0 (R) be a function such that on the support of Φ 2 the condition (5.7) is fulfilled. We have the identity The phase ϕ satisfies the eikonal equation . We choose ϕ 1 = ρ and one determines ϕ k , k ≥ 2, from the equation k+j=K (k + 1)(j + 1)ϕ k+1 ϕ j+1 + ∂ y d ϕ K + K = F (ϕ 1 , ..., ϕ K ) with 1 = −1, K = 0 for K ≥ 2. Next we choose a 0,0 = Φ 2 (η 1 ), a k,0 = 0 for k ≥ 1 and the functions a k,ν are determined form the equations 2i ν j=0 (j + 1)(ν + 1 − j)ϕ ν+1−j a k,j+1 + (ν + 1)(ν + 2)a k−1,ν+2 + i∂ y d a k,ν Therefore Lemma 6.1, 6.2, 6.3, 6.4 in [14] hold without any change since the signe before t in the form of P 0 is not involved. Thus, as in Section 6 of [14], for a neighborhood Y of a point in R d−1 we obtain Proposition 5.3. Assume the conditions (5.6), (5.7) fulfilled. Then for all s ≥ 0 we have the estimates Now we will study the case when µ and η d satisfy the condition and we will construct a parametrix for the problem . For the construction we need some estimates for the Airy function A(z) = Ai(e i2π/3 z). Here Ai(z) is the Airy function defined for s ∈ R by In the following the branch −π < arg z < π will be used and z 1/2 = |z| 1/2 e i arg z/2 . Notice also that The function A(z) satisfies the equalities It is well known that A(z) has for | arg z − π 3 | ≥ δ > 0 the representation (see [9], [8]) where ω = e 2πi/3 and In the same domain in C one has also an asymptotic expansion for the derivatives of A(z) by taking differentiation term by term (see [9]). Introduce the function Then for | arg z − π/3| ≥ δ > 0 we have the expansion For large |z| and Im z < 0 we have the estimate |F (z)| ≤ C|z| 1/2 , while for bounded |z| and Im z < 0 one obtaons |F (z)| ≤ C 1 . Consequently, |F (z) ≤ C 0 (|z| + 1) 1/2 , Im z < 0.
For the derivatives F (k) (z) = ∂ k F ∂z k (z) (see Chapter 5 in [8]) we get the following Lemma 5.4. For Im z < 0 and every integer k ≥ 0 we have the estimate Given an integer k ≥ 0, set Φ 0 (z) = 1, Taking the derivatives in the above equality and using (5.13), by induction in k one obtains Lemma 5.5. For Im z < 0 and all integers k ≥ 1, l ≥ 0, we have the bound (5.14) For t ≥ 0 and Im z < 0, set The next Lemma is an analogue of Lemma 3.3 in [14].
On the other hand, (5.12) implies the equality Next the construction of the parametrix goes without any changes as in Section 5 in [14] applying Lemma 5.4, 5.5 and 5.6 instead of Lemma 3.1, 3.2 and 3.3 in [14]. Thus as an analogue of Theorem 5.7 in [14] we get the following Proposition 5.7. For all s ≥ 0, we have the bounds Combining Proposition 5.3 and Proposition 5.7, we obtain, as in [14], Theorem 5.2.
After this preparation we pass to the analysis of an eigenvalues-free region when Re z = 1, h 1− ≤ Im z ≤ h , 0 < 1.
Let ρ = √ 1 − r 0 + i Im z. As in the previous section, we examine the equation Consider the partition of the unity χ + /2 + χ 0 /2 + χ − /2 = 1 on T * (Γ) introduced in the beginning of this section. Applying Theorem 5.2, we have Taking into account Theorem 5.1 for the operators N (z, h)χ ± /2 , one deduces We write Clearly, The same estimates holds for d 1 , hence d 1 ∈ S Combining this with (5.23), for small h we conclude as in Section 4, that f = 0.
The Dirichlet problem for −h 2 ∆ − z with z = 1 + i Im h 2/3 w, |w| ≤ C 0 , has been investigated by Sjöstrand (see Chapters 9 and 10 in [12]). For 0 ≤ w ≤ 1 this covers the region D. In [12] the exterior Dirichlet-to-Neumann map N ext is defined with respect to the outgoing solution of the problem (1.5). Notice that for Im z > 0 the outgoing solutions are in H 2 h (Ω) and we have N ext = N (z, h). We recall some results in Chapter 10 of [12]. The operator N (z, h) is a h−pseudodifferential operator with symbol n ext (x , ξ , h). Introduce the glancing set G = {(x , ξ ) ∈ T * (Γ) : r 0 (x , ξ ) = 1}.
Combining this with (5.25), one deduces that the left hand side of (5.24) is greater than (η 3 − C 1 h 1/3 ) f 2 L 2 (Γ) . For small h and small δ 0 (depending on η 3 ) we obtain a contradiction with the estimate of the right hand side of (5.24). This completes the proof of Theorem 1.2.