David Lafontaine
email: david.lafontaine@math.univ-toulouse.fr

Euan A Spence
email: e.a.spence@bath.ac.uk

SHARP BOUNDS ON HELMHOLTZ IMPEDANCE-TO-IMPEDANCE MAPS AND APPLICATION TO OVERLAPPING DOMAIN DECOMPOSITION

We prove sharp bounds on certain impedance-to-impedance maps (and their compositions) for the Helmholtz equation with large wavenumber (i.e., at high-frequency) using semiclassical defect measures. The paper [GGG + 22] recently showed that the behaviour of these impedance-to-impedance maps (and their compositions) dictates the convergence of the parallel overlapping Schwarz domain-decomposition method with impedance boundary conditions on the subdomain boundaries. For a model decomposition with two subdomains and sufficientlylarge overlap, the results of this paper combined with those in [GGG + 22] show that the parallel Schwarz method is power contractive, independent of the wavenumber. For strip-type decompositions with many subdomains, the results of this paper show that the composite impedanceto-impedance maps, in general, behave "badly" with respect to the wavenumber; nevertheless, by proving results about the composite maps applied to a restricted class of data, we give insight into the wavenumber-robustness of the parallel Schwarz method observed in the numerical experiments in [GGG + 22].

1. Introduction 1.1. Motivation and outline. Over the last 30 years, there has been sustained interest in computing approximations to solutions of the Helmholtz equation ∆u + k 2 u = 0 with wavenumber k 1 using domain-decomposition (DD) methods; see the recent review papers [START_REF] Gander | A class of iterative solvers for the Helmholtz equation: factorizations, sweeping preconditioners, source transfer, single layer potentials, polarized traces, and optimized Schwarz methods[END_REF][START_REF] Gander | Schwarz methods by domain truncation[END_REF]. However, it still remains an open problem to provide a k-explicit convergence theory, valid for arbitrarily-large k, for any practical DD method for computing approximations to Helmholtz solutions.

Working towards this goal, the paper [GGG + 22] studied a parallel overlapping Schwarz method for the Helmholtz equation, where impedance boundary conditions are imposed on the subdomain boundaries. This method can be thought of as the overlapping analogue of the parallel nonoverlapping method introduced in Després' thesis [START_REF] Després | Méthodes de décomposition de domaine pour les problémes de propagation d'ondes en régime harmonique[END_REF][START_REF] Benamou | A domain decomposition method for the Helmholtz equation and related optimal control problems[END_REF], which was the first method in the Helmholtz context to demonstrate the benefits of using impedance boundary conditions on the subdomain problems (the analogous algorithm for Laplace's equation, with Robin boundary conditions on the subdomains, was introduced by P.-L. Lions [START_REF] Lions | On the Schwarz alternating method. III: a variant for nonoverlapping subdomains[END_REF]). Indeed, with or without overlap, the parallel Schwarz method with Dirichlet boundary conditions on the subdomains need not converge when applied to the Helmholtz equation (see, e.g., [DJN15, §2.2.1]), and is not even well posed if k 2 is a Dirichlet eigenvalue of the Laplacian on one of the subdomains. In contrast, the method with impedance boundary conditions is well posed, and always converges in the nonoverlapping case by [BD97, Theorem 1] (see, e.g., [DJN15, §2.2.2]).

The paper [GGG + 22] studied the parallel overlapping Schwarz method at the continuous level, i.e., without discretisation of the subdomain problems, with then the follow-up paper [START_REF] Gong | Convergence of Restricted Additive Schwarz with impedance transmission conditions for discretised Helmholtz problems[END_REF] showing that, at least for certain 2-d decompositions and provided the discretisation is sufficiently fine, the discrete method inherits the properties of the method at the continuous level.

The main result of [GGG + 22] was the expression of the error-propagation operator (i.e., the operator describing how the error propagates from one iteration to the next) in terms of certain impedance-to-impedance maps. This result paved the way for k-explicit results about the convergence of the DD method to be obtained from k-explicit bounds on the norms of the impedanceto-impedance maps (and/or their compositions), and the present paper provides such k-explicit bounds for a model set-up in 2-d.

The structure of the paper is as follows. §1.2 defines the impedance-to-impedance maps for a certain 2-d model set-up, and §1.3 states bounds on these maps that are sharp in the limit k → ∞. §1.4 states sharp bounds on compositions of these maps. §2 discusses the implications of the main results for the parallel Schwarz method, recapping the necessary material from [GGG + 22]; §2 also recaps other literature on impedance-to-impedance maps in the context of domain-decomposition, including the recent paper [START_REF] Beck | Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs[END_REF] (see Remark 2.11). §3 recaps known material about semiclassical defect measures (mainly from [START_REF] Miller | Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary[END_REF]). §4 proves propagation results at the level of measures see Figure 1.

Γ b Γ r Γ t Γ l Γ i d l d r h Figure 1
. The domain D, the boundaries Γ l , Γ r , Γ t , and Γ b , and the interior interface Γ i

We are interested in the two following model problems.

(Model 1)

     (∆ + k 2 )u = 0 in D ( 1 i ∂ x1 + k)u = g on Γ l , u is outgoing near Γ s , ( Model 2)
         (∆ + k 2 )u = 0 in D ( 1 i ∂ x1 + k)u = g on Γ l , (-1 i ∂ x1 + k)u = 0 on Γ r , u is outgoing near Γ s ,
where the outgoingness near Γ s (resp. Γ s ) is defined in Definition 1.4 below. Since the outgoing boundary condition is a high-frequency (k → ∞) assumption, there might exist more than one solution to such a model for a given k; however, all solutions will have the same high-frequency behaviour. Indeed, for both models, an admissible (in the sense of Definition 7.1 below) solution operator g → u exists, and two solutions coming from distinct admissible solution operators always coincide modulo O(k -∞ ) (see Lemmas 1.5 and 1.6 below). Since we are interested in the highfrequency behaviour of the solutions, we therefore fix from now on an admissible solution operator to both models.

Observe that in Model 2, the impedance data (-1 i ∂ n + k)u is specified on both Γ l and Γ r , where ∂ n is the outward normal derivative to D. If g ∈ L 2 (Γ l ) and u is the associated solution of (Model 1) in D, we let

I - 1 (h, d l , k)g := γ Γi 1 i ∂ x1 -k u, I + 1 (h, d l , k)g := γ Γi 1 i ∂ x1 + k u;
i.e., I - 1 and I + 1 are the two different impedance traces on Γ i , with the plus/minus superscripts correspond to the plus/minus in the impedance traces, where γ Γi denotes the trace operator onto L 2 (Γ i ). In the same way, if g ∈ L 2 (Γ l ), and u is the associated solution of (Model 2) in D, we let

I - 2 (h, d l , d r , k)g := γ Γi 1 i ∂ x1 -k u, I + 2 (h, d l , d r , k)g := γ Γi 1 i ∂ x1 + k u.
1.3. Upper and lower bounds on the impedance operators.

Theorem 1.1 (Upper and lower bounds for (Model 1)). Let h, d l > 0 and let θ max ∈ (0, π/2) be defined by θ max (h, d l ) := arctan h d l .

(1) For any > 0, there exists k 0 ( ) > 0 such that, for all k ≥ k 0

I - 1 (h, d l , k) L 2 →L 2 ≤ 1 -cos(θ max ) 1 + cos(θ max ) +
In addition,

lim sup k→∞ I - 1 (h, d l , k) L 2 →L 2 ≥ 1 -cos(θ max ) 1 + cos(θ max ) .
(2) For any > 0, there exists k 0 ( ) > 0 such that, for all k ≥ k 0

I + 1 (h, d l , k) L 2 →L 2 ≤ 1 + In addition, lim sup k→∞ I + 1 (h, d l , k) L 2 →L 2 ≥ 1.
The heuristic behind Theorem 1.1. In Fourier space, 1 i ∂ x1 ±k acts as multiplication by ξ 1 ±k, where ξ 1 is the first component of the dual variable to x (that is, the frequency); thus I - 1 is governed by (ξ 1k)/(ξ 1 + k) and I + 1 is governed by (ξ 1 + k)/(ξ 1 + k). This ratio is one in the case of I + 1 , leading to the lower bound in Part 2 of Theorem 1.1. To bound the ratio sharply in the case of I - 1 , we observe the following: since the solution is outgoing on the borders of the cell, the mass reaching Γ i comes necessarily from Γ l . To go from Γ l to Γ i at frequency k, this mass must travel with a direction ξ = k(cos θ, sin θ), where 0 ≤ θ ≤ θ max (with mass travelling from the very bottom of Γ l to the very top of Γ i traveling with angle θ max ). This direction ξ therefore satisfies |ξ 1 -k| = k| cos θ -1| = k 1cos θ ≤ k 1cos(θ max ) and |ξ 1 + k| = k| cos θ + 1| = k 1 + cos θ ≥ k 1 + cos(θ max ) , from which the upper bounds in Theorem 1.1 follow. Our proof implements these heuristic arguments in a rigorous way, using so-called semiclassical defect measures. In particular, to prove the lower bounds in Theorem 1.1, we take data g from a so-called coherent state, chosen so that all the mass of the solution concentrates on a ray coming from one point x 0 ∈ Γ l and travels in one direction ξ 0 = k(cos θ 0 , sin θ 0 ); we then take x 0 to be very close to the origin (i.e., the bottom of Γ l ) and θ 0 to be as close as θ max as possible.

Theorem 1.2 (Lower bounds for (Model 2)). Let h, d l , d r > 0. Then,

lim sup k→∞ I ± 2 (h, d l , d r , k) L 2 →L 2 ≥ 1.
The heuristic behind Theorem 1.2. The difference between Model 2 and Model 1 is that now rays are reflected from Γ r . Arguing as above, the map I + 2 is bounded below by one on rays travelling directly from Γ l to Γ i . Provided these rays are not horizontal (i.e., ξ 2 = 0), their subsequent reflections from Γ r and Γ l do not interfere with the initial ray (they move either up or down in the vertical direction, until being absorbed by the outgoing conditions at the top and bottom), and thus I + 2 is bounded below by one. To see that I - 2 is bounded below by one, we need to consider the first reflected ray. When the wave exp(iξ • x) with |ξ| = k hits the impedance boundary on the right, a reflected wave R exp(-iξ 1 x 1 + iξ 2 x 2 ) is created, where the reflection coefficient R := (ξ 1k)/(ξ 1 + k). For a wave exp(-iξ 1 x 1 + iξ 2 x 2 ), I - 2 is governed by (-ξ 1k)/(-ξ 1 + k), and thus the contribution to

I - 2 from the first reflected ray is |R(-ξ 1 -k)/(-ξ 1 + k)| = 1.
As discussed in the previous paragraph, provided that ξ 2 = 0, further reflections do not interfere with this first reflected ray, and thus I -

2

is bounded below by one.

1.4. The behaviour of the composite impedance map in (Model 2). Given h, d l + , d r + , d l -, d r -> 0, we consider arbitrary compositions of the two maps (1.1)

I - 2 (h, d l -, d r -, k) and I + 2 (h, d l + , d r + , k);
we allow the two maps I - 2 and I + 2 to have different arguments d l and d r because of the application of these results in domain decomposition -see Remark 2.9 below.

An arbitrary composition of the two maps (1.1) can be written as the following: given n ≥ 0 and σ ∈ {+, -} n , let

I σ (h, d + l , d r + , d - l , d r -, k) := =0,••• ,n I σ( ) 2 (h, d l σ( ) , d r σ( ) , k),
where the product denotes composition of the maps.

In addition, for any λ > 0, we define the projection λ-away from zero frequency Π k λ as

Π k λ g := F -1 k 1 -ψ • λ F k g , F k g(ζ) := k 2π e -ikyζ g(y) dy, where ψ ∈ C ∞ c (R; [0, 1]
) is equal to one on [-1, 1], and F k is the Fourier transform at scale k. Written with F 1 , the non-scaled Fourier-transform,

Π k λ g = F -1 1 1 -ψ • λk F 1 g ;
that is, Π k λ is a projection λk-away from zero in the Fourier variable. The projection Π k λ is applied below to impedance data; the heuristic interpretation of this is the following: since a Helmholtz solution is, in the high-frequency limit, supported in Fourier space where |ξ 1 | 2 +|ξ | 2 = k 2 (with ξ 1 the dual variable of x 1 and ξ of x ), truncating the impedance data λk-away from ξ = 0 produces a solution supported at high frequencies where

|ξ 1 | 2 ≤ k 2 (1 -λ 2 ), hence away from the horizontal direction corresponding to |ξ 1 | 2 = k 2 . Theorem 1.3 (The composite impedance map). Let h, d + l , d r + , d - l , d r -> 0.
(1) For any n ≥ 1 and any σ ∈ {+, -} n , lim sup k→∞

I σ (h, d + l , d r + , d - l , d r -, k) L 2 →L 2 ≥ 1.
(2) Let σ ∈ {+, -} N and, for any n ≥ 1,

σ n := (σ(1), • • • , σ(n)) ∈ {+, -} n . Given λ > 0, let n 0 (λ) := h(min(d + l , d - l )) -1 λ -1 √ 1 -λ 2 . Then for all n ≥ n 0 (λ), lim k→∞ I σn (h, d + l , d r + , d - l , d r -, k)Π k λ L 2 →L 2 = 0.
The heuristic behind Theorem 1.3. The main idea behind Theorem 1.3 is that, in the high-frequency limit, the impedance-to-impedance map associated with (Model 2) pushes the mass emanating from Γ l with an angle θ to the horizontal up and down by a distance proportional to θ -1 , while preserving its mass. Therefore, if g creates a Helmholtz solution emanating from Γ l with angles > 0 to the horizontal, all its mass is pushed off the domain in a finite number of iterations. After applying Π k λ , the data creates a solution emanating from Γ l with angles ≥ arctan λ √ 1-λ 2 > 0; hence the high-frequency nilpotence of the composite impedance map, i.e., Part (2) of the theorem. On the other hand, the same idea allows us to construct the lower bound in Part (1): taking coherent-state data g creating a Helmholtz solution concentrating all its mass on an arbitrarily small angle to the horizontal, the image by the composite impedance map after the corresponding, arbitrarily high, number of iterations, is still in the domain and hence has order one mass. 1.5. The semiclassical notation and definition of outgoingness. It is convenient to work with the semiclassical small parameter := k -1 . In addition, we let D • := 1 i ∂ • . Then, (Model 1) and (Model 2) become, with g replaced by g,

(M1)      (-2 ∆ -1)u = 0 in D ( D x1 + 1)u = g on Γ l , u is outgoing near Γ s . (M2)          (-2 ∆ -1)u = 0 in D ( D x1 + 1)u = g on Γ l , u is outgoing near Γ s , (-D x1 + 1)u = 0 on Γ r .
We can now define the outgoingness near the border of the cell: Definition 1.4. We say that a -family of solutions u to (-2 ∆ -1)u = 0 in D is outgoing near Γ ∈ {Γ s , Γ s } if there exists an open set D + ⊃ D ∪ Γ (independent of > 0) with ∂D + ∩ (∂D\Γ) = ∂D\Γ, such that u can be extended to a -tempered solution of (-2 ∆ -1)u = 0 in D + and

∀(x, ξ) ∈ WF u ∩ {x ∈ Γ}, ξ • n(x) ≥ 0,
where the normal n(x) points out of the domain D depicted in Figure 1.

The wavefront set WF of an -tempered family of functions is defined in Definition 3.1 below. It describes where the non-negligible mass of an -dependent family of functions lies in phase-space (that is, in both position and direction) in the high-frequency limit → 0; Definition 1.4 therefore means the solution u has only mass pointing outside the cell -hence outgoing.

1.6. Wellposedness results. We say that S = S( ) : The admissibility condition of Definition 7.1 corresponds to requiring that any solution can be extended in a slightly bigger domain, where it is bounded, and has bounded traces where an impedance boundary condition is imposed (Γ l for (M1) and Γ l ∪ Γ r for (M2)). All admissible solution operators have the same high-frequency behavior in the following sense, where f

g ∈ L 2 (Γ l ) → u ∈ H 1 (D)
H s := f L 2 + s f Ḣs .
Lemma 1.6. If S 1 , S 2 are two admissible solutions operators, then, for any bounded g ∈ L 2 (Γ l ), any N > 0, and any χ ∈ C ∞ c (D), there exists

C N > 0 such that χ(S 1 g -S 2 g) H N ≤ C N N .

Implications of the main results for the parallel overlapping Schwarz method

The plan of this section is to • define the parallel overlapping Schwarz method studied in [GGG + 22] ( §2.1),

• summarise the performance of the parallel overlapping Schwarz method as illustrated in the numerical experiments in [GGG + 22] ( §2.2), • show how impedance-to-impedance maps govern the behaviour of the error of this method ( § §2.3-2.5) and define these impedance-to-impedance maps for general decompositions ( §2.6), • show how the maps of Models 1 and 2 in §1.2 are the relevant impedance-to-impedance maps for 2-d strip decompositions when the boundary condition on the whole domain (approximating the Sommerfeld radiation condition) is the outgoing condition described in §1.5 ( §2.7- §2.9), and • explain the relevance of Theorems 1.1, 1.2, and 1.3 to the analysis of the parallel Schwarz method ( §2.10).

2.1.

Ω ⊂ R d d ≥ 2, f ∈ L 2 (Ω), and g ∈ L 2 (∂Ω), find u ∈ H 1 (Ω) satisfying (∆ + k 2 )u = -f on Ω and 1 i ∂ ∂n -k u = g on ∂Ω (2.1)
(where ∂/∂n denotes the outward normal derivative on ∂Ω) and considers its solution via the following parallel overlapping Schwarz method with impedance transmission conditions. Let {Ω j } N j=1 form an overlapping cover of Ω with each Ω j ⊂ Ω and Lipschitz polyhedral. If u solves (2.1), then u j := u| Ωj satisfies

(∆ + k 2 )u j = -f in Ω j , (2.2) 1 i ∂ ∂n j -k u j = 1 i ∂ ∂n j -k u on ∂Ω j \∂Ω, (2.3) 1 i ∂ ∂n j -k u j = g on ∂Ω j ∩ ∂Ω, (2.4)
where ∂/∂n j denotes the outward normal derivative on ∂Ω j . 1 Let {χ j } N j=1 be such that χ j ∈ C 1,1 (Ω; [0, 1]), χ j ≡ 0 in Ω ∩ ((Ω j ) c ) (and thus, in particular, on ∂Ω j \ ∂Ω) and N j=1 χ j (x) = 1 for all x ∈ Ω. The parallel Schwarz method is: given an iterate u n defined on Ω, let u n+1 j be the solution of

(∆ + k 2 )u n+1 j = -f
in Ω j , (2.5)

1 i ∂ ∂n j -k u n+1 j = 1 i ∂ ∂n j -k u n on ∂Ω j \∂Ω, (2.6) 1 i ∂ ∂n j -k u n+1 j = g on ∂Ω j ∩ ∂Ω; (2.7)
finally, let

u n+1 := χ u n+1 . (2.8)
This method is well-defined since if u n ∈ U (Ω) then u n+1 ∈ U (Ω), where

U (Ω) := u ∈ H 1 (Ω) : (∆ + k 2 )u ∈ L 2 (Ω), (-i∂/∂n -k)u ∈ L 2 (∂Ω) ; see [GGG + 22, Theorem 2.12].
We highlight that the impedance boundary condition enters in (2.2)-(2.8) in two ways

(1) as the boundary condition on ∂Ω ((2.4), (2.7)), and

(2) as the boundary conditions on the subdomains (2.6).

Regarding 1: the motivation for imposing an impedance boundary condition on ∂Ω is that it is the simplest-possible approximation to the Sommerfeld radiation condition, and the interior impedance problem is a ubiquitous model problem in the numerical analysis of the Helmholtz equation (see, e.g., the discussion and references in [GLS21a, §1.1]). However, the analysis in [GGG + 22] is, in principle, applicable to other boundary conditions on ∂Ω, and we discuss this further in Remark 2.5 below.

Regarding 2: as discussed in §1.1, the advantage of using impedance boundary conditions on the subdomains was recognised in Després' thesis [START_REF] Després | Méthodes de décomposition de domaine pour les problémes de propagation d'ondes en régime harmonique[END_REF][START_REF] Benamou | A domain decomposition method for the Helmholtz equation and related optimal control problems[END_REF], and (2.2)-(2.8) is the overlapping analogue of the non-overlapping method in [Des91, BD97] (see, e.g., the discussion in [DJN15, §2.3]).

We see later (in §2.9) that the impedance-to-impedance maps in Theorems 1.1-1.3 are those dictating the behaviour of the parallel Schwarz method in the idealised case where the boundary condition on ∂Ω is the outgoing condition. The rationale for considering this idealised case is that it allows us to focus on the impedance boundary conditions imposed in the domain decomposition method itself (i.e., in Point (2) above), and ignore the influence of the impedance boundary condition imposed as an approximation of the Sommerfeld radiation condition (i.e., in Point (1) above). (1) Strip decompositions (described in §2.7 below) in 2-d rectangular domains with height one and maximum length 64/3 with k ∈ [20, 80] (so that at the highest k there were approximately 267 wavelengths in the domain). (2) Uniform ("checkerboard") and non-uniform (created by the mesh partitioning software METIS) decompositions of the 2-d unit square with k ∈ [40, 160] (so that at the highest k there were approximately 25 wavelengths in the domain).

Summary of the numerical experiments in [GGG

The experiments in [GGG + 22, §6] showed the following three features of the parallel Schwarz method.

1 The impedance boundary conditions in [GGG + 22] are written in the form ∂nik (with this form more commonly-used in numerical analysis). In this section we write the results of [GGG + 22] using the impedance condition 1 i ∂nk; the two conventions are equivalent up to multiplication/division by i of the data g on ∂Ω.

(a) For a fixed number of subdomains with fixed overlap proportional to the subdomain length, the number of iterations required to achieve a fixed error tolerance decreases as k increases (in the ranges above) -this was shown for the strip decompositions in [GGG + 22, Experiment 6.2 and Table 2] and for the square in [GGG + 22, Tables 7, 8, 10, 11] (with a similar result seen for a different parallel DD method in [GSZ20, Table 3]). (b) For the strip decomposition with fixed number of subdomains, the convergence rate of the method increases as the length of subdomains increases with the overlap proportional to the subdomain length (so that the overall length of the domain increases) [GGG + 22, Experiment 6.1 and Figure 5]. (c) For the strip decomposition with an increasing number of subdomains and fixed subdomain length and overlap (so the length of domain increases), at fixed k, one needs roughly O(N ) iterations to obtain a fixed error tolerance; see [GGG + 22, Experiment 6.2]. We highlight that using the method with a fixed number of subdomains, as in (a), is not completely practical, since the subproblems have the same order of complexity as the global problem. Nevertheless, this situation provides a useful starting point for methods based on recursion; see the discussion in [GSZ20, Section 1.4]. Furthermore, we see in §2.10 how the results of the present paper imply that analysing the method even in this idealised case is very challenging.

2.3. The error propagation operator T . We consider the vector of errors

e n = (e n 1 , e n 2 , . . . e n N )
, where e n := uu n = u| Ωu n , = 1, . . . , N. (2.9) By the definition of u n (2.8) and the fact that {χ } N =1 is a partition of unity,

e n := u -u n = χ u| Ω - χ u n = χ e n .
(2.10) Thus, subtracting (2.5)-(2.7) from (2.2)-(2.4), we obtain

(∆ + k 2 )e n+1 j = 0 in Ω j , (2.11) 1 i ∂ ∂n j -k e n+1 j = 1 i ∂ ∂n j -k e n = 1 i ∂ ∂n j
k χ e n , on ∂Ω j \∂Ω, (2.12)

1 i ∂ ∂n j
k e n+1 j = 0 on ∂Ω j ∩ ∂Ω. (2.13)

The map from e n to e n+1 can be written in a convenient way using the operator-valued matrix T = (T j, ) N j, =1 , defined as follows. For v ∈ U (Ω ), and any j ∈ {1, . . . , N }, let T j, v ∈ U (Ω j ) be the solution of

(∆ + k 2 )(T j, v ) = 0 in Ω j , (2.14) 1 i ∂ ∂n j -k (T j, v ) = 1 i ∂ ∂n j -k (χ v ) on ∂Ω j \∂Ω, (2.15) 1 i ∂ ∂n j -k (T j, v ) = 0 on ∂Ω j ∩ ∂Ω. (2.16)
Therefore,

e n+1 j =
T j, e n , and thus e n+1 = T e n . (2.17) Observe that (i) if Ω j ∩ Ω = ∅, then T j, = 0 (since the right-hand side of (2.15) is zero on ∂Ω j \ ∂Ω), (ii) since χ vanishes on ∂Ω \ ∂Ω, (-i∂/∂nk)(χ v ) vanishes on ∂Ω \ ∂Ω, and thus T , ≡ 0 for all .

It is convenient here to introduce the notation Γ j, := (∂Ω j \ ∂Ω) ∩ Ω , (2.18) so that (2.15) holds on Γ j, and (2.16) holds on ∂Ω j \Γ j, .

Remark 2.1. Using the definition of T , the parallel Schwarz method (2.5)-(2.7) can be written as

u n+1 = T u n + F ,
where (u n ) j := u n j , (F ) j := F j , where

F j ∈ U (Ω j ) satisfies (∆ + k 2 )F j = -f in Ω j , 1 i ∂ ∂n j -k F j = 0 on ∂Ω j \ ∂Ω, and
1 i ∂ ∂n j -k F j = g on ∂Ω j ∩ ∂Ω.
2.4. The goal: proving power contractivity of T . The paper [GGG + 22] sought to prove that T M is a contraction, for some appropriate M ≥ 1, in an appropriate norm. The motivation for this is that, in 1-d with a strip decomposition (i.e., the 1-d analogue of the decompositions considered in §2.7 below), T N = 0 where N is the number of subdomains; see [NRdS94, Propositions 2.5 and 2.6]. This property holds because, in 1-d, the impedance boundary condition is the exact Dirichlet-to-Neumann map for the Helmholtz equation.

To define an appropriate norm, let

U 0 (Ω j ) := u ∈ H 1 (Ω j ) : (∆ + k 2 )u = 0, (-i∂/∂ n -k)u ∈ L 2 (∂Ω j ) ⊂ U (Ω j ).
Since e n j ∈ U 0 (Ω j ) for each j, [GGG + 22] analyses convergence of (2.17) in the space U 0 := N =1 U 0 (Ω ). Lemma 2.2 (Norm on U 0 (Ω j )). For Ω j a bounded Lipschitz domain, let • 1,k,∂Ωj be defined by

v 2 1,k,∂Ωj := ∂v ∂n 2 L 2 (∂Ωj ) + k 2 v 2 L 2 (∂Ωj ) , (2.19)
where ∂/∂n denotes the outward normal derivative on ∂Ω j . Then • 1,k,∂Ωj is a norm on U 0 (Ω j ) and

v 2 1,k,∂Ωj = 1 i ∂v ∂n -kv 2 L 2 (∂Ωj ) = 1 i ∂v ∂n + kv 2 L 2 (∂Ωj ) , (2.20)
The norm on U 0 is then defined by

v 2 1,k,∂ := N =1 v 2 1,k,∂Ω for v ∈ U 0 . (2.21)
For the proof of Lemma 2.2, see [GGG + 22, Lemma 3.3]. We highlight that (i) the norm (2.19) was used in the non-overlapping analysis in [START_REF] Després | Méthodes de décomposition de domaine pour les problémes de propagation d'ondes en régime harmonique[END_REF][START_REF] Benamou | A domain decomposition method for the Helmholtz equation and related optimal control problems[END_REF] (see [START_REF] Benamou | A domain decomposition method for the Helmholtz equation and related optimal control problems[END_REF]Equation 12]) and (ii) the relation (2.20) is a well-known "isometry" result about impedance traces; see, e.g., [Spe15, Lemma 6.37], [CCJP23, Equation 3].

2.5. How impedance-to-impedance maps arise in studying T M for M ≥ 1. When studying T M for M ≥ 1, compositions such as T j, T ,j naturally arise. Indeed,

(T 2 ) j,j = T j, T ,j , (2.22)
where the sum is over all ∈ {1, 2, . . . , N }\{j, j }, with neither Γ j, nor Γ ,j empty (equivalently, neither Ω j ∩ Ω nor Ω ∩ Ω j empty).

To condense notation, let

imp j := 1 i ∂ ∂n j -k, so that the boundary condition (2.15) becomes (2.23) imp j (T j, v ) = imp j (χ v ) on Γ j, .
A useful expression for the action of (2.22) can be obtained by inserting v = T ,j z j , with z j ∈ U (Ω j ), into the boundary condition (2.23), to obtain imp j T j, T ,j z j = imp j χ T ,j z j = χ imp j T ,j z j + ∂χ ∂n j (T ,j z j on Γ j, (2.24)

Observe that to find the first term on the right-hand side of (2.24) one (i) finds T ,j z j , i.e., the unique function in U 0 (Ω ) with impedance data given by imp (χ j z j ) on Γ ,j and zero otherwise, (ii) evaluates imp j (T ,j z j ) on Γ j, and (iii) multiplies the result by χ . The combination of steps (i) and (ii) naturally lead us to consider impedance-to-impedance maps.

2.6. The impedance-to-impedance map for general decompositions. Definition 2.3 (Impedance map). Let , j, j ∈ {1, . . . , N } be such that neither Γ j, nor Γ ,j is empty (equivalently, neither Ω j ∩ Ω nor Ω ∩ Ω j is empty). Given g ∈ L 2 (Γ ,j ), let v be the unique element of U 0 (Ω ) with impedance data

imp (v ) = g on Γ ,j 0 on ∂Ω \Γ ,j . (2.25)
Then the impedance-to-impedance map I Γ .j →Γ j, : L 2 (Γ ,j ) → L 2 (Γ j, ) is defined by

I Γ .j →Γ j, g := imp j (v ), on Γ j, .
(2.26)

Ω j ′ Ω ℓ Ω j Γ ℓj ′ Γ jℓ (a) Ωj ∩ Ω j = ∅ Ω j ′ Ω ℓ Ω j Γ ℓj ′ Γ jℓ (b) Ωj ∩ Ω j = ∅ Ω j Ω ℓ Γ ℓj Γ jℓ (c) Ωj = Ω j Figure 2.
Illustrations of the domain (red) and co-domain (blue) of I Γ .j →Γ j, in 2d (in the case when none of the subdomains Ω j , Ω j , and Ω touch ∂Ω)

Figure 2 illustrates the domain (in red) and co-domain (in blue) of the impedance-to-impedance map, with arrows indicating the direction of the normal derivative.

The following is a simplified version of [GGG + 22, Theorem 3.9].

Lemma 2.4 (Connection between T 2 and the impedance-to-impedance map). Let , j, j ∈ {1, . . . , N } be such that Ω ∩ Ω j = ∅ and Ω ∩ Ω j = ∅. If z j ∈ U (Ω j ), then imp j (T j, T ,j z j ) = χ I Γ .j →Γ j, imp (T ,j z j ) + ∂χ ∂n j (T ,j z j ) on Γ j, . (2.27)

Proof. Since T ,j z j ∈ U 0 (Ω ) and imp (T ,j z j ) vanishes on ∂Ω \Γ ,j , the definition of I Γ .j →Γ j, implies that imp j T ,j z j = I Γ .j →Γ j, imp T ,j z j on Γ j, ; the result then follows from (2.24).

Remark 2.5 (Imposing other boundary conditions on ∂Ω). §2.1 mentioned how, in principle, the analysis of [GGG + 22] can be repeated with boundary conditions on ∂Ω other than the impedance boundary condition. Suppose the boundary condition in (2.1) is replaced by Bu = g and (2.7) replaced by Bu n+1 j = g where B is an arbitrary operator. Proceeding formally (i.e., ignoring the issue of identifying the correct analogue of U (Ω) in which the method is well-posed) we now outline the changes to the above. The analogues of equations (2.14)-(2.16) are the following

(∆ + k 2 )(T j, v ) = 0 in Ω j , imp j (T j, v ) = imp j (χ v ) on ∂Ω j \∂Ω, B(T j, v ) = 0 on ∂Ω j ∩ ∂Ω.
Just under (2.18) above, we split the boundary condition on ∂Ω j \∂Ω into two (using the support properties of χ ) to obtain the boundary conditions (2.28)

imp j T j, v = imp j χ v on Γ j, , imp j T j, v = 0 on (∂Ω j \∂Ω)\Ω , imp j T j, v = 0 on ∂Ω j ∩∂Ω,
and then combined the latter two as one boundary condition on ∂Ω j \ Γ j, . Now we cannot do this combination, and so have to keep the following analogue of (2.28)

imp j T j, v = imp j χ v on Γ j, , imp j T j, v = 0 on (∂Ω j \∂Ω)\Ω , B(T j, v ) = 0 on ∂Ω j ∩∂Ω,
The boundary conditions for the function v in the definition of the impedance-to-impedance map (Definition 2.3) then become

imp (v ) = g on Γ ,j , 0 on (∂Ω \ ∂Ω)\Ω j , B(v ) = 0 on ∂Ω ∩ ∂Ω;
i.e., the absorbing boundary condition B(v ) = 0 is always imposed on the part of any subdomain boundary that intersects ∂Ω.

Γ - ℓ Γ + ℓ Γ - ℓ+1 Γ + ℓ-1 δ ℓ δ ℓ+1 Ω ℓ Ω ℓ-1 Ω ℓ+1
Γ , -1 , Γ -1, , Γ +1, , Γ , +1 and Γ -, Γ + -1 , Γ - +1 , Γ + .
Under this set-up, the functions {χ j } N j=1 defined in §2.1 satisfy

χ | Γ + -1 = 1 = χ | Γ - +1 , (2.30)
and thus Theorem 2.4 simplifies to the following (with this result a simplified version of [GGG + 22, Corollary 4.3]).

Corollary 2.6. Let ∈ {1, . . . , N } and j, j

∈ { -1, + 1}. If z j ∈ U (Ω j ), then imp j (T j, T ,j z j ) = I Γ .j →Γ j, (imp (T ,j z j )) on Γ j, . (2.31)
Recall from the definition of T j, (2.14)-(2.16) that T , = 0 and T j, = 0 if Ω j ∩ Ω = ∅. Therefore, for these strip decompositions, T takes the block tridiagonal form

(2.32) T =          0 T 1,2 T 2,1 0 T 2,3 T 3,2 0 T 3,4 . . . . . . . . . T N -1,N -2 0 T N -1,N T N,N -1 0         
.

The impedance-to-impedance maps for 2-d strip decompositions.

There are now four impedance-to-impedance maps associated with Ω :

I Γ . -1 →Γ -1, = I Γ -→Γ + -1 , I Γ . +1 →Γ +1, = I Γ + →Γ - +1 , I Γ . -1 →Γ +1, = I Γ -→Γ - +1 , I Γ . +1 →Γ -1, = I Γ + →Γ + -1 ; (2.33)
there are four maps because ∂Ω \ ∂Ω has two disjoint components, Γ -and Γ + , and there are two "interior interface" in Ω , Γ + -1 and Γ - +1 . These four maps can in turn be expressed in terms of two maps.

Definition 2.7. ("Canonical" impedance-to-impedance maps [GGG + 22, Definition 4.9]) 1). Let u be the solution to

For L > 0, let D := [0, L] × [0, 1], let Γ l := {0} × [0, 1] and let Γ i := {d l } × [0, 1] for d l ∈ (0, 1) (compare to Figure
(2.34) (∆ + k 2 )u = 0 in D, - 1 i ∂ x1 -k u = g on Γ l , 1 i ∂ n -k u = 0 on ∂ D\Γ l .
Let

I -g := 1 i ∂ x1 -k u, I + g := 1 i ∂ x1 + k u on Γ i , (2.35) and let (2.36) ρ(k, d l , L) = sup g∈L 2 (Γ l ) I -g L 2 (Γi) g L 2 (Γ l ) , γ(k, d l , L) = sup g∈L 2 (Γ l ) I + g L 2 (Γi) g L 2 (Γ l ) .
The maps I Γ -→Γ +

I Γ -→Γ + -1 L 2 →L 2 = ρ(k, δ , L ), I Γ + →Γ - +1 L 2 →L 2 = ρ(k, δ +1 , L ), (2.37) and I Γ -→Γ - +1 L 2 →L 2 = γ(k, L -δ +1 , L ), I Γ + →Γ + -1 L 2 →L 2 = γ(k, L -δ , L ) (2.38)
(where δ and δ +1 are shown in Figure 3, and L is the width of Ω ). 2.9. The relation of the impedance-to-impedance maps in Definition 2.7 to those in §1.2. The maps in §1.2 are related to the impedance-to-impedance maps in Definition 2.7 for the 2-d strip decomposition when the boundary condition on ∂Ω (approximating the Sommerfeld radiation condition) is the outgoing condition described in §1.5. Indeed, following Remark 2.5, the maps I ± in Definition 2.7 correspond to I ± 2 when the subdomain is not Ω 1 or Ω N (i.e., when the subdomain does not touch the left-or right-ends of the strip) and I ± 1 when the subdomain is one of Ω 1 or Ω N . In particular, in this set-up with only two subdomains (i.e., N = 2), the maps I ± are I ± 1 . As mentioned in §2.3, the rationale for considering the idealised case where the boundary condition on ∂Ω is the outgoing condition is that it allows us to focus on the impedance boundary conditions imposed in the domain decomposition method itself, and ignore the influence of the impedance boundary condition imposed as an approximation of the Sommerfeld radiation condition.

Perfectly matched layers approximate outgoing Helmholtz solutions with accuracy increasing exponentially in the limit k → ∞; see [START_REF]Perfectly-matched-layer truncation is exponentially accurate at high frequency[END_REF]. We therefore expect the behaviour of the maps in §1.2 to be similar to the analogous maps with a perfectly matched layer on the top and bottom of D.

2.10. The relevance of Theorems 1.1-1.3 to the analysis of the parallel Schwarz method for 2-d strip decompositions.

We now give five conclusions and one conjecture about the analysis of the parallel Schwarz method for 2-d strip decompositions. The justification/discussion of each is given after its statement.

Conclusion 1: with outgoing boundary condition on ∂Ω, the parallel Schwarz method with two subdomains is power contractive for sufficiently-large subdomain overlap.

By considering powers of T given by (2.32), using (2.31), (2.33), and the definitions of ρ (2.37) and γ (2.38), [GGG + 22, Corollary 4.14] obtains the following sufficient condition for T N to be a contraction Theorem 2.10. (Informal statement of [GGG + 22, Corollary 4.14]) If ρ is sufficiently small relative to γ and N , then

T N 1,k,∂ < 1.
Theorem 1.1 and Corollary 2.8 show that, with outgoing boundary conditions on ∂Ω and two subdomains (i.e., N = 2), the analogue of ρ (2.36) in this set-up can be made arbitrarily small by increasing the overlap of the subdomains, and the analogue of γ is bounded below by one; i.e., in this set up, T N is a contraction for sufficiently-large overlap (which necessarily means that the length of Ω must be sufficiently large relative to its height).

Conclusion 2: one cannot obtain a k-explicit convergence theory, valid for arbitrarily-large k, of the parallel Schwarz method with more than 2 subdomains by looking at the norms of single impedance-to-impedance maps.

Theorem 1.2 and Corollary 2.8 show that, with outgoing boundary conditions on ∂Ω and more than two subdomains, the analogues of ρ and γ are both bounded below by one. Thus Theorem 2.10 or its precursor [GGG + 22, Theorem 4.13], which are based on bounding T N by norms of single impedance-to-impedance maps, cannot be used to prove that T N is a contraction. Conclusion 3: the values of k chosen in the computations of impedance-to-impedance maps in [GGG + 22] were not large enough to see the asymptotic behaviour of ρ (the norm of I -).

The lower bounds on I ±

2 in Theorem 1.2 do not immediately give lower bounds for ρ and γ in Definition 2.7; this is because ρ and γ contain contributions from additional reflections from the impedance boundary conditions on the top and bottom, which are not present in I ± 2 . Although it is possible that these additional reflections create fortunate cancellation, this would be highly non-generic, and we therefore expect the lower bounds in Theorem 1.2 to carry over to ρ and γ; i.e., we expect both ρ and γ to be bounded below by one at high frequency.

[GGG + 22, Table 1] computes ρ(k, L/3, L) and γ(k, L/3, L) for k in the range [10, 80] and L in the range [1,16]. At k = 80, ρ ≤ 0.3 and γ ≥ 0.97, showing that k = 80 is large enough to see the asymptotic behaviour of γ, but not the asymptotic behaviour of ρ.

Conclusion 4: one cannot obtain a k-explicit convergence theory, valid for arbitrarily-large k, of the parallel Schwarz method by looking at the norms of composite impedance-to-impedance maps on the whole of the space on which they are defined.

Recognising that estimating T N 1,k,∂ via norms of single impedance-to-impedance maps might be too crude, [GGG + 22, §4.4.5] outlined how to estimate T N 1,k,∂ via norms of compositions of impedance-to-impedance maps; essentially this involves using (2.31) iteratively and relating the impedance-to-impedance maps to the canonical maps via (2.33).

However, Part (1) of Theorem 1.3 shows that the norm of any composition of I ± is bounded below by one for sufficiently-large k, meaning that the strategy outlined in [GGG + 22, §4.4.5] will not be able to show that T N 1,k,∂ is small for arbitrarily-large k Conclusion 5: the lower bound on the composite map in Theorem 1.3 is only reached for very large k, explaining why the computed values of the composite map in [GGG + 22, Table 3] are small.

As discussed in §1.4, the lower bound in Part (1) of Theorem 1.3 is based on choosing particular data corresponding to rays leaving Γ l close to horizontal; this is clear both from the proof of Part (1) of Theorem 1.3, and also from Part (2) of Theorem 1.3, which shows that if one projects away from zero frequency on the boundary (with zero frequency on the boundary corresponding to rays leaving horizontally) then the norm of the composite map on such data is zero.

Rays leaving Γ l close to the horizontal hit Γ r close to the horizontal, and thus have small reflection coefficient (since rays hitting horizontally, i.e., normally, have zero reflection coefficient). Therefore, the asymptotic lower bound on the map in Part (1) of Theorem 1.3 will only be reached for very large k. This is consistent with the computations of a particular composite impedance map in [GGG + 22, Tables 5 and6]; even when the subdomains have small overlap, the norms of a composition of four impedance-to-impedance maps were less than 0.1 for k ∈ {10, 20, 40, 80}; see [GGG + 22, Table 6].

Conjecture 1: a convergence theory for the parallel Schwarz method with k 1 can be obtained by combining Part (2) of Theorem 1.3 with the fact that normally incident waves are those for which the exact DtN map is the impedance boundary condition.

For a 2-d strip decomposition with more than two subdomains and outgoing boundary condition on Ω, a possible route to prove that T M 1,k,∂ < 1 for some M > 0 is the following. Combine (i) the expression for T M in terms of composite impedance-to-impedance maps (outlined in [GGG + 22, §4.4.5]), (ii) Part (2) of Theorem 1.3, to show that the contribution from non-normally incident waves is small for M large enough, and (iii) the fact that the normally incident waves not covered by Part (2) of Theorem 1.3 are those for which the exact DtN map is the impedance boundary condition (i.e., the impedance boundary condition on the subdomains is the "correct" one for these waves). We expect the optimal M will come from balancing the fact that to use (iii) we want λ to be small (so that only rays very close to horizontal need to be dealt with via this mechanism), but the smaller λ, the larger the n 0 (λ) in Theorem 1.3 (since the smaller the λ, the closer to the horizontal the rays can be, and a larger number of reflections is needed to push their mass into the outgoing layers). Since implementing these steps would require a more-refined analysis of the parallel Schwarz method than in [GGG + 22], this will be investigated elsewhere.

Remark 2.11. (Discussion of other literature on impedance-to-impedance maps in the context of domain decomposition) Whereas the paper [GGG + 22] showed how impedance-to-impedance maps govern the error propagation in the non-overlapping parallel Schwarz method, but do not appear explicitly in the formulation of the method, impedance-to-impedance maps appear explicitly in the formulation of certain fast direct solvers [START_REF] Gillman | A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media[END_REF], [START_REF] Beams | A parallel shared-memory implementation of a high-order accurate solution technique for variable coefficient Helmholtz problems[END_REF] and non-overlapping domain decomposition methods [START_REF] Pedneault | Schur complement domain decomposition methods for the solution of multiple scattering problems[END_REF].

The impedance-to-impedance maps associated with the fast direct solver in [START_REF] Gillman | A spectrally accurate direct solution technique for frequency-domain scattering problems with variable media[END_REF] were recently analysed in [START_REF] Beck | Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs[END_REF] The prototypical example of the maps studied in [START_REF] Beck | Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs[END_REF] is the following. Given a square with zero impedance data on three sides, the considered map is the map from impedance data on the fourth side to the impedance data on that same side but with the sign swapped (i.e., the map from ∂ n u+iku to ∂ n u-iku, where u is a Helmholtz solution); see [BCM22, Definition 1.1]. The main goal of [START_REF] Beck | Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs[END_REF] is to study I -R 1 R 2 , where R 1 , R 2 are two of these impedanceto-impedance maps on squares. If I -R 1 R 2 is invertible, then the impedance-to-impedance map on a 2 × 1 rectangle can be recovered from the impedance-to-impedance maps on two non-overlapping square subdomains (see [BCM22, Equations 7-9 and the surrounding text]).

Whereas I -R 1 R 2 was assumed invertible in [GBM15], [BCM22, Theorem 1.2] proved a k-explicit bound on (I -R 1 R 2 ) -1 .
The techniques used in [START_REF] Beck | Quantitative bounds on Impedance-to-Impedance operators with applications to fast direct solvers for PDEs[END_REF] are quite different from those in the present paper; indeed [BCM22, Theorem 1.2] is proved using vector-field arguments and bounds on the Neumann Green's function. Nevertheless, the fact that the bound on (I -R 1 R 2 ) -1 in [BCM22, Theorem 1.2] allows for its norm to grow as k → ∞ is thematically similar to the "bad" behaviour as k → ∞ of our impedance-to-impedance maps proved in Theorem 1.2 and Part (2) of Theorem 1.3.

Definition and useful properties of defect measures

3.1. The local geometry and associated notation.

3.1.1. The geometry. For D + as in Definition 1.4, we denote Γ + i := {d l } × R ∩ D + . We denote the dual variables to (x 1 , x ) by (ξ 1 , ξ ). Near the interfaces Γ = Γ l , Γ = Γ l , and Γ = Γ + i , we use Riemannian/Fermi normal coordinates (s, x ), in which Γ is given by {s = 0} and [0, d l + d r ] × [0, h] is given by {s > 0} for both Γ l and Γ r , and [0, d l ] × [0, h] is given by {s > 0} for Γ i ; therefore

s = x 1 on Γ l , s = -x 1 on Γ r , s = -x 1 on Γ + i .
The conormal variable to s is denoted by ζ; so that

ζ = ξ 1 on Γ l , ζ = -ξ 1 on Γ r , and ζ = -ξ 1 on Γ + i . In addition, on T * (Γ l ∪ Γ + i ∪ Γ r ), we let r(x , ξ ) := 1 -|ξ | 2 ,
in such a way that, near

Γ l ∪ Γ + i ∪ Γ r , -2 ∆ -1 is given by -2 ∆ -1 = ( D x1 ) 2 -r(x , D x ).
We define the hyperbolic set in

T * (Γ l ∪ Γ + i ∪ Γ r ) by H := (x , ξ ) ∈ T * (Γ l ∪ Γ + i ∪ Γ r ), r(x , ξ ) > 0 , and for (x, ξ ) ∈ H, we denote ζ in (x , ξ ) := -r(x , ξ ), ζ out (x , ξ ) := + r(x , ξ ).
Finally, ϕ t is the Hamiltonian flow associated to the equation, that is, for (x 0 , ξ 0 ) ∈ T * R 2 and t ∈ R, ϕ t (x 0 , ξ 0 ) = (x(t), ξ(t)) is the solution to

x(0) = x 0 , ξ(0) = ξ 0 , ξ(t) = 0, ẋ(t) = 2ξ(t).
3.1.2. Quantisation and semiclassical wavefront set. For b ∈ C ∞ c (T * R 2 ), we recall that the (standard) semiclassical quantisation of b, which we denote by b(x, D x ) or Op (b), is defined by

(3.1) b(x, D x )v (x) := (2π ) -2 T * R 2 e i(x-y)•ξ/ a(x, ξ)v(y) dydξ, see, for example, [Zwo12, §4.1], [DZ19, Page 543]. In addition, for Γ ∈ {Γ i , Γ l , Γ r }, if a ∈ C ∞ c (T * Γ),
we define, as an operator acting on L 2 (Γ) a(x , D x )v (x ) := (2π ) -1 T * Γ e i(x -y )•ξ / a(x, ξ)v(y) dy dξ .

We can now introduce the semiclassical wavefront-set of an -tempered family of functions. An -family of functions v( ) ∈ L 2 loc is -tempered if for each χ ∈ C ∞ c there exists C, N such that χv( ) L 2 ≤ C -N . Definition 3.1 (Semiclassical wavefront set). For an -tempered family of functions v( ) ∈ L 2 loc (D + ), we say that (x 0 , ξ 0 ) ∈ T * D + is not in the semiclassical wavefront-set of v( ) and write

(x 0 , ξ 0 ) / ∈ WF (v) if for any b ∈ C ∞ c (T * D +
) supported sufficiently close to (x 0 , ξ 0 ), and for any N, s, there is

C s,N > 0 such that b(x, D x )v H s ≤ C s,N N ,
for the semi-classical Sobolev norm

(3.2) f H s := f L 2 + s f Ḣs
(where the second term on the right-hand side is the H s semi norm).

Semiclassical defect measures.

Theorem 3.2 (Existence of defect measures). Let v( k ) ∈ L 2 (D + ).

(1) If there exists C > 0 so that v L 2 (D + ) ≤ C, then, there exists a subsequence h k → 0 and a non-negative Radon measure µ on

T * R 2 , such that for any b ∈ C ∞ c (T * R 2 ) b(x, h k D x )v, v → b dµ,
where v has been extended by zero to R 2 .

(2

) Let Γ ∈ {Γ + i , Γ l , Γ r }. If there exists C > 0 so that v L 2 (Γ) + ∂ n v L 2 (Γ) ≤ C,
then, there exists a subsequence h k → 0, non-negative Radon measures ν d , ν n , and a Radon measure ν j on T * R such that for any a ∈ C ∞ c (T * R),

a(x, h k D x )v, v Γ → a dν d , a(x, h k D x )h k D s v, v Γ → a dν j , a(x, h k D x )h k D s v, h k D s v Γ → a dν n ,
where the traces of v have been extended to R by zero.

Reference for the proof. See [Zwo12, Theorem 5.2].

In the rest of this section, we assume that v satisfies

(-2 ∆ -1)v = 0 in D +
and has defect measure µ, and boundary measures ν d , ν n , ν j on Γ.

We now introduce the outgoing and incoming boundary measures on Γ. To do so, we define geodesic coordinates on the flow-out and the flow-in of a subset of H:

Definition 3.3. For V ⊂ H, let B out (V), B in (V) be defined by B out (V) := (x ,ξ )∈V ϕ t 0, x , ξ , ζ = ζ out (x , ξ ) , t ∈ R ∩ T * D + B in (V) := (x ,ξ )∈V ϕ t 0, x , ξ , ζ = ζ in (x , ξ ) , t ∈ R ∩ T * D + . In B out (V), we work in geodesic coordinates (ρ, t) ∈ π -1 Γ V ∩ ζ = ζ out × R, defined by B (x, ξ) = ϕ t (ρ)
; and in B in (V), we can in the same way work in geodesic coordinates (ρ, t) ∈

π -1 Γ V ∩ ζ = ζ in × R.
We recall that, for f : X 1 → X 2 and a measure α on X 1 , the pushforward measure f * α on X 2 is defined by f * α(B) := α(f -1 (B)), and the change of variable formula

(3.3) X2 g d(f * α) = X1 g • f dα
holds. In addition, we let a ⊗ b denote the product measure of a and b, and we let π Γ be the projection map defined by

(3.4) π Γ : s = 0, x , ζ, ξ ∈ T * R 2 ∩ {x ∈ Γ} → (x , ξ ) ∈ T * Γ.
Lemma 3.4 (Relationship between boundary measures and the measure in the interior). There exists positive measures µ in and µ out on T * Γ and supported in H so that, if V ⊂ H, then, in geodesic coordinates defined above,

µ = p out * (2 √ rµ out ) ⊗ dt in B out (V), µ = p in * (2 √ rµ in ) ⊗ dt in B in (V),
where p in and p out are defined by

p out/in : (x , ξ ) ∈ H -→ x , ξ , ζ = ζ out/in (x , ξ ) ∈ π -1 Γ V ∩ ζ = ζ out/in . Proof.
Existence of µ in and µ out is proven in [Mil00, Proposition 1.7], and the rest of the result is proved in [GLS21a, Lemma 2.16]. In [GLS21a, Lemma 2.16], B out/in is defined only with t ≷ 0. When Γ ∈ {Γ l , Γ r }, the definitions in [START_REF] Galkowski | Local absorbing boundary conditions on fixed domains give order-one errors for high-frequency waves[END_REF] coincide with Definition 3.3, and then the result follows immediately from [GLS21a, Lemma 2.16]. When Γ = Γ + i , applying [GLS21a, Lemma 2.16] once with measures taken from the left, and once with measures taken from the right gives the result for all t ∈ R.

The relationship between the outgoing and incoming boundary measures and the boundary measures ν d , ν n , ν j is given in the following lemma. Both here and in the rest of the paper we use the notation (as in [START_REF] Miller | Refraction of high-frequency waves density by sharp interfaces and semiclassical measures at the boundary[END_REF]) that aµ(B) := B adµ for a measure µ and a Borel set B. Lemma 3.5. [Mil00, Proposition 1.10] The incoming and outgoing measures satisfy the following.

(1) In H,

2µ out = r(x , ξ )ν d + 2 ν j + 1 r(x , ξ ) ν n , 2µ in = r(x , ξ )ν d -2 ν j + 1 r(x , ξ ) ν n .
(2) If µ in = 0 on some Borel set B ⊂ H, then, on B

µ out = 2 ν j = 2 r(x , ξ )ν d = 2 r(x , ξ ) ν n .
(3) If µ out = 0 on some Borel set B ⊂ H, then, on B

µ in = -2 ν j = 2 r(x , ξ )ν d = 2 r(x , ξ ) ν n . (4) If -2 ν j = ( α)ν d = 4( α)|α| -2 ν n
on some Borel set B ⊂ H for α a complex valued function such that α + 2 r(x , ξ ) is never zero on B then

µ out = Rµ in ,
where

R := 2 r(x , ξ ) -α 2 r(x , ξ ) + α 2 on B, If instead, α -2 √ r is never zero, then R -1 µ out = µ in .
Notation 3.6. When they exist, we denote ν x d , ν x n , ν x j , µ x in , µ x out the measures of v on Γ x with x ∈ {l, r}, and on Γ + i when x = i. In these definitions, we take the measures on Γ + i from the left. When all these objects exist, we succinctly say that u admits defect measures and boundary measures.

Invariance properties of outgoing Helmholtz solutions

In this section, we assume that v satisfies

(-2 ∆ -1)v = 0 in D,
and we investigate the consequences of v being outgoing near Γ ∈ {Γ s , Γ s }. In the following, D + is as in Definition 1.4.

We first prove two lemmas about location of WF v.

Lemma 4.1. Assume that v is outgoing near Γ ∈ {Γ s , Γ s } (in the sense of Definition 1.4) and let Γ := ∂D\Γ. Then,

WF v ∩ T * D ∩ D + ⊂ (x, ξ), ∃t < 0, x + tξ ∈ Γ .
Proof. If it is not the case, there exists (x 0 , ξ 0 ) ∈ WF v ∩ T * D ∩ D + so that ∀t < 0, x 0 + tξ 0 / ∈ Γ. Therefore, there exists τ < 0 so that x 1 := x 0 + τ ξ 0 ∈ Γ. As x 0 ∈ D, we have necessarily ξ 0 • n(x 1 ) < 0 (where, as in Definition 1.4, n(x 1 ) is the normal at x 1 pointing out of D). On the other hand, as the wavefront set is invariant under the Hamilton flow (this follows, for example, from propagation of singularities, [DZ19, Theorem E.47]), (x 1 , ξ 0 ) ∈ WF v. This is a contradiction with the outgoingness of v (Definition 1.4). Lemma 4.2. Assume that v is outgoing near Γ ∈ {Γ s , Γ s } (in the sense of Definition 1.4). Then, for any M > 0 and > 0, there exists c > 0 so that

(4.1) WF (v) ∩ T * D + ∩ T * ( , d l + d r -) × (-M, h + M ) ⊂ (x, (ξ 1 , ξ )), |ξ 1 | ≥ c .
Proof. By a similar proof to the one of Lemma 4.1, there exists c( , M ) > 0 so that WF (v) ∩

T * D + ∩ T * ( , d l + d r -) × (-M, h + M ) ⊂ {(x, (ξ 1 , ξ )), |ξ 1 |/|ξ| ≥ c}. The result follows using the additional fact that WF v ∩ T * D + ⊂ {|ξ| 2 = 1} since (-2 ∆ -1)v = 0 in D + .
By the definition of (WF v) c and the definition of µ, supp µ ⊂ WF v (if (x 0 , ξ 0 ) / ∈ WF v, taking b as in Definition 3.1 gives µ(b) = 0 using the fact that b(x, D x )v = O( ∞ ) in L 2 , and hence (x 0 , ξ 0 ) / ∈ supp µ). Therefore, an immediate consequence of Lemma 4.1 is the following, where we recall the definition of B out from Definition 3.3. We now define, for (x, y) ∈ {i, r, l} 2

B x y→x := ρ ∈ T * Γ x ∩ H, ∃t < 0, π x ϕ t (p in )(ρ) ∈ Γ y , B x x→y := ρ ∈ T * Γ x ∩ H, ∃t > 0, π x ϕ t (p out )(ρ) ∈ Γ y ,
with p in and p out exchanged in the case (x, y) = (i, r).

The sets B x y→x and B x x→y are, respectively, the elements of T * Γ x reached by the flowout of Γ y , and the elements of T * Γ x whose flowout reaches Γ y . The reason why we have to define the case (x, y) = (i, r) separately is because the measures are taken from [0, h] × [0, d l ] (i.e., from the left) on Γ i , hence when working in [0, h] × [d l , d l + d r ] the roles of ζ in and ζ out on T * Γ i (and therefore of p in and p out ) are exchanged.

Lemma 4.4.

(1) If v is outgoing near Γ s (as in Model 1), then, for any defect measures of v, (a) µ l in = 0, (b) µ i out = 0, (c) µ r out = 0, (2) If v is outgoing near Γ t ∪ Γ b (as in Model 2), then, for any defect measures of v, for any (x, y) ∈ {(r, l), (l, r), (i, l), (l, i), (r, i)} (a) supp µ x in ⊂ B x y→x , (b) supp µ x out ⊂ B x x→y , with supp µ i in,out replaced with T * Γ i ∩ supp µ i in,out in the case x = i. In addition, (c) T * Γ i ∩ supp µ i out ⊂ B i r→i . Proof. Part (1a) follows from Corollary 4.3 together with Lemma 3.4. Part (1b) follows from the same results and the additional observation that

B out (T * Γ i ∩ H) ∩ B out (T * Γ l ∩ H) = ∅,
and Part (1c) is proven in the same way. Part (2a) and (2b) for (x, y) ∈ {(r, l), (l, r)} follow again from Corollary 4.3 and Lemma 3.4, together with the facts that

   B x y→x = π Γx B in (T * Γ x ∩ H) ∩ B out (T * Γ y ∩ H) ∩ {x ∈ Γ x } , B x x→y = π Γx B out (T * Γ x ∩ H) ∩ B in (T * Γ y ∩ H) ∩ {x ∈ Γ x } .
The case (x, y) = (r, i) follows from the case (x, y) = (r, l) and the observations that B r l→r ⊂ B r i→r and B r r→l ⊂ B r r→i . The cases (x, y) ∈ {(i, l), (l, i)}, as well as Part (2c), are shown in the same way and using the additional facts that

B out (T * Γ l ∩ H) ∩ B out (T * Γ i ∩ H) = ∅, B out (T * Γ r ∩ H) ∩ B in (T * Γ i ∩ H) = ∅,
while also exchanging the roles of µ in and µ out in the case of Part (2c).

We now investigate how defect measures propagate from one interface to another. To do so, we let, for (x, y) ∈ {r, i, l} 2 , d l := 0, d r := d l + d r , d i := d l , and

Φ in x→y : T * ({d x } × R) -→T * ({d y } × R) (x , ξ ) -→ π T * ({dy}×R) {ϕ t (0, x , ξ , ζ in (x , ξ )), t < 0} ∩ {x ∈ {d y } × R} ,
where ζ in is replaced by ζ out in the case (x, y) = (i, r) (i.e., Φ in x→y maps mass on the interface x to mass on the interface y under the backward flow from incoming mass on x) and

Φ out x→y : T * ({d x } × R) -→T * ({d y } × R) (x , ξ ) -→ π T * ({dy}×R) {ϕ t (0, x , ξ , ζ out (x , ξ )), t > 0} ∩ {x ∈ {d y } × R}
where ζ out is replaced by ζ in in the case (x, y) = (i, r) (i.e., Φ out x→y maps mass on the interface x to mass on the interface y under the forward flow from outgoing mass on x). Once again, the reason why we had to define the case (x, y) = (i, r) separately is because the measures on Γ i are taken from the left.

In the following lemma, we recall the notation that aµ(C) := C adµ for a measure µ.

Lemma 4.5. For any defect measure of v and any (x, y) ∈ {r, i, l} × {r, i, l},

µ x out = µ y in • Φ out x→y on B x x→y , µ x in = µ y out • Φ in x→y on B x y→x , with µ i
out and µ i in exchanged in the cases (x, y) ∈ {(i, r), (r, i)}. In particular, for any a ∈ C ∞ (T * R) not depending on x aµ x out (B) = aµ y in (Φ out x→y B), for all B ⊂ B x x→y , aµ x in (B) = aµ y out (Φ in x→y B), for all B ⊂ B x y→x , with µ i out and µ i in exchanged in the cases (x, y) ∈ {(i, r), (r, i)}. Proof. We prove the lemma for Φ x→y := Φ out x→y , the proof for Φ in x→y being similar. In addition, we assume that (x, y) ∈ {(r, l), (l, r), (i, l), (l, i)}; the proof in the cases (x, y) ∈ {(i, r), (r, i)} is the same with the roles of ζ in and ζ out exchanged on T * Γ i . Observe that

B out (B x x→y ) = B in (B y x→y ). In V := B out (B x x→y ) = B in (B y
x→y ), we work in geodesic coordinates given by Definition 3.3, (ρ

x , t x ) ∈ π -1 Γ B x x→y ∩ ζ = ζ out ×R + , and in geodesic coordinates (ρ y , t y ) ∈ π -1 Γ B y x→y ∩ ζ = ζ in ×R -. Observe that t y = t x -τ (ρ 1 ), ρ y = p in Φ x→y π Γx (ρ x ),
for some τ (ρ 1 ) > 0, thus dt y = dt x =: dt. Now, by Lemma 3.4, in (ρ x , t x ) and (ρ y , t y ) coordinates, µ can be written as

µ = p out * (2 √ rµ x out ) (ρ x ) ⊗ dt = p in * (2 √ rµ y in ))(ρ y ⊗ dt. It follows that p in * (2 √ rµ y in ) = Φ x→y * p out * (2 √ rµ x out )
, where Φ x→y := p in Φ x→y π Γx . Let now a be arbitrary and b :=

1 2 √ r a. Then Φx→yB a dµ y in = Φx→yB b d 2 √ rµ y in = p in (Φx→yB) b • π Γy d p in * (2 √ rµ y in ) = p in (Φx→yB) b • π Γy d Φ x→y * p out * (2 √ rµ x out ) = p out (B) b • π Γy • Φ x→y d p out * (2 √ rµ x out ) = B b • π Γy • Φ x→y • p out d(2 √ rµ x out ) = B b • Φ out x→y d(2 √ rµ x out ) = B 2 √ r × b • Φ out x→y dµ x out , (4.2)
where we used the change-of-variable formula (3.3), the fact that

Φ x→y p out B = p in (Φ x→y B),

and

π Γy • Φ x→y • p out = Φ out x→y . Now, as the Hamiltonian flow ϕ t consists of straight lines and Γ x and Γ y are parallel (straight) segments, we have, using the fact that r depends only on ξ ,

2 √ r × b • Φ out x→y = (2 √ rb) • Φ out x→y = a • Φ out
x→y , and the first part of the result follows by (4.2). The second part follows again from the fact that the Hamiltonian flow ϕ t consists of straight lines and Γ x and Γ y are parallel segments.

To finish this section, we define auxiliary measures in the following way.

Definition 4.6. If v admits defect measures and boundary defect measures, we define the auxiliary measure µ as µ := (ξ 1 + 1) 2 µ.

In addition, we let

µ x out := (+ √ r + 1) 2 µ l out for x = l, (- √ r + 1) 2 µ x out for x ∈ {i, r}, µ x in := (- √ r + 1) 2 µ l in for x = l, (+ √ r + 1) 2 µ x in for x ∈ {i, r},
and define, for x ∈ {l, i, r}, ν x d to be the Dirichlet measure associated to ( D x1 + 1)u on Γ x . The heuristic is that these measures correspond to measures of ( D x1 + 1)u. Although they are not defined in this way through Theorem 3.2, we now show that they satisfy all the expected properties.

Lemma 4.7. Assume that v is outgoing near Γ ∈ {Γ s , Γ s } and admits defect measures and boundary defect measures.

( 2 µ r in , and thus µ r out ( √ r = 1) = 0, and by the interpretation of boundary measures given by Lemma 3.4 we have {ξ 1 = -1} ∩ supp µ = ∅ as well. It follows that supp µ = supp µ. Hence, µ satisfies the conclusions of Corollary 4.3. By construction, it also satisfies the conclusions of Lemma 3.4. Therefore, since these were the only properties used in the proofs of Lemma 4.4 and 4.5, the auxiliary measures satisfy the conclusions of Lemma 4.4 and 4.5 (with the exact same proofs as for µ).

We now show (2), for example with x = l. By definition of the defect measures, on B

0 = ν l d = ν l d -2 ν l j + ν l n .
By the Cauchy-Schwarz inequality and a similar reasoning as in the proof of [GSW20, Lemma 3.3],

ν l j ≤ ν l d ν l n ,
Therefore, on B, ν l n = ν l d = -ν l j . Hence, from Lemma 3.5 (with α = 2), on B,

µ l out = √ r -1 √ r + 1 2 µ l in ,
which, by Definition 4.6, is µ l in = µ l out . Finally, (3) is a direct consequence of the similar property for µ from Lemma 3.5 together with the fact that {ξ 1 = -1} ∩ supp µ = ∅.

Finally, by propagation of defect measures given by Lemma 4.5 together with Lemma 4.4, Part (2a) (recalling Part (1) of Lemma 4.7), with (x, y) = (i, l),

µ i in = µ l out • Φ in i→l in T * Γ i ,
and the result (5.3) follows.

A first application of Proposition 5.3 is the following.

Corollary 5.4. Assume that u = Sg solves (M1) with g L 2 ≤ C, and admits defect measures and boundary defect measures given by Lemma 5.2, and let u := ( D x1 + 1)u. Then, Proof. We first prove the upper bound (5.1), then the lower bound (5.2). Let ι ∈ {-1, 1}.

lim →∞ ( D x1 + ι)u( ) 2 L 2 (Γi) = B l l→i ι + √ r 1 + √ r 2 d ν l d , ι = 1, -1.
The upper bound. If the upper bound (5.1) does not hold, then there exists → 0, g (h ) ∈ L 2 and u( ) = Sg( ) solving (M1) and

(5.7)

( D x1 + ι)u( ) L 2 (Γi) ≥ 1 + ι cos(θ max ) 1 + cos(θ max ) + g ( ) L 2 .
By rescaling, we can assume that (5.8) g ( ) L 2 = 1.

Then, by Lemma 5.2, up to extraction of a subsequence, u( ) admits defect measure and boundary defect measures. Now, by Corollary 5.4, (5.9) lim

→∞ ( D x1 + ι1)u( ) 2 L 2 (Γi) = B l l→i ι + √ r 1 + √ r 2 d ν l d But, √ r + ι √ r + 1 2 = 1 -ξ 2 + ι 1 -ξ 2 + 1 2
which is non-decreasing for ξ ∈ [0, 1] and |ξ | ≤ sin(θ max ) on B l l→i . Therefore, by (5.9)

lim →∞ ( D x1 + ι)u( ) 2 L 2 (Γi) ≤ ι + cos(θ max ) 1 + cos(θ max ) 2 ν l d (B l i→l ) ≤ ι + cos(θ max ) 1 + cos(θ max ) 2 ν l d (T * Γ l ) ≤ 1 + ι cos(θ max ) 1 + cos(θ max ) 2 lim sup →∞ g ( ) 2 L 2 = 1 + ι cos(θ max ) 1 + cos(θ max ) 2 ,
where we used (5.8) on the last line. This last inequality contradicts (5.7) and thus the upper bound (5.1) holds.

The lower bound. Let y 0 = (0, z 0 ) ∈ Γl and ξ 0 ∈ T * Γ l to be fixed later. Let η > 0 be such that

z 0 ∈ [η, L -η], and χ ∈ C ∞ c (R) be such that χ = 1 near z 0 and supp χ ⊂ [ 1 2 η, L -1 2 η]. We define g( ) ∈ L 2 (Γ l ), supp g Γ l by
(5.10) g( ) := (π ) -1/4 e i(y-y0)•ξ 0 / e -i(y-y0) 2 /2 χ(y).

Let u be the associated solution of (M1), and let u := ( D x1 + 1)u. By Lemma 5.2 (using the fact that g L 2 ≤ C) any sequence u( ) admits, up to extraction of a subsequence, defect measure and boundary defect measures on Γ l and Γ + i . Integrating by parts (see for example by [Zwo12, Page 102, Example 1]), we have that, (5.11) ν l d = δ ξ =ξ 0 ,y=y0 , and in addition, as → 0, (5.12) g( ) L 2 → 1.

We show that, for any sequence → 0, up to a subsequence, (5.13) lim

→∞ ( D x1 + ι)u( ) L 2 (Γi) g( ) L 2 (Γ l ) = 1 + ι cos(θ max -) 1 + cos(θ max -) ,
from which the result follows; indeed, if (5.2) fails, there exists a sequence → 0 so that

lim inf →∞ ( D x1 + ι)u( ) L 2 (Γi) g( ) L 2 (Γ l ) - 1 + ι cos(θ max -) 1 + cos(θ max -) > 0,
which contradicts (5.13) Take an arbitrary sequence → 0. By Corollary 5.4 together with (5.11),

(5.14) lim

→∞ ( D x1 + ι)u( ) 2 L 2 (Γi) = B l l→i ι + √ r 1 + √ r 2 dδ ξ =ξ 0 ,y=y0 .
Let ξ 0 := sin θ with θ := θ max -.

Since θ < θ max , there exists y 0 ∈ Γl so that (ξ 0 , y 0 ) ∈ B l i→l . With the data g given by (5.10) with these (ξ 0 , y 0 ), the desired result (5.13) follows from (5.14), (5.12), and the fact that r(ξ ) = 1ξ 2 . 6. Proof of the results involving Model 2; i.e. Theorem 1.2 and Theorem 1.3

Similarly to the start of §5, we fix admissible solution operators S(h, d l + , d r + ) : g → u and S(h, d l -, d r -) : g → u to (M2) (with the two sets of parameters (h, d l ± , d r ± )) throughout this section.

6.1. The impedance trace on Γ i in term of measures for Model 2. Proposition 6.1. Assume that u satisfies (M2) and has defect measure and boundary defect measures. For ι ∈ {1, -1}, let u := ( D x1 + ι)u. Then, up to a subsequence, u admits a Dirichlet boundary measure ν i d on Γ + i and (6.1)

supp ν i d ∩ T * Γ i ⊂ supp µ l out • Φ in i→l ∪ supp µ r out • Φ in i→r . If, in addition, (6.2) supp µ l out • Φ in i→l ∩ supp µ r out • Φ in i→r ∩ T * Γ i = ∅, then, in T * Γ i , (6.3) ν i d = 1 2 √ r ι + √ r 1 + √ r 2 µ l out • Φ in i→l + 1 2 √ r -ι + √ r -1 + √ r 2 µ r out • Φ in i→r .
The support property (6.1) reflects the fact that there are two contributions to the impedance trace on Γ i : mass coming from the left boundary Γ l , and mass coming from the right boundary Γ r . Both of these contributions a priori interact. However, when they don't interact, i.e., under the disjoint supports assumption (6.2), the impedance trace on Γ i corresponds precisely to the sum of these contributions, as expressed by (6.3), which is the analogue of (5.3) in the present case of (M2).

Together with (6.8) and (6.5), (6.12) and (6.13) give (6.3), and the proof is complete. Proposition 6.1 is our key tool in the rest of this section. Throughout the section, we repeatedly use the fact that if u = Sg solves (M2) with g L 2 ≤ C, then Lemma 5.2 allows us to extract defect measures and boundary defect measures for u. 6.2. Effect of the composite impedance map on data concentrating as a Dirac: Proof of Theorem 1.3, (1). We say that a family of Diracs on T * Γ l is (h, d l , d r )-non interacting if none can be obtained as the image under the flow of another one, and if the family produce rays whose intersections with T * Γ i are all distinct. (Strictly speaking, it is the beam-like Helmholtz solutions corresponding to the Dirac data on the boundary that are non-interacting, but since we work on the boundary we talk about the family of Diracs as non-interacting.) To introduce such a definition, we denote, for (x , ξ ) ∈ T * ({0} × R) and q ≥ 1,

Φ 1 →i (x , ξ ) := Φ out l→i (x , ξ ), Φ q+1 →i (x , ξ ) := Φ q →i • Φ out r→l • Φ out l→r (x , ξ ), Φ 1 i← (x , ξ ) := Φ out r→i • Φ out l→r (x , ξ ), Φ q+1 i← (x , ξ ) := Φ q i← • Φ out r→l • Φ out l→r (x , ξ
). In other words, Φ q →i (x , ξ ) is the point of T * Γ i attained by (x , ξ ) ∈ T * Γ l after q -1 reflections on the left and right boundaries of the domain and coming from the left, whereas Φ q ←i (x , ξ ) is the point of T * Γ i attained by (x , ξ ) ∈ T * Γ l after q reflections on the left and right boundaries of the domain and coming from the right. We can now define the notion of (h, d l , d r )-non interacting Diracs. Definition 6.2. We say that (δ x ,ξ ) 0≤ ≤N , for (x , ξ

) ∈ T * Γ l , is a family of (h, d l , d r )-non inter- acting Diracs if for all 0 ≤ 1 = 2 ≤ N, for all n ≥ 0, (x 1 , ξ 1 ) = Φ in r→l • Φ in l→r n (x 2 , ξ 2 ), (6.14) and for all 1 , 2 ∈ {→ i, i ←}, 0 ≤ 1 , 2 ≤ N, q 1 , q 2 ≥ 0 such that Φ q1 1 (x 1 , ξ 1 ) ∈ T * Γ i and Φ q2 2 (x 2 , ξ 2 ) ∈ T * Γ i , ( 1 , 1 , q 1 ) = ( 2 , 2 , q 2 ) =⇒ Φ q1 1 (x 1 , ξ 1 ) = Φ q2 2 (x 2 , ξ 2 ) (6.15)
The following proposition is a consequence of Proposition 6.1. It describes exactly the impedanceto-impedance maps in the high frequency limit when applied to data concentrating as a sum of non-interacting Diracs. Proposition 6.3. Let ι ∈ {1, -1}. Assume that u = Sg solves (M2) with g L 2 ≤ C and let u := ( D x1 + 1)u. If

ν l d = N =0 a (ξ )δ x ,ξ ,
where a ∈ C ∞ (R), a ≥ 0 and (δ x ,ξ ) 0≤ ≤N is a family of (h, d l , d r )-non interacting Diracs, then u := ( D x1 + ι)u admits Dirichlet boundary measure on

Γ + i , satisfying, in T * Γ i , (6.16) ν i d = ι + √ r 1 + √ r 2 =0•••N q≥1 Φ q →i (x ,ξ )∈T * Γi a 1 - √ r 1 + √ r 2(q-1) δ Φ q →i (x ,ξ ) + -ι + √ r -1 + √ r 2 =0•••N q≥1 Φ q i← (x ,ξ )∈T * Γi a 1 - √ r 1 + √ r 2q δ Φ q i← (x ,ξ ) .
Proof. Let Φ l→r := Φ in l→r , Φ r→l := Φ in r→l , and

Φ := Φ r→l • Φ l→r .
We claim that (6.17)

1 2 √ r µ l out = =0•••N k≥0 a √ r -1 √ r + 1 2k δ Φ k (y ,ξ ) 1 Φ k (y ,ξ )∈T * Γ l , and
(6.18) 1 2 √ r µ r out = =0•••N k≥0 a √ r -1 √ r + 1 2(k+1) δ Φ l→r •Φ k (y ,ξ ) 1 Φ l→r •Φ k (y ,ξ )∈T * Γr .
Once this claim is established, the result follows directly from Proposition 6.1, as the non-interaction assumption implies the disjoint-support assumption (6.2). We therefore only need to show (6.17) and (6.18). As (-D x1 + 1) u = 0 on Γ r , Lemma 3.5 together with Definition 4.6 implies that (6.19)

µ r out = √ r -1 √ r + 1 2 µ r in .
In addition, observe that if B ∩ ( (x , ξ )) = ∅, then ν l d (B) = 0 and thus also ν l j (B) = 0 by the Cauchy-Schwarz inequality. Hence by Lemma 4.7, (3), (6.20)

µ l out (B) = µ l in (B), for all B ⊂ T * Γ l with B ∩ =0•••N (x , ξ ) = ∅.
Finally, by Lemma 4.7, (1) and Lemma 4.5, for any a ∈ C ∞ (T * R) not depending on x, (6.21) a µ l in (B) = a µ r out (Φ l→r B), for all B ⊂ B l r→l , and (6.22) a µ r in (B) = a µ l out (Φ r→l B), for all B ⊂ B r l→r . Observe that we have the disjoint union

T * Γ l ∩ H = 0≤k≤∞ R k , R k := ρ ∈ T * Γ l ∩ H, inf : Φ (ρ) ∩ T * Γ l = ∅ = k ;
i.e., each R k consists of points of T * Γ l ∩ H that are no longer in T * Γ l after exactly k iterations of the map Φ (going from left to right and back again). Let now B ⊂ R k0 . We first assume that k 0 < ∞. Then, by definition,

Φ k0 (B) ∩ T * Γ l = ∅.
It follows that Φ l→r • Φ k0-1 (B) ∩ B r l→r = ∅. As supp µ r in ⊂ B r l→r by Part (2a) of Lemma 4.4 (recall Lemma 4.7, (1)), we have using (6.19) that

µ r out (Φ l→r • Φ k0-1 (B)) = 0. If, for all k, Φ k (B) ∩ =0•••N (x , ξ ) = ∅,
it follows by (6.19), (6.20), (6.21) and (6.22) that (6.23)

0 = µ r out (Φ l→r • Φ k0-1 (B)) = µ l in (Φ k0-1 (B)) = µ l out (Φ k0-1 (B)) = µ r in (Φ l→r • Φ k0-2 (B)) = √ r -1 √ r + 1 -2 µ r out (Φ l→r • Φ k0-2 (B)) = • • • = √ r -1 √ r + 1 -2k0 µ l out (B), hence (6.24) if, for all k, Φ k (B) ∩ =0•••N (x , ξ ) = ∅, then µ l out (B) = 0.
Assume now that there exists k 1 so that Φ k1 (B)

∩ =0•••N (x , ξ ) = ∅.
By the non-interaction assumption, if B is small enough, such a k 1 is unique, and there exists a unique so that Φ k1 (B) (y , ξ ). Following the same argument as in (6.23

) up to k = k 1 , we get µ l in (Φ k1 B) = 0. It follows by Lemma 3.5 that 1 2 √ r µ l out (Φ k1 B) = ν l d (Φ k1 B). Hence 1 2 √ r µ l out (Φ k1 B) = a (ξ )
. We now follow the same argument as in (6.23) from k = k 1 down to k = 0. We get (6.25)

a (ξ ) = 1 2 √ r µ l out (Φ k1 B) = 1 2 √ r µ r in (Φ l→r • Φ k1-1 (B)) = √ r -1 √ r + 1 -2 1 2 √ r µ r out (Φ l→r • Φ k1-1 (B)) = 1 2 √ r √ r -1 √ r + 1 -2 1 2 √ r µ l in (Φ k1-1 (B)) = • • • = √ r -1 √ r + 1 -2k1 1 2 √ r µ l out (B).
From (6.24) and (6.25), it follows that (6.17) is verified for any B ⊂ 0≤k<∞ R k . It remains to verify (6.17) for B ∈ R ∞ . Observe that such a set satisfies B ⊂ { √ r = 1}. Hence, using (6.22) and (6.19)

µ l in (B) = µ r out (Φ l→r (B)) = 0 × µ r in (Φ l→r (B)) = 0. It follows by Lemma 3.5 that µ l out (B) = 2 √ r ν l d (B) = 2 √ rδ ξ =ξ 0 ,y=y0 (
B). Thus (6.17) is verified also for B ⊂ R ∞ , hence (6.17) holds. The proof of (6.18) follows the same lines.

To prove Theorem 1.3, Part (1), we iterate Proposition 6.3 by taking a boundary data producing a Dirac Dirichlet measure at high frequency, whose image with mass one is still in the domain after n iterations. To do so, we need to know that a suitable family of (h, d l , d r )-non interacting Diracs exists; this is given by the following lemma, in which we let ←π be the canonical projection from T * Γ i to T * Γ l (using that T * Γ i T * ({0} × (0, h)) = T * Γ l ). Lemma 6.4. For any N ≥ 1 and any σ ∈ {→ i, i ←} N , there exists (x σ , ξ σ ) ∈ T * Γ l with ξ σ = 0 so that (1)

Φ 1 σ(N ) =1•••N -1 ← - π Φ 1 σ( ) (x σ , ξ σ ) ∈ T * Γ i
(where the product denotes composition of the maps) (2) for any 0 ≤ ≤ N -1, the family X of points of T * Γ l defined by

   X 0 := (x σ , ξ σ ) , X +1 := ← - π Φ q (x , ξ ), s.t. (x , ξ ) ∈ X , q ≥ 1, ∈ {→ i, i ←}, Φ q (x , ξ ) ∈ T * Γ i
forms a corresponding family of (h, d l , d r )-non interacting Diracs, and X / ∈ {0, h} for any .

Proof. We take (for example) x σ := 1 2 h. Observe that the property (1) is satisfied as soon as |ξ σ | is small enough (depending on N ). Hence, it remains to show that we can construct ξ σ so that the property (2) holds as well. To do so, it suffices to show that, given a finite set of points (x j ) 1≤j≤n ∈ Γ l , the set A := (ξ j ) 1≤j≤n ⊂ B(0, 1) n , so that (x j , ξ j ) 1≤j≤n does not satisfy P is discrete, where we say that a set X of points of T * Γ l satisfies P if ←π Φ q (x , ξ ), s.t. (x , ξ ) ∈ X, q ≥ 1, ∈ {→ i, i ←}, Φ q (x , ξ ) ∈ T * Γ i forms a corresponding family of (h, d l , d r )-non interacting Diracs and X / ∈ {0, h} for any . Indeed, if it is the case, the set of ξ = 0 so that (x σ , ξ ) does not satisfy (2) is discrete, and hence it suffices to take ξ σ small enough.

We therefore show that for any n and any set of points X = (x j ) 1≤j≤n ∈ Γ l , A is discrete. Assume that, for some (ξ j ) 1≤j≤n , (x j , ξ j ) 1≤j≤n does not satisfies P. Let C 1 (X) be the subset of indices of 1 ≤ • • • ≤ n so that (6.14) fails, C 2 (X) so that (6.15) fails, and C 3 so that X ∈ {0, h}. If j ∈ C 1 (resp. C 2 or C 3 ), taking ξ j = ξ j with | ξ jξ j | small enough and X := X\(x j , ξ j ) ∪ (x j , ξ j ), we have

C 1 ( X) ⊂ C 1 (X)\{j}, C 2 (X) ⊂ C 2 ( X) and C 3 (X) ⊂ C 3 ( X) (resp. C 2 ( X) ⊂ C 2 (X)\{j} with C 1 (X) ⊂ C 1 ( X) and C 3 (X) ⊂ C 3 ( X); or C 3 ( X) ⊂ C 3 (X)\{j} with C 1 (X) ⊂ C 1 ( X) and C 2 (X) ⊂ C 2 ( X))
, which shows, acting repetitively, that A is indeed discrete.

We can now conclude.

Proof of Theorem 1.2 and Theorem 1.3, Part (1). Let N ≥ 1 and σ ∈ {+, -} N be arbitrary; in the case of Theorem 1.2, N := 1. Let σ ∈ {→ i, i ←} N be associated to σ by

σ(n) = i ← if σ(n) = +, → i if σ(n) = -,
and let (x σ , ξ σ ) ∈ T * Γ l be given by Lemma 6.4. Moreover, let χ ∈ C ∞ c (Γ l ) be equal to one near x σ . We define g( ) ∈ L 2 (Γ l ) by g( )(y) := χ(y)(πh) -1/4 e i(y-x σ )•ξ σ / e -i(y-x σ ) 2 /2 , and (u n ) 1≤n≤N to be the cascade of solutions of (M2) associated with σ and g by S(h, d l ± , d r ± ), that is so that

         (-2 ∆ -1)u n = 0 in D ( D x1 + 1)u n = γ Γi(d σ(n-1) ) ( D x1 ± σ(n-1) 1)u n-1 on Γ l , u n is outgoing near Γ s , (-D x1 + 1)u n = 0 on Γ r (d r σ(n) ),
where we denoted ± σ(n) := σ(n) to improve readability, and

         (-2 ∆ -1)u 1 = 0 in D ( D x1 + 1)u 1 = g( ) on Γ l , u 1 is outgoing near Γ s , (-D x1 + 1)u 1 = 0 on Γ r (d r σ(1) ).
In addition, denote

u n := ( D x1 + 1)u n , u n := ( D x1 ± σ(n) 1)u n .
We claim that there is a sequence so that, as → ∞

(6.26) u N ( ) L 2 (Γi) → c ≥ 1.
Since g( ) L 2 (Γ l ) → 1 (by direct computation), showing (6.26) ends the proof. As g is locally uniformly bounded in L 2 , by Lemma 5.2, there is a sequence so that u 1 admits defect measures and boundary defect measures. In addition, direct computation shows that

ν 1,l d = δ (x σ ,ξ σ ) .
Hence by Proposition 6.3, ν 1,l d exists and is given by (6.16). In addition, observe that the existence of ν 1,i d implies that, up to a subsequence, γ Γi(d σ(1) ) ( D x1 ± σ(1) 1)u 1 is locally uniformly bounded in L 2 , hence we can iterate the argument thanks to the non self-interaction property of Lemma 6.4, Part (2). By Lemma 6.4, Part (1), repeated use of Proposition 6.3 gives in particular

ν N,i d ≥ δ (x 1 ,ξ 1 ) , (x 1 , ξ 1 ) := Φ 1 σ(N ) =1•••N -1 ← - π Φ 1 σ( ) (x σ , ξ σ ) ∈ T * Γ i ,
hence (6.26) follows thanks to (7.21) (from Lemma 5.2).

6.3. High-frequency nilpotence of the impedance-to-impedance map away from zerofrequency: Proof of Theorem 1.3, Part (2). The following Proposition reflects the fact that, in the high-frequency limit, the impedance-to-impedance map associated with (M2) pushes the mass emanating from Γ l with an angle λ to the horizontal up and down by a distance proportional to λ -1 . We denote -→ π the canonical projection from T * Γ l to T * Γ i .

Proposition 6.5. Let ι ∈ {1, -1}, 0 < λ < 1, and

X + λ , X - λ ⊂ T * Γ l so that X + λ ⊂ {ξ ≥ λ}, X - λ ⊂ {ξ ≤ -λ}. Assume that u = Sg solves (M2) with g L 2 ≤ C and let u := ( D x1 + 1)u. If supp ν l d ⊂ X + λ ∪ X - λ ,
then, up to a subsequence u := ( D x1 + ι)u admits Dirichlet defect measure on Γ + i , satisfying

supp ν i d ∩ T * Γ i ⊂ - → π X + λ + d l λ √ 1 -λ 2 , 0 ∪ - → π X - λ + - d l λ √ 1 -λ 2 , 0 .
Proof. Observe that, by Proposition 6.1,

supp ν i d ∩ T * Γ i ⊂ supp µ l out • Φ in i→l ∪ supp µ r out • Φ in i→r .
Therefore, the result follows if we can show that (6.27)

supp µ l out • Φ in i→l ∪ supp µ r out • Φ in i→r ∩ T * Γ i ⊂ - → π X + λ + d l λ √ 1 -λ 2 , 0 ∪ - → π X - λ + - d l λ √ 1 -λ 2 , 0 .
To show (6.27), we focus on the part of the statement concerning µ l out ; the proof for µ r out is similar. To do so, let B 0 ⊂ T * Γ i be such that (6.28)

B 0 ∩ - → π X + λ + d l λ √ 1 -λ 2 , 0 ∪ - → π X - λ + - d l λ √ 1 -λ 2 , 0 = ∅.
Our goal is to show that µ l out (Φ in i→l B 0 ) = 0. We denote Φ l→r := Φ in l→r , Φ r→l := Φ in r→l , and

Φ := Φ r→l • Φ l→r .
By simple geometric optics, a ray starting from (x , ξ ) ∈ T * Γ makes an angle sin ξ to the horizontal. Using that tan arcsin λ = λ √ 1-λ 2 , we then have that

(x , ξ ) / ∈ - → π X + λ + d l λ √ 1 -λ 2 , 0 ∪ - → π X - λ + - d l λ √ 1 -λ 2 , 0 =⇒ for all n ≥ 0, Φ n • Φ in i→l (x , ξ ) / ∈ X + λ ∪ X - λ .
Hence, (6.28) together with the support assumption gives that (6.29) for all n ≥ 0, ν l d Φ n • Φ in i→l B 0 = 0. As (-D x1 + 1) u = 0 on Γ r , by Lemma 3.5 together with Definition 4.6, (6.30)

µ r out = √ r -1 √ r + 1 2 µ r in .
In addition, by Lemma 4.7, (3), (6.31) µ l out (B) = µ l in (B), for all B ⊂ T * Γ l s.t. ν l d (B) = 0. Finally, by Lemma 4.5 (recall Lemma 4.7, (1)), for any a ∈ C ∞ (T * R) not depending on x, (6.32) a µ l in (B) = a µ r out (Φ l→r B), for all B ⊂ B l r→l , and (6.33) a µ r in (B) = a µ l out (Φ r→l B), for all B ⊂ B r l→r . We denote B 0 := Φ in i→l B 0 , consider again the disjoint union

T * Γ l ∩ H = 0≤k≤∞ R k , R k := ρ ∈ T * Γ r ∩ H, inf , Φ (ρ) ∩ T * Γ l = ∅ = k ,
and decompose

B 0 = 0≤k≤∞ B k 0 .
We show that (6.34) for all 0 ≤ k ≤ ∞, µ l out ( B k 0 ) = 0, which completes the proof. We first assume that k < ∞. Then,

Φ k ( B k 0 ) ∩ T * Γ l = ∅. It follows that Φ l→r • Φ k-1 ( B k 0 ) ∩ B r l→r = ∅.
As, by Lemma 4.4, (2a) (and Lemma 4.7, (1) again), supp µ r in ⊂ B r l→r , we have using (6.30)

µ r out (Φ l→r • Φ k-1 ( B k 0 )) = 0. It follows that for all n ≥ h min(d + l , d - l ) √ 1 -λ 2 λ , supp ν n,i d ∩ T * Γ i = ∅.
Thanks to (7.21), this implies (6.35) and therefore ends the proof.

7.

The wellposedness results i.e. proof of Lemma 1.5 and 1.6, and interior trace bounds 7.1. Definition of the admissible solution operators. Let us first define extensions of our models. For > 0, let

D 1 := (0, d l + d r + ) × (-, h + ), D 2 := (0, d l + d r ) × (-, h + ), let Γ 1,2
x, denote the corresponding parts of the boundaries of D 1,2 , and g ∈ L 2 (Γ l ) being given, define g as its extension by zero to L 2 (Γ 1 l, ) = L 2 (Γ 2 l, ). Denote also Γ 1 b, := Γ 1 l, and Γ 2 b, := Γ 2 l, ∪ Γ 2 r, . We define the extended models (M1 ) and (M2 ) in the following way.

(M1 )      (-2 ∆ -1)u = 0 in D 1 ( D x1 + 1)u = g on Γ 1 l, , u is outgoing near Γ 1 s, . (
M2 )          (-2 ∆ -1)u = 0 in D 2 ( D x1 + 1)u = g on Γ 2 l, , u is outgoing near Γ 2 s, , (-D x1 + 1)u = 0 on Γ 2
r, . We also define

D 1 := (-∞, d l + d r + ) × (-, h + ), D 2 := (-∞, ∞) × (-, h + ),
and, if u ∈ L 2 (U ), with U ⊂ R 2 , we denote by u its extension by zero to R 2 . Definition 7.1. A solution operator S associated to model (M1), resp. (M2), is said to beadmissible if there exists , C, 0 > 0 such that for any g ∈ L 2 (Γ l ), Sg can be extended to a solution of (M1 ), resp. (M2 ), that

(1) is bounded: u L 2 (D 1,2 ) ≤ C g L 2 (Γ l ) for all 0 < ≤ 0 , and

(2) has bounded traces:

u L 2 (Γ 1,2 b, ) + ∂ n u L 2 (Γ 1,2 b, ) ≤ C g L 2 (Γ l
) for all 0 < ≤ 0 . In §7.2 and §7.3 below, we use some results about semiclassical pseudo-differential calculus in R d , in particular, the elliptic parametrix construction. We refer for example to [DZ19, Appendix E] for a precise definition of the pseudo-differential operator classes and the statement of this result (alternatively, a brief introduction to these techniques can be found in [LSW22, §4]). 7.2. Proof of Lemma 1.5 and 1.6.

Proof of Lemma 1.5. We prove existence for (M2); the proof for (M1) is similar.

Existence of a solution to (M2 ). For > 0, let Θ = Θ 1 ∪ Θ 2 be the union of two smooth, connected, convex subsets

Θ 1 , Θ 2 ⊂ R 2 so that (7.1) Θ ∩ (0, d r + d l ) × R = ∅, Θ 1 ∩ {0} × R = Γ 2 l, , and
Θ 2 ∩ {d l + d r } × R = Γ 2 r, .
We extend g to ∂Θ by 0, as a new function g ∈ L 2 (∂Θ). Now, let w be solution the exterior impedance problem with the Sommerfeld radiation condition in

Ω := R 2 \Θ (7.2)      (-2 ∆ -1)w = 0 in Ω, (-D n + 1)w = g on ∂Θ, (-D r + 1)w = o(r -1 2 ) as r → ∞,
where n is the normal pointing into Θ (i.e., the normal pointing out of D + on Γ l, and Γ r, ). Noting the sign of the normal, the solution of (7.2) is unique by, e.g., [CK83, Theorem 3.37], and exists by Fredholm theory for all > 0. In addition, observe that w is -tempered with

(7.3) w H 1 (K∩(R 2 \Θ)) ≤ C -1/4 g L 2
for any compact K ⊂ R 2 by [Spe14, Theorem 1.8]. We now claim that

(7.4) WF w ∩ T c ⊂ ρ ∈ T * (R 2 \Θ), ∃t < 0, π x ϕ t (ρ) ∈ supp g ,
where T consists of the trapped rays between Θ 1 and Θ 2 (observe that T ⊂ {ξ • n(x) = 0} in the notation of Definition 1.4), and ϕ t (x, ξ) denotes the point of the phase-space attained following, for a time t, the ray of geometrical optics in R 2 \Θ starting from (x, ξ). Observe that the above shows the existence of a solution to (M2 ) by taking

u := w| D 2 and any D + ⊂ (0, d r + d l ) × R satisfying D + ⊃ D 2 ∪ Γ 2 so that ∂D + ∩ (∂D 2 \Γ 2 ) = ∂D 2 \Γ 2 .
We therefore show (7.4). Observe that, for A > 0 so that B(0, A) Θ

(7.5) WF w ∩ T * B(0, A) c ⊂ ξ • x > 0 .
The inclusion (7.5) is proven similarly as in [START_REF] Burq | Semi-classical estimates for the resolvent in nontrapping geometries[END_REF], observing that, for Ψ ∈ C ∞ (R 2 ) such that Ψ = 0 near Θ and Ψ = 1 near B(0, A) c , Ψw is outgoing (in the standard sense) and solves the following Helmholtz equation in R 2 ,

(-2 ∆ -1)Ψw = -[ 2 ∆, Ψ]w, hence (
x ϕ t (ρ), t ∈ [0, τ ] ∩ supp g = ∅ =⇒ ϕ τ (ρ) / ∈ WF w.
By (7.6) and (7.7), we obtain (7.4). We define Sg := u |D , and it remains to show the points (1) and (2) of Definition 7.1.

The trace bound in Part (2) of Definition 7.1. If we prove that (7.8)

w L 2 (∂Θ) + ∂ n w L 2 (∂Θ) g L 2 (Γ l ) .
then this implies the bound in Part (2) of Definition 7.1. For R 1, let Ω R := B(0, R)\Θ. Pairing the equation by v and integrating by parts on Ω R , we get (7.9)

2 ∇w 2 L 2 (Ω R ) -w 2 L 2 (Ω R ) = i w 2 L 2 (∂Ω R ) + D n w -w, w L 2 (∂Ω R ) .
Taking the imaginary part, we obtain by Cauchy-Schwarz inequality and the inequality 2ab ≤ a 2 +b 2 for a, b ≥ 0,

w 2 L 2 (∂Ω R ) ≤ D n w -w L 2 (∂B(0,R)) w L 2 (∂B(0,R)) + 1 2 g 2 L 2 + 1 2 w 2 L 2 (∂Θ)
Using the Sommerfeld radiation condition, we obtain that

w 2 L 2 (∂Ω R ) ≤ 2 D n w -w L 2 (∂B(0,R)) w L 2 (∂B(0,R)) + g 2 L 2 ≤ 2 (R)R -1 2 1 L 2 (∂B(0,R)) w L 2 (∂B(0,R)) + g 2 L 2 ≤ 2 (R) w L 2 (∂B(0,R)) + g 2 L 2 , where (R) → 0 as R → ∞ for fixed > 0. If, > 0 being fixed, w L 2 (∂B(0,R)) ≤ w 2
L 2 (∂B(0,R)) , (7.8) follows by taxing R fixed big enough so that 2 (R) ≤ 1 2 . If is not the case, then w L 2 (∂B(0,R)) ≤ 1, and (7.8) follows by letting R → ∞, since w 2 L 2 (∂Θ) ≤ w 2 L 2 (∂Ω R ) .

An oscillatory property. Before showing Part (2) of Definition 7.1, we obtain an oscillatory bound ((7.8) below) on solutions w.

Let χ ∈ C ∞ c (R 2 ), ψ ∈ C ∞ (R 2
) be bounded and so that ψ = 0 in B(0, 2), and denote by w the extension of w to R 2 by zero.

First, observe that, as (-2 ∆-1) is semiclassicaly elliptic on WF ψ( D x ), there exists E ∈ Ψ -2 so that (7.10)

ψ( D x ) = E(-2 ∆ -1) + O( ∞ ) Ψ -∞ . Now, (-2 ∆ -1)χw = 1 x∈D 2 (-2 ∆ -1)χw + -δ(x 1 ) ⊗ ∂ x1 (χw) |x1=0 + hδ (x 1 ) ⊗ (χw) |x1=0 = 1 x∈D 2 [-2 ∆, χ]w + -δ(x 1 ) ⊗ ∂ x1 (χw) |x1=0 + hδ (x 1 ) ⊗ (χw) |x1=0 = [-2 ∆, χ]w -δ(x 1 ) ⊗ (∂ x1 χ)w |x1=0 + (-δ(x 1 ) ⊗ ∂ x1 (χw) |x1=0 + hδ (x 1 ) ⊗ χw |x1=0 ). (7.11) Let z := δ(x 1 ) ⊗ (∂ x1 χ)w |x1=0 + -δ(x 1 ) ⊗ ∂ x1 χw |x1=0 + hδ (x 1 ) ⊗ (χw) |x1=0 .
Pairing with a test function ϕ and using the Cauchy-Schwarz inequality and then trace inequalities, we find that

| z, ϕ | ≤ C ϕ L 2 ({x1=0}) + 2 ∂ n ϕ L 2 ({x1=0}) w L 2 (∂Θ) + ∂ n w L 2 (∂Θ) ≤ C ϕ H 1 2 +η + 2 ϕ H 3 2 +η w L 2 (∂Θ) + ∂ n w L 2 (∂Θ) ≤ C 1 2 -η ϕ H 3 2 +η w L 2 (∂Θ) + ∂ n w L 2 (∂Θ) , for η > 0. Hence z H -3 2 -η ≤ C 1 2 -η w L 2 (∂Θ) + ∂ n w L 2 (∂Θ) .
By combining the above with (7.11), (7.10) and (7.3), and using the fact that E

[-2 ∆, χ] ∈ Ψ -1 ⊂ Ψ 0 , we get (7.12) ψ( D x )χw L 2 ≤ C 1 2 -η w L 2 (∂Θ) + ∂ n w L 2 (∂Θ) + g L 2 (Γ l ) .
Together with (7.8), this gives the oscillatory bound

(7.13) ψ( D x )χw L 2 ≤ C 1 2 -η g L 2 (Γ l ) .
The resolvent estimate in Part (1) of Definition 7.1. We show that, for any χ ∈ C ∞ c (R 2 ), there is C > 0 so that for 0 > 0 small enough and any 0

< ≤ 0 χw L 2 ≤ C g L 2 .
In particular, this gives Part (1) of Definition 7.1. Without loss of generality, we assume that Θ supp χ. If the estimate fails, there exists sequences h → 0, g ∈ L 2 (Γ ), and w ( ) (that we will denote w( ) to lighten the notations) so that (7.14) χw( ) L 2 ≥ g L 2 .

Rescaling, we may assume that (7.15) χw( ) L 2 = 1.

As Θ supp χ, it follows from (7.15) and the properties of the free outgoing resolvent that w( ) is bounded in L 2 loc (see for example [Bur02, Lemma 3.1]). Together with (7.14), the trace bound (7.8) and the analogue of Theorem 3.2 in R 2 \Θ (e.g. [GLS21a, Theorem 2.3]), up to extracting a subsequence, we can assume that u( ) admits defect measure and boundary defect measures.

Thanks to the oscillatory property (7.12), it follows from (7.15) that µ(χ 2 ) = 1 (see, e.g., [GSW20, Proof of Lemma 4.2]). To obtain a contradiction, we show that µ = 0. We focus on the most challenging part of the phase space T * D 2 ∩ D + , that contains the trapped trajectories T (recall (7.1)), and show that µ(T * D 2 ∩ D + ) = 0. The fact that µ vanishes on the other parts of the phase space is obtained in the same way, in the spirit of [START_REF] Burq | Semi-classical estimates for the resolvent in nontrapping geometries[END_REF] and [START_REF] Galkowski | Optimal constants in nontrapping resolvent estimates[END_REF], by combining the invariance of the measure with the outgoingness of the wavefront-set (with the invariance of the measure on the glancing set G in our case coming from the facts that, in our particular setting of the exterior of two convex obstacles, the generalized bicharacteristics through G are bicharacteristics of H p and H p µ = δ(x 1 ) ⊗ (µ inµ out ) everywhere, e.g., by [GLS21a, Lemmas 2.10, 2.13, 2.14]). We therefore work in D 2 from now on. Observe that all the results of §3 and §4 apply to solutions of (M2 ) in D 2 , and we use these results in the corresponding case without further mention. We will show that (7.16) µ l out = µ l in = 0, µ r out = µ r in = 0.

Observe that, by (7.14), g L 2 (Γ l ) → 0.

It follows from the above together with the boundary condition on Γ l, that ν l d + 2 ν l j + ν l n = 0.

Similarly as in the proof of Lemma 4.7, this implies by Lemma 3.5 that (7.17 We have the disjoint union

T * Γ l ∩ H = 0≤k≤∞ R k , R k := ρ ∈ T * Γ l ∩ H, inf , Φ (ρ) ∩ T * Γ l = ∅ = k ,
where Φ := Φ r→l •Φ l→r . Let now B ⊂ R k0 . We first assume that k 0 < ∞. Then, Φ k0 (B)∩T * Γ l = ∅. It follows that Φ l→r • Φ k0-1 (B) ∩ B r l→r = ∅. As, by Lemma 4.4, (2a), supp µ r in ⊂ B r l→r , we have µ r in (Φ l→r • Φ k0-1 (B)) = 0, and using (7.17), (7.18), (7.19) and (7.20) repetitively, it follows that µ l out (B) = 0. Therefore

µ l out 0≤k<∞
R k = 0.

In addition, as √ r = 1 on R ∞ , (7.17) yields to µ l out (R ∞ ) = 0 as well. Hence µ l out = 0. The same arguments lead to µ r out = 0, and hence, using (7.19) and (7.20) again, we conclude that (7.16) holds. Using Lemma 3.4 together with Corollary 4.3, it follows from (7.16) that µ(T * (D 2 ∩ D + )) = 0.

Proof of Lemma 1.6. Similarly as in the proof of Lemma 1.5, since S 1 g -S 2 g solves (M1), resp (M2) with zero impedance boundary condition on Γ b , propagation of singularities together with the outgoing condition give

WF (S 1 g -S 2 g) ∩ T * D = ∅.
The result follows by definition of the wavefront set together with Part (2) of Lemma 7.6.

Remark 7.2. The propagation results of Melrose and Sjöstrand [MS78, MS82] are proven for homogeneous (that is, non semiclassical) second-order operators, with wavefront-set defined from homogeneous pseudo-differential operators. In the spirit of [START_REF] Lebeau | Contrôle de l'équation de Schrödinger[END_REF], to use their results in our semiclassical setting, for any sequences = ( k ) k≥1 , u = (u k ) k≥1 , with (-2 k ∆ -1)u k = 0, if we define Θ(u) := k e -it -1 k u k , then for any (x, ξ, t, τ ) ∈ T * Ω×T * R, (x, ξ) ∈ WF (u) iff (x, ξ, t, τ ) ∈ WF (Θ(u)) (see [START_REF] Lebeau | Contrôle de l'équation de Schrödinger[END_REF]), and Θ(u) solves the homogeneous wave equation (∂ 2 t -∆)Θ(u) = 0 (with damping boundary condition in the setting of this paper), to which [MS78, MS82] apply; we therefore obtain propagation of singularities with -wavefront-sets for u. For a more general presentation of semiclassical propagation of singularities up to the boundary, we refer the interested reader for example to [START_REF] Ivrii | Microlocal analysis and precise spectral asymptotics[END_REF][START_REF]Microlocal analysis, sharp spectral asymptotics and applications[END_REF]. the above to χf and using the pseudo-locality of the semiclassical pseudo-differential calculus, we get

(7.22) Aχf L 2 (Γ i, ) ≤ C χAχf L 2 + Ch -1 χ(-2 ∆ -1)Aχf L 2 + O( ∞ ) χf L 2 .
By Lemma 4.2, there exists A ∈ Ψ 0 (R 2 ) so that WF A ∩ S * Γ = ∅ and WF (χ Since, in addition, WF (χv) ⊂ {|ξ| 2 = 1}, it follows from (7.23) that (7.24) WF (χv) ⊂ {1 ≥ r(ξ ) ≥ c 2 }. Now, let ψ ∈ C ∞ c be so that supp ψ ⊂ [c 2 /2, 2] and ψ = 1 near {1 ≥ z ≥ c 2 }, and define Ψ(ξ ) := ψ(r(ξ )). From (7.24) together the definition of the wavefront-set and Lemma 7.6, (2), for any s and any N there exists C s , N > 0 so that

D
(1 -Ψ( D x ))χv H s ≤ C s,N N .
Hence, by Sobolev trace estimates, (1 -Ψ( D x ))v L 2 (Γi) → 0. It follows that, by definition of defect measures, ν i d (1 -Ψ( D x )) 2 = 0. As (1 -Ψ( D x )) 2 = 1 near H c , the Lemma follows.

  is a solution operator associated to model (M1)/(M2), if S is linear and for any g ∈ L 2 (Γ l ), u( ) := S( )g is solution to model (M1)/(M2). The following results are shown in §7.Lemma 1.5. There exists an admissible (in the sense of Definition 7.1) solution operator to (M1)/(M2).

  + 22] on the performance of the parallel Schwarz method. The experiments in [GGG + 22, §6] considered two situations.

Figure 3 .

 3 Figure 3. Three overlapping subdomains in the 2-d strip decomposition

- 1 and

 1 I Γ + →Γ - +1 can be written in terms of I -and I + , as in the following result from [GGG + 22]. Corollary 2.8. ([GGG + 22, Corollary 4.11]) For the 2-d strip decomposition described above,

  Remark 2.9 (Why the maps in §1.4 involve two different values of d l and d r ). If the subdomains all have equal lengths L and equal overlaps δ, then the map I -appears in Corollary 2.8 with d l = δ and d r = L-δ and the map I + appears with d l = L-δ and d r = δ. This is why in §1.4 we consider I + (d + , d r + ) and I -(d + , d r + ) with d + not necessarily equal to d -(and similar for d r ± ).

Corollary 4. 3 .

 3 Assume that v is outgoing near Γ ∈ {Γ s , Γ s } and let Γ := ∂D\Γ. Then, for any defect measure of v, supp µ ∩ T * D ∩ D + ⊂ B out (T * Γ ∩ H).

1 )

 1 The auxiliary measures µ x out/in and ν x d satisfy the conclusions of Lemma 4.4 and 4.5. (2) For x ∈ {l, r}, if ν x d = 0 on some Borel set B ⊂ H, then, on B, µ x in = µ x out . (3) If µ in = 0 on some Borel set B ⊂ H, then, on B, µ out = 2 √ r ν d . Similarly, if µ out = 0 on some Borel set B ⊂ H, then, on B, µ in = 2 √ r ν d . Proof. We first show (1). Observe that supp µ ⊂ supp µ ∪ {ξ 1 = -1}. We now show that {ξ 1 = -1} ∩ supp µ = ∅. If v is outgoing near Γ s , this follows directly from Corollary 4.3. If v is outgoing near Γ s , by Lemma 3.5, µ r out = √ r-1 √ r+1

Proof.

  Part (1) of Lemma 4.7 and Part (1) of Lemma 4.4 imply that µ l in = 0, and thus Part (3) of Lemma 4.7 implies that, in H, µ l out = 2 √ r ν l d ; the result then follows by Proposition 5.3 together with (7.21) from Lemma 5.2. 5.2. Proof of Theorem 5.1.

  same way, as (-D x1 + 1)u = 0 on Γ r , Lemma 4.5, denoting Φ l→r := Φ in l→r , Φ r→l := Φ in r→l , for any a ∈ C ∞ (T * R) not depending on x, (7.19) aµ l in (B) = aµ r out (Φ l→r B), for all B ⊂ B l r→l , and (7.20) aµ r in (B) = aµ l out (Φ l→r B), for all B ⊂ B r l→r .

  x1 u) ∩ WF (I -A) = ∅. In addition, let ψ ∈ C ∞ (R d ) so that ψ = 0 in B(0, 2) and ψ = 1 outside B(0, 3). Observe that χ D x1 u = Aχ D x1 u + (1 -A)ψ( D x )χ D x1 u + (1 -A)(1ψ( D x ))χ D x1 u, hence, by definition of the wavefront-set, as (1 -A)(1ψ( D x ))χ has a compactly supported symbol, and using Part (2) of Lemma 7.6, χ D x1 u = Aχ D x1 u + O( ∞ ) χu in any H m space. Hence, applying (7.22) to D x1 u (and using a trace estimate to deal with the O( ∞ ) error), we obtain the trace bound on ∂ n u using the fact that [-2 ∆, Aχ] ∈ Ψ 1 , Part (1) of Lemma 7.6, and the resolvent bound of Part (1) of Definition 7.1. The trace bound on u is obtained in the same way.Proof of Lemma 7.4. Let ψ ∈ C ∞ (R) be so that ψ = 0 in B(0, 4) and ψ = 1 outside B(0, 5), and ψ ∈ C ∞ (R 2 ) be so that ψ = 0 in B(0, 2) and ψ = 1 outside B(0, 3). By definition of the Dirichlet boundary measure, it suffices to show thatψ( D x )χγ Γ i, ( D x1 + ι)u → 0. for any χ ∈ C ∞ c (R), such that χ = 1 near {d l } × [0, h] and (say) supp χ ⊂ {d l } × [-4 , h + 4 ]. Let χ ∈ C ∞ c (R), such that χ = 1 near supp χ and supp χ ⊂ {d l } × [-2 , h + 2 ].By pseudo-locality of the semiclassical pseudo-differential calculus (e.g., together with a trace estimate and Part 1 of Definition 7.1), it suffices to show thatχψ( D x )χγ Γi, ( D x1 + ι)u → 0.Extend χ and χ as elements of C ∞ c (R 2 ), so that χ = 1 on supp χ, χ = 1 near Γ i , and supp χ ⊂ D . As χψ( D x )χ is a tangential operator, it commutes with the trace and it suffices to show thatχψ( D x )χ( D x1 + ι)u L 2 (Γi, ) → 0.Now, by a trace estimate,χψ( D x )χ( D x1 + ι)u L 2 (Γi, ) -1 2 ψ( D x )χ( D x1 + ι)u H 1 . Decompose ψ( D x )χ( D x1 + ι)u = ψ( D x ) ψ( D x )χ( D x1 + ι)u + ψ( D x )(1ψ( D x ))χ( D x1 + ι)u.By Part (2) of Lemma 7.6,-1 2 ψ( D x ) ψ( D x )χ( D x1 + ι)u H 1 → 0.On the other hand, as supp(1-ψ(ξ))∩supp ψ(ξ ) = ∅, we haveψ( D x )(1-ψ( D x )) = O( ∞ ) Ψ -∞ ,and the result follows using the bound in Part (1) of Definition 7.1.Proof of Lemma 7.5. We show the result for ν i d ; the proofs for ν i d and ν i j are the same. Shrinking D + if necessary, let D + ⊃ D + be such that Definition 1.4 holds with D + replaced by D + , and let χ ∈ C ∞ c be such that supp χ ⊂ D + and χ = 1 near Γ + i . By Lemma 4.2, there exists c > 0 so that (7.23) WF (χv) ⊂ {|ξ 1 | ≥ c}.

  7.5) follows from the analogous property for the free resolvent; see [Bur02, Proposition 2.2]. From (7.5) and invariance of the wavefront set by the Hamiltonian flow in the interior (this follows, for example, from propagation of singularities, [DZ19, Theorem E.47]),

	On the other hand, from propagation of singularities up to the boundary (see, e.g., [MS82] and
	Remark 7.2 below), we have, for any τ > 0
	(7.7)	ρ / ∈ WF w and π

(7.6) WF w ⊂ (x, ξ) ∈ T * (R 2 \Θ), ∃t < 0, x + tξ ∈ ∂Θ .

for the models we are interested in. § §5- §6 prove the main results. §7 shows wellposedness results for our models.1.2. Definition of the impedance-to-impedance maps. Given cartesian coordinates (x 1 , x ) in R 2 and h, d l , d r > 0, we letD := (0, d l + d r ) × (0, h) Γ l := {0} × [0, h], Γ i := {d l } × [0, h], Γ r := {d l + d r } × [0, h], Γ t := (0, d l + d r ) × {h}, Γ b := (0, d l + d r ) × {0}, Γ s := Γ t ∪ Γ b ∪ Γ r , Γ s := Γ t ∪ Γ b ;

Acknowledgements

DL and EAS acknowledge support from EPSRC grant EP/R005591/1, thank Jeffrey Galkowski (University College London) for useful discussions, and thank Ivan Graham (University of Bath) for reading and commenting on an earlier draft of the paper.

Proof of result involving Model 1, i.e., Theorem 1.1

We fix an admissible solution operator S : g → u to (M1). The aim of this section is to prove the following theorem, which implies Theorem 1.1.

Theorem 5.1. Let ι ∈ {1, -1}. Then

(1) For any > 0, there exists 0 ( ) > 0 such that, for any -family of data (g( )) >0 , if u( ) = S( )g( ) is the associated solution to (M1), then for all 0 < ≤ h 0

(5.1)

(2) For all 0 < < θ max , there exists g( ) ∈ L 2 (Γ l ) so that, the associated solution u( ) = S( )g( ) to (M1) satisfies

Both in this section and in §6, we use the following consequence of the results of §7 (see §7.3).

Lemma 5.2. Let g( ) ∈ L 2 (Γ l ) be such that g L 2 ≤ C, and u( ) = S( )g( ) the associated solution to (M1) (resp. (M2)). Then, up to a subsequence, u( ) has a defect measure and boundary defect measures on Γ l and

5.1. The impedance trace on Γ i in terms of measures for Model 1. Proposition 5.3. Assume that u satisfies

and, for ι ∈ {1, -1}, let u := ( D x1 + ι)u. If u is outgoing near Γ s and admits defect measure and boundary defect measures, then u admits a Dirichlet boundary measure on Γ + i , and in T * Γ i we have

Proof. By Part (1) of Lemma 4.4, µ i out = 0. Thus, by Part (3) of Lemma 3.5,

(5.4)

On the other hand, for u := ( D x1 + 1)u, the definition of u and u in terms of u imply that u and u have Dirichlet boundary defect measures on Γ + i given by (this notation coincide with Definition 4.6 in the case of ν i d )

(where we recall that s = -x 1 on Γ i ). Hence, by (5.4), in H

and thus, in H

(5.5)

On the other hand, by Part (1) of Lemma 4.7 and Part (1) of Lemma 4.4 again, µ i out = 0 and thus, by Part (3 of Lemma 4.7, (5.6)

By Lemma 7.5, ν i d is supported in H; thus the combination of (5.5) and (5.6) implies that

Proof. By the definition of u in terms of u and the definition of defect measures, u has a Dirichlet boundary measure in Γ + i and (6.4)

(exactly the same as at the beginning of the proof of Proposition 5.3). Hence,

By Lemma 7.5, ν i d , ν i j , and ν i n are all supported in H, and thus (6.5)

We now seek to relate µ i out/in to µ i out/in using the relationship between µ and µ and Lemma 3.4.

2 µ r in , and thus µ r out ( √ r = 1) = 0. Therefore, by the interpretation of boundary measures given by Lemma 3.4 that {ξ 1 = -1} ∩ supp µ = ∅. Hence supp µ ⊂ supp µ, from which, by Lemma 3.4 again, supp µ i in/out ⊂ supp µ i in/out . Therefore, from (6.5) we get (6.6)

Finally, observe that, by Part (1) of Lemma 4.7 and propagation of defect measures given by Lemma 4.5 together with Lemma 4.4, Parts (2c) and (2a) with (x, y) = (i, l), (6.7)

Then (6.7) combined with (6.6) gives (6.1). We now assume that (6.2) holds and we show (6.3). The previous support arguments show that (6.8)

In other words

Therefore, by Lemma 3.5, (6.9)

and

(6.10)

In addition, observe that, by the definition of u in terms of u and the definition of defect measures, (6.11)

From (6.4) and (6.11) together with (6.9) we get

Therefore, as in the same way

which gives, by (6.7), (6.12)

In the same way, using (6.4) and (6.11) together with (6.10), then (6.7), we obtain (6.13)

It follows by (6.30), (6.31), (6.32) and (6.33) combined with (6.29) that

hence (6.34) holds for all k < ∞. It remains to show that µ l out ( B ∞ 0 ) = 0. Observe that such a set satisfies B ∞ 0 ⊂ { √ r = 1}. Hence, using (6.33) and (6.30)

)) = 0, from which, by (6.31) combined with (6.29)

out ( B ∞ 0 ) = 0, which ends the proof.

We can now conclude.

Proof of Theorem 1.3, Part (2). Let σ ∈ {+, -} N and λ > 0 and assume that the claim fails. Then, for any A > 0, there exists n ≥ A, > 0, g ∈ L 2 (Γ l ) with g L 2 = 1 and → 0 so that for all ,

We show that, for any n big enough, we have, up to a subsequence in , as → ∞ (6.35)

which gives a contradiction and completes the proof. We denote (u n ) n≥1 the cascade of solutions of (M2) associated with σ and f by S(h, d l ± , d r ± ), that is so that

where we denoted ± σ(n) := σ(n), and

In addition, denote

As f is uniformly bounded in L 2 , by Lemma 5.2, u 1 admit defect measures and boundary defect

λ , it follows by Proposition 6.5 that, up to a subsequence u 1 admit Dirichlet boundary defect measure on Γ i , satisfying

In addition, observe that the existence of ν 1,i d implies that, up to a subsequence, γ Γi(d σ(1) ) ( D x1 ± σ(1) 1)u 1 is locally uniformly bounded in L 2 , hence we can iterate the argument. We obtain that, for any n, up to a subsequence in , u n admits Dirichlet boundary defect measure on Γ i , satisfying

7.3. Interior trace estimates. The purpose of this paragraph is to show the three following lemmas about the trace of the solution on the interior surface Γ i .

Lemma 7.3. Let S be an -admissible solution operator to (M1), resp. (M2), and 0 ≤ < . Then, there exists 0 > 0 and C > 0 such that if 0 < ≤ 0 then, given g ∈ L 2 (Γ l ), any solution u = Sg of (M1), resp. (M2) satisfies,

Lemma 7.4. Let S be an -admissible solution operator to (M1), resp. (M2), ι ∈ {-1, 1}, g ∈ L 2 (Γ l ) be such that g L 2 ≤ C, u = Sg and ν i d = ν i d -2ι ν i j + ν i n a Dirichlet boundary measure of u := ( D x1 u + ι)u on Γ i, for some 0 < < . Then, (7.21)

). Lemma 7.5. If v is outgoing near Γ s , and admits defect measures ν i d , ν i n , ν i j on Γ + i , then they are supported in H.

In particular, Lemma 5.2 is a consequence of Lemma 7.3 and 7.4 together with Part (1) of Definition 7.1.

Proof of Lemma 5.2. Existence of the defect measure and boundary measures is a direct consequence Lemma 7.3, Parts (1) and (2) of Definition 7.1, and Theorem 3.2. The relationship (7.21) follows from Lemma 7.4 using the fact that, given an -admissible solution operator, D + can be taken such that Γ + i Γ i, ; indeed, for 0 < < so that Γ + i Γ i, , the boundary measures of u on Γ + i and of u on Γ i, coincide on T * Γ i . We need the following result, quantifying in the interior of the domain the fact that an Helmholtz solution is oscillating at frequency ∼ -1 . Lemma 7.6. Let U ⊂ R d be an open set and v ∈ L 2 loc (U ) be so that (-2 ∆ -1)v = 0 in U . Then, for any χ ∈ C ∞ c (U ), and any χ ∈ C ∞ c (U ) so that χ = 1 on supp χ, the following holds.

(1) For any m ≥ 0 there exists C m > 0 such that

(2) For any bounded function ψ ∈ C ∞ (R d ) so that ψ = 0 in B(0, 2) and any N ≥ 1 there exists C N > 0 such that

U ) be so that χ 1 = 1 on supp χ and χ = 1 on supp χ 1 . Observe that χ 1 (-2 ∆ -1) is semiclassically elliptic on WF ψ( D x )χ, hence, by the elliptic parametrix construction, there exists E ∈ Ψ -2 so that

Applying χ to the right of the above, we get

and hence (2) follows applying this identity to v1 U . Let now ψ 1 ∈ C ∞ (R d ) be such that 1ψ 1 is compactly supported and ψ 1 = 0 on B(0, 2). Since (1

Together with (2), this gives (1).

Proof of Lemma 7.3. By [GLS21a, Lemma 3.7], if Γ is a smooth closed curve and A ∈ Ψ m (R 2 ) is such that WF A ∩ S * Γ = ∅, there exists C > 0 such that

Let , δ > 0 be so that 0 < < < + 4δ < . Next, let Γ ⊂ [ d l 2 , d l +dr 2 ] × [--4δ, h + + 4δ] be a smooth closed curve so that and Γ ∩ {d l } × R = {d l } × [--3δ, h + + 3δ]. Next, let χ ∈ C ∞ c (R 2 ) be so that supp χ ∩ Γ = {d l } × [-δ, h + + δ] and χ = 1 near Γ i, ; and χ ∈ C ∞ c (R 2 ) be so that χ = 1 on supp χ and supp χ ∩ Γ = {d l } × [--2δ, h + + 2δ]. Applying