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Resolvent expansions and continuity of the scattering matrix at embedded thresholds: the case of quantum waveguides

We present an inversion formula which can be used to obtain resolvent expansions near embedded thresholds. As an application, we prove for a class of quantum waveguides the absence of accumulation of eigenvalues and the continuity of the scattering matrix at all thresholds.


Introduction
During the recent years, there has been an increasing interest in resolvent expansions near thresholds and their various applications. These developments were partially initiated by the paper of A. Jensen and G. Nenciu [7] in which a general framework for asymptotic expansions is presented and then applied to potential scattering in dimension 1 and 2. The key point of that paper is an inversion formula which provides an efficient iterative method for inverting a family of operators A(z) as z → 0 even if ker A(0) = {0}. Corrections or improvements of this inversion formula can be found in [2,Lemma 4], [5,Prop. 3.2] and [8,Prop. 1]. However, in all these papers either it is assumed that A(0) is self-adjoint, or the construction relies on a Riesz projection which is not always convenient to deal with. These features are harmless in these works, since the threshold considered always lies at the endpoints of the spectrum of the underlying operator. However, once dealing with embedded thresholds, these features turn out to be critical (see the comment at the end of Section 2.2).
Our aim in the present paper is thus twofold. On the one hand, we revisit the mentioned inversion formula, and on the other hand we show how its revised version can be used for proving the continuity of a scattering matrix at embedded thresholds. The abstract part of our results is presented in Section 2, and consists first in a reformulation of the inversion formula which does not require that A(0) is self-adjoint or that the projection is a Riesz projection (see Proposition 2.1). We then discuss two natural choices for the projection : either the Riesz projection defined in terms of the resolvent of A(0) if 0 is an isolated point in the spectrum of A(0), or the orthogonal projection on ker A(0) if A(0) has a non-negative imaginary part. If both conditions hold, we also discuss the relations between these two projections, and provide sufficient conditions for their equality. This situation often takes place in applications even without the assumption that A(0) is self-adjoint (see Corollary 2.8).
In the second part of the paper (Section 3), we present an application of our abstract results to scattering theory for quantum waveguides. Quantum waveguides provide a particularly good model of study since their Hamiltonians possess an infinite number of embedded thresholds (with a change of multiplicity at each threshold) but give rise to a simple scattering theory taking place in a one-Hilbert space setting. We refer to [11] for basic results and earlier references on the spectral and scattering theory for quantum waveguides.
For a straight quantum waveguide with a compactly supported potential V , we derive an asymptotic expansion of the resolvent in a neighbourhood of each embedded threshold. More precisely, if the potential is written as V = v uv with v non-negative and u unitary and self-adjoint, and if H 0 is the Dirichlet Laplacian for the waveguide, then we give an expansion of the operator u + v (H 0 − z) −1 v −1 as z converges to any threshold z 0 (see Proposition 3.2). Note that the operator v (H 0 − z 0 ) −1 v (once properly defined) has a non-trivial imaginary part. This fact automatically prevents the use of any approach assuming the self-adjointness of A(0), as mentioned above. We then deduce two consequences of this asymptotic expansion. First, we prove in Corollary 3.3 that the possible point spectrum of the operator H := H 0 + V does not accumulate at thresholds. Since the thresholds are the only possible accumulation points for such a model, we thus rule out this possibility. Second, we characterize for all scattering channels corresponding to the transverse modes of the waveguide the behavior of the scattering matrix for the pair {H 0 , H} at embedded thresholds. More precisely, we show that the scattering matrix is continuous at the thresholds if the channels we consider are already open, and that the scattering matrix has a limit from the right at the thresholds if a channel precisely opens at these thresholds (see Proposition 3.8 for a precise formulation of this result). Up to our knowledge, these types of results are completely new since the analysis of the behavior of a scattering matrix at embedded thresholds has apparently never been performed. We also show the continuity of the scattering matrix at embedded eigenvalues which are not located at thresholds. But in this case, similar results were already known for other models, see for example [4,Prop. 10] or [12,Prop. 6.7.11] (see also [3] where propagation estimates at embedded thresholds are obtained for a Schrödinger operator with time periodic potential).
As a final comment, we stress that we fully describe all possible behaviors at thresholds since we do not assume any condition on the absence of bound states or resonances at thresholds. Based on the expressions obtained in this paper, a Levinson's type theorem for quantum waveguides could certainly be derived, and deserves further investigations.
Acknowledgements. The authors thank A. Jensen for useful discussions.

Inversion formula
In this section, we adapt the inversion formula [8,Prop. 1] to the case of an arbitrary projection, and then discuss two possible choices for this projection. The symbol H stands for an arbitrary Hilbert space with norm · and scalar product · , · , and B(H) denotes the algebra of bounded operators on H with norm also denoted by · .
with A 0 ∈ B(H) and A 1 (z) uniformly bounded as z → 0. Let also S ∈ B(H) be a projection such that : Then, for |z| > 0 small enough the operator B(z) : SH → SH defined by

and in this case one has
Proof. For z ∈ O with |z| > 0 small enough, one has the equality So, the condition (ii) implies the second equality in (2.1). The second part of the claim is a direct application of the inversion formula [7, Lemma 2.1].
The choice of the projection S plays an important role in the previous proposition. For example, if 0 is an isolated point in the spectrum σ(A 0 ) of A 0 , a natural candidate for S is the Riesz projection associated with this value, which is the choice made in [2,7,8]. Another natural candidate is the orthogonal projection on the kernel of A 0 . However, for both choices additional conditions are necessary in order to verify conditions (i) and (ii). Below, we first discuss the case of the Riesz projection and then the case of the orthogonal projection.

Riesz projection
In this section, we assume that 0 is an isolated point in σ(A 0 ) and write S r for the corresponding Riesz projection. In that case, A 0 S r = S r A 0 = S r A 0 S r and A 0 + S r is invertible with bounded inverse (see [9,Chap. III.6.4]). The condition (ii) above, namely S r (A 0 + S r ) −1 S r = S r , is more complicated to check. However, if one assumes that A 0 S r = 0, or the stronger condition that A 0 is self-adjoint, then the equalities S r (A 0 + S r ) −1 = S r = (A 0 + S r ) −1 S r hold, and thus condition (ii) is satisfied (note that in that case a small simplification takes place on the r.h.s. of (2.1)). However, the condition A 0 S r = 0 does not always hold since A 0 S r is in general only quasi-nilpotent [9, Sec. III.6.5]. Fortunately, the condition A 0 S r = 0 holds if A 0 has a particular form, as shown in the following lemma (which is an extension of [8,Prop. 2]). or equivalently Im(J) ≥ 0 in S r H. Since J is quasi-nilpotent [9, Eq. (III.6.28)] and trace-class, and since quasi-nilpotent trace-class operators have trace 0 [10, p. 32], it follows that 0 = Tr(J) = Tr Re(J) + i Tr Im(J) .
This equality together with the inequality Im(J) ≥ 0 imply that Im(J) = 0. Thus, J is self-adjoint and quasi-nilpotent, which means that J = 0.
We now list a series of consequences of the previous result. Proof. The inclusion S r H ⊂ ker(A 0 ) follows from the equality A 0 S r = 0, and the inclusion S r H ⊃ ker(A 0 ) is standard.
We finally present a simple result which holds under the assumptions of Lemma 2.2, but can be proved in a slightly more general context. The norms and scalar products of the different Hilbert spaces are written with the same symbols.
Lemma 2.5. Let G be an auxiliary Hilbert space, take Z n ∈ B(H, G), and assume that the sum n Z * n Z n is weakly convergent. Let also A 0 = X + i n Z * n Z n , with X a bounded self-adjoint operator in H, and suppose that S is a projection satisfying A 0 S = 0 and SA 0 = 0. Then, Z m S = 0 and SZ * m = 0 for each m.

Orthogonal projection on the kernel
In this section, we assume from the beginning that A 0 = X + i Y , with X, Y bounded self-adjoint operators and Y ≥ 0. In that case, one has ker(A 0 ) = ker(X)∩ker(Y ) = ker(A * 0 ). Also, if S o denotes the orthogonal projection on ker(A 0 ), the relations Thus, if one shows that A 0 + S o is invertible with bounded inverse, then the conditions (i) and (ii) of Proposition 2.1 would follow. So, we concentrate in the sequel on this invertibility condition.
Since A 0 is reduced by the orthogonal decomposition , the only remaining question concerns the boundedness of the inverse A −1 0 on Ran(A 0 ). In the following two lemmas, we exhibit conditions under which this question can be answered affirmatively.
Lemma 2.6. Assume that A 0 = X +i Y , with X, Y bounded self-adjoint operators and Y ≥ 0, and suppose that 0 is an isolated point in σ(A 0 ). Let S r denote the corresponding Riesz projection, and assume that S r A 0 S r is a trace-class operator. Then, A 0 is invertible in ker(A 0 ) ⊥ with bounded inverse if and only if S r is an orthogonal projection.
Before giving the proof, we recall that if S r is an orthogonal projection, then it automatically follows from Corollary 2.4 that S r = S o .
Proof. Sufficient condition : Assume that S r is an orthogonal projection (and thus equal to S o ). Since A 0 is invertible in S ⊥ r H with bounded inverse by [9, Thm. III. 6.17], one infers that A 0 is invertible in Necessary condition : Suppose by absurd that S r is not an orthogonal projection, or more precisely that S ⊥ r H = S ⊥ o H (since we already know that S r H = ker(A 0 ) = S o H by Corollary 2.4). Then, if there exists ϕ ∈ S ⊥ r H \ {0} with ϕ ∈ S ⊥ o H, one has S o ϕ = 0 and S ⊥ o ϕ = 0, and for any z ∈ C \ {0} with |z| small enough we know from [9, Thm. III. 6.17] that the l.h.s. has a limit in H as z → 0. But since S o ϕ ∈ ker(A 0 ), the first term on the r.h.s. does not have a limit as z → 0. Therefore, the second term on the r.h.s. neither has a limit as z → 0, and thus the operator In this case, the second term on the r.h.s. does have a limit in H as z → 0, but the first term on the r.h.s. does not. Therefore, the l.h.s. does not have a limit in H as z → 0, and thus the operator Under the assumptions of Lemma 2.7, the value 0 is an isolated point in σ(A 0 ). Thus, the Riesz projection S r is well defined, and one obtains the following by combining the two previous lemmas : Proof. We know from Lemma 2.7 that A 0 is invertible in ker(A 0 ) ⊥ with bounded inverse. Thus, it follows from Lemma 2.6 that S r = S o and that the conditions (i) and (ii) of Proposition 2.1 are verified for S = S r = S o if S r A 0 S r is a trace-class operator. But, the operator S r A 0 S r is clearly trace-class if A 0 is a finite-rank operator. On the other hand, if A 0 = U + K with U unitary and K compact, then the isolated eigenvalue 0 is of finite multiplicity, S r H is finite-dimensional [9, Remark III.6.23], and S r A 0 S r is also trace-class.
We close this section with a comment on the usefulness of Corollary 2.8 for the iterative procedure of the next section. If we use a Riesz projection S r without knowing that it is orthogonal, this is harmless at the first step of the iteration (as illustrated in [8]), but this becomes more and more annoying at each step of the iteration. Indeed, conjugation by Riesz projections does not preserve positivity, and thus any argument based on positivity can hardly be invoked. Therefore, Corollary 2.8 leads to various simplifications in the iterative procedure since it provides conditions guaranteeing that S r is orthogonal.

Quantum waveguides
We introduce in this section the model of quantum waveguide we use and recall some of its basics properties. Much of the material is borrowed from [11] to which we refer for further information.
We consider a bounded open connected set Σ ⊂ R d −1 with d ≥ 2, and let −∆ Σ D be the Dirichlet Laplacian on Σ acting in L 2 (Σ). This operator has a purely discrete spectrum τ := {λ n } n≥1 consisting in eigenvalues λ 1 ≤ λ 2 ≤ · · · repeated according to multiplicity. The corresponding set of eigenvectors is denoted by {f n } n≥1 and the corresponding set of one-dimensional orthogonal projections is denoted by {P n } n≥1 . Sometimes, we omit for simplicity to stress that n ≥ 1.
(a) If s > 1/2, then the limit and is independent of the sequence ζ ′ → ζ. Moreover, the limit is a Hilbert-Schmidt operator with Hilbert-Schmidt norm where the constant may depend on ε but not on λ and ζ.
Proof. The first claim follows from (3.2). For the second one, one has to compute the integral kernel of Q −s R 0 (λ + ζ) − R 0 (λ) Q −s , taking into account the following equalities with y = |x − x ′ | and x, x ′ ∈ R : is measurable with bounded support. We impose the boundedness of the support for simplicity, but we note that our results would also hold for potentials V decaying sufficiently fast at infinity (see for example the seminal papers [6,7] for precise conditions on the decay of V at infinity). Following the standard idea of decomposing the perturbation into factors, we define the functions v : Then, the operator u + v R 0 (z) v has a bounded inverse in H for each z ∈ C\ R and the resolvent equation may be written as Since the following equality holds: deriving expansions in z for the resolvent (H − z) −1 amounts to deriving expansions in z for the operator u + v R 0 (z)v −1 , as we shall do in the section.

Asymptotic expansion at embedded thresholds or eigenvalues
We derive in this section an asymptotic expansion in z for the operator u +v R 0 (z)v −1 . As a by-product, we show the absence of accumulation of eigenvalues of H. For this, we first adapt a convention of [7] by considering values z = λ − κ 2 with κ belonging to the sets Then, the main result of this section reads as follows : Proof. For each λ ∈ R, ε > 0 and κ ∈ O(ε), one has Im(λ − κ 2 ) = 0. Thus, (3.3) implies that the operator u + v R 0 (λ − κ 2 ) v −1 belongs to B H) and is continuous in κ ∈ O(ε). For the other claims, we distinguish the cases λ ∈ τ and λ ∈ σ p (H) \ τ , treating first the case λ ∈ τ . All the operators defined below depend on the choice of λ, but for simplicity we do not always mention these dependencies.
(i) Assume that λ ∈ τ , take ε > 0, set N := {n ≥ 1 | λ n = λ}, and write P := n∈N P n for the corresponding orthogonal projection (of dimension greater or equal to 1). Then, (3.1) implies for κ ∈ O(ε) that (3.2)) implies that the continuous function Also, Lemma 3.1(a) implies the existence and the unicity in B(H) of the limits Therefore, one has for κ ∈ O(ε) that with the operators Our goal thus reduces to derive an asymptotic expansion for I 0 (κ) −1 as κ → 0. Since I 0 (0) = N 0 is a finite-rank operator, 0 is not a limit point of σ(N 0 ). Also, N 0 is self-adjoint, therefore the orthogonal projection S 0 on ker(N 0 ) is equal to the Riesz projection of N 0 associated with the value 0. We can thus apply Proposition 2.1, and obtain for κ ∈ O(ε) with ε > 0 small enough that the operator I 1 (κ) : S 0 H → S 0 H defined by is uniformly bounded as κ → 0. Furthermore, I 1 (κ) is invertible in S 0 H with bounded inverse satisfying the equation It follows that for κ ∈ O(ε) with ε > 0 small enough, one has with the first term vanishing as κ → 0.
To describe the second term of u + v R 0 (λ − κ 2 ) v −1 as κ → 0, we recall the equality I 0 (0) + S 0 −1 S 0 = S 0 , which (together with (3.5)) implies for κ ∈ O(ε) with ε > 0 small enough that with and Then, we observe that [7, Lemma 5.1(i)] implies that N 2 (κ) admits a finite limit as κ → 0. Also, we note that Lemma 3.1(b) implies that the second term in (3.8) vanishes as κ → 0. Therefore, M 2 (κ) B(S0H) is uniformly bounded as κ → 0. Now, we recall that with u unitary and self-adjoint, N 1 (0) self-adjoint and compact, and with the last term compact with non-negative imaginary part (the last property holds for weighted resolvents on the real axis). So, since S 0 is an orthogonal projection with finite-dimensional kernel, the operator I 1 (0) = S 0 M 1 (0)S 0 acting in the Hilbert space S 0 H can also be written as the sum of a unitary and self-adjoint operator, a self-adjoint and compact operator, and a compact operator with non-negative imaginary part. Thus, Corollary 2.8 applies with S 1 the finite-rank orthogonal projection on ker I 1 (0) , and the iterative procedure of Section 2 can be applied to I 1 (κ) as it was done for I 0 (κ). Thus, for κ ∈ O(ε) with ε > 0 small enough, the operator I 2 (κ) : S 1 H → S 1 H defined by is uniformly bounded as κ → 0. Furthermore, I 2 (κ) is invertible in S 1 H with bounded inverse satisfying the equation This expression for I 1 (κ) −1 can now be inserted in (3.6) in order to get for κ ∈ O(ε) with ε > 0 small enough with the first two terms bounded as κ → 0. Let us concentrate on the last term and check once more that the assumptions of Proposition 2.1 are satisfied. For that purpose, we recall that I 1 (0) + S 1 −1 S 1 = S 1 , and observe that for κ ∈ O(ε) with ε > 0 small enough The inclusion M 3 (κ) ∈ O(1) follows from standard estimates and from the fact that 1 κ N 2 (κ) − N 2 (0) admits a finite limit as κ → 0 (see [7, Lemma 5.1(i)]). Note also that the kernel of N 2 (0) is given by Now, as already observed, one has M 1 (0) = X + i Z * Z, with X, Z bounded self-adjoint operators in H. Therefore it follows that I 1 (0) = S 0 M 1 (0)S 0 = S 0 XS 0 + i (ZS 0 ) * (ZS 0 ), and one infers from Corollary 2.5 that ZS 0 S 1 = 0 and S 1 S 0 Z * = 0. Since S 1 S 0 = S 1 = S 0 S 1 , it follows that ZS 1 = 0, that S 1 Z * = 0, and also that So, this operator is self-adjoint, and thus one infers from (3.11) and (3.12) that I 2 (0) = S 1 M 2 (0)S 1 is the sum of two bounded self-adjoint operators in S 1 H. Since S 1 H is finite-dimensional, 0 is not a limit point of the spectrum of I 2 (0). So, the orthogonal projection S 2 on ker I 2 (0) is a finite-rank operator, and Proposition 2.1 applies to I 2 (0) + κ M 3 (κ). Thus, for κ ∈ O(ε) with ε > 0 small enough, the operator I 3 (κ) : S 2 H → S 2 H defined by is uniformly bounded as κ → 0. Furthermore, I 3 (κ) is invertible in S 2 H with bounded inverse satisfying the equation This expression for I 2 (κ) −1 can now be inserted in (3.9) in order to get for κ ∈ O(ε) with ε > 0 small enough Fortunately, the iterative procedure stops here. The argument is based on the relation (3.14) So, if we replace u + v R 0 (λ − κ 2 ) v −1 by the expression (3.13) and if we take into account that all factors of the form I j (κ) + S j −1 have a finite limit as κ → 0, we infer from (3.14) that lim sup Therefore, it only remains to show that this relation holds not just for κ = ε 2 (1 − i ) but for all κ ∈ O(ε). For that purpose, we consider I 3 (κ) once again, and note that (3.16) The precise form of M 3 (0) can be computed explicitly, but is irrelevant. Now, since I 3 (0) acts in a finitedimensional space, 0 is an isolated eigenvalue of I 3 (0) if 0 ∈ σ I 3 (0) , in which case we write S 3 for the corresponding Riesz projection. Then, the operator I 3 (0) + S 3 is invertible with bounded inverse, and (3.16) implies that I 3 (κ) + S 3 is also invertible with bounded inverse for κ ∈ O(ε) with ε > 0 small enough. In addition, one has . By the inversion formula given in with bounded inverse and that the following equalities hold This implies that (3.15) holds for some κ ∈ O(ε) if and only if the operator (ii) Assume now that λ ∈ σ p (H) \ τ , take ε > 0, let κ ∈ O(ε), and set J 0 (κ) := T 0 + κ 2 T 1 (κ) with Then, one infers from Lemma 3.1(b) that T 1 (κ) B(H) is uniformly bounded as κ → 0. Also, the assumptions of Corollary 2.8 hold for the operator T 0 , the Riesz projection S associated with the value 0 ∈ σ(T 0 ) is an orthogonal projection, and Proposition 2.1 applies for J 0 (κ). It follows that for κ ∈ O(ε) with ε > 0 small enough, the operator J 1 (κ) : SH → SH defined by is uniformly bounded as κ → 0. Furthermore, J 1 (κ) is invertible in SH with bounded inverse satisfying the equation It follows that for κ ∈ O(ε) with ε > 0 small enough one has Fortunately, the iterative procedure already stops here. Indeed, the argument is similar to the one presented above once we observe that Therefore, (3.18) implies that the function We now give a result on the possible embedded eigenvalues. Since it is already known that the eigenvalues of H in σ(H) \ τ are of finite multiplicity and can accumulate at points of τ only (see [11,Thm. 3.4(b)]), we show that such accumulations do not take place : Proof. To show the absence of local accumulation of eigenvalues, suppose by absurd that there is an accumulation of eigenvalues at some point λ ∈ τ . Then, the validity of the expansion (3.17) at the point λ contradicts the validity of the expansion (3.19) which would take place at each of these eigenvalues. Thus, there is no accumulation of eigenvalues at points of τ , and the claim is proved.
We end up this section with some auxiliary results which will be useful later on. All notations and definitions are borrowed from the proof of Proposition 3.2. The only change is that we extend by 0 the operators defined originally on subspaces of H to get operators defined on all of H. Lemma 3.4. Take 2 ≥ j ≥ k ≥ 0 and κ ∈ O(ε) with ε > 0 small enough. Then, one has in B(H) Proof. The fact that S j is the orthogonal projection on the kernel of I j (0) and the relations S k S j = S j = S j S k imply that [S k , S j ] = 0 and [I k (0), S j ] = 0. Thus, one has the equalities which implies the claim.
Lemma 3.5. Let λ ∈ τ and let G be an auxiliary Hilbert space.
(b) For each n / ∈ N and B n ∈ B(H, G) such that B * n B n = Im v P n ⊗R 0 (λ−λ n ) v , one has S 1 B * n = 0 and B n S 1 = 0.
Proof. The first claim follows from the fact that S 0 is the orthogonal projection on ker v (P ⊗ 1) v . The second claim follows from Lemma 2.5 applied with Z n = B n S 0 and For what follows, we recall that Q is the multiplication operator by the variable in L 2 (R).

Scattering theory and spectral representation
In this section, we recall some basics on the scattering theory for the pair {H 0 , H} and on the spectral decomposition for H 0 . As before, we assume that V ∈ L ∞ (Ω; R) has bounded support.
Under this assumption, it is a well-known that the wave operators exist and are complete (see [11,Cor. 3.5(b)]). As a consequence, the scattering operator S := W * + W − is a unitary operator in H which commutes with H 0 , and thus S is decomposable in the spectral representation of H 0 . So, in order to proceed, we start by recalling the spectral representation of H 0 . For that purpose, we define for each λ ∈ [λ 1 , ∞) the finite set N(λ) := n ≥ 1 | λ n ≤ λ and the finite-dimensional space with λ n and P n as in Section 3. Note that H (λ) is naturally embedded in H (∞) := n≥1 P n L 2 (Σ) ⊕ P n L 2 (Σ) . Now, for any ξ ∈ R, we let γ(ξ) : S (R) → C be the trace operator given by γ(ξ)f = f (ξ), with S (R) the Schwartz space on R. Also, we define for each λ ∈ [λ 1 , ∞) \ τ the operator T (λ) : Some regularity properties of the map λ → T (λ) have been established in [11,Lemma 2.4], and additional properties are derived below for the related map λ → F 0 (λ) which we now define.
The operator F 0 (λ; n, σ) : L 2 (Σ) ⊙ S (R) → P n L 2 (Σ) is defined on a slightly larger set of λ than the operator F 0 (λ) : L 2 (Σ) ⊙ S (R) → H (λ). Also, we define the sets , and we let L 2 s (R) be the domain of Q s , s ∈ R, endowed with the graph norm. Then, given λ ∈ τ ∪ σ p (H), we consider for each κ ∈ ∂O(ε) with ε > 0 small enough the asymptotic expansion in κ of the operator F 0 (λ − κ 2 ; n, σ). If λ n < λ, one has for κ ∈ ∂O(ε) with ε > 0 small enough Similarly, if s > 0 is big enough and if σ ∈ {+, −}, one has in B L 2 As a consequence, we have in B L 2 (Σ) ⊗ L 2 s (R); P n L 2 (Σ) On the other hand, if λ = λ n ∈ τ and −κ 2 > 0, then one obtains in B L 2 (Σ) ⊗ L 2 s (R), P n L 2 (Σ) with γ j (n) : L 2 (Σ) ⊗ L 2 s (R) → P n L 2 (Σ) the operator given by With these expansions at hand, we can start the study of the regularity properties of the scattering matrix at thresholds or at embedded eigenvalues. Before that, we just need to give a final auxiliary result. Recall that the orthogonal projections S 0 and S 1 have been introduced in the proof of Proposition 3.2.

Continuity of the scattering matrix
Since the scattering operator S commutes with H 0 , it follows from the spectral decomposition of H 0 that where S(λ), the scattering matrix at energy λ, is defined and is a unitary operator in H (λ) for almost every λ ∈ [λ 1 , ∞ and that the map is a k-times continuously differentiable, for any k ≥ 0.
Since the regularity of the map λ → S(λ) is already known when λ ∈ [λ 1 , ∞) \ {τ ∪ σ p (H)}, we now describe the behavior of S(λ) as λ approaches points of τ ∪ σ p (H). To do this, we decompose the scattering matrix S(λ) into a collection of channel scattering matrices corresponding to the transverse modes of the waveguide. Namely, for λ ∈ [λ 1 , ∞) \ {τ ∪ σ p (H)}, for n, n ′ ≥ 1 such that λ n < λ and λ n ′ < λ, and for σ, σ ′ ∈ {+, −}, we define the operators S(λ; n, σ, n ′ , σ ′ ) ∈ B P n ′ L 2 (Σ), P n L 2 (Σ) by with δ nσn ′ σ ′ := 1 if (n, σ) = (n ′ , σ ′ ), and δ nσn ′ σ ′ := 0 otherwise. We consider separately the continuity at thresholds and the continuity at embedded eigenvalues, starting with the thresholds. Note that for each λ ∈ τ , a channel can either be already open (in which case one has to show the existence and the equality of the limits from the right and from the left), or can open at the energy λ (in which case one has only to show the existence of the limit from the right).
Before giving the proof, we define for 2 ≥ j ≥ k ≥ 0 the operators We know from Lemma 3.4 that C j k (κ) ∈ O(κ), but the formulas (3.4), (3.7) and (3.10) imply in fact that C ′ j k (0) := lim κ→0 1 κ C j k (κ) exists in B(H). In other cases, we use the notation F (κ) ∈ O as (κ n ) for an operator F (κ) ∈ O(κ n ) such that lim κ→0 κ −n F (κ) exists in B(H). Finally, we note that (3.17) can be rewritten as The interest in this formulation is that the projections S j (which lead to simplifications in the proof) have been put into evidence at the beginning or at the end of each term.