Scattering by a periodic tube in R3: part i. The limiting absorption principle

Scattering of time-harmonic waves from periodic structures at some fixed real-valued wave number becomes analytically difficult whenever there arise surface waves: These non-zero solutions to the homogeneous scattering problem physically correspond to modes propagating along the periodic structure and clearly imply non-uniqueness of any solution to the scattering problem. In this paper, we consider a medium, described by a refractive index which is periodic along the axis of an infinite cylinder in and constant outside of the cylinder. We prove that there is a so-called limiting absorption solution to the associated scattering problem. By definition, such a solution is the limit of a sequence of unique solutions for artificial complex-valued wave numbers tending to the above-mentioned real-valued wave number. By the standard one-dimensional Floquet–Bloch transform and the introduction of the exterior Dirichlet–Neumann map we first reduce the scattering problem to a class of periodic problems in a bounded cell, depending on the wave number k and the Bloch parameter . We use a functional analytic singular perturbation result to study this problem in a neighborhood of a singular pair . This abstract result allows us to derive explicitly a representation for the limiting absorption solution as a sum of a decaying part (along the axis of the cylinder) and a finite sum of propagating modes.

Keywords: Helmholtz equation, limiting absorption principle, periodic wave guide (Some figures may appear in colour only in the online journal)

Introduction
Periodic non-absorbing surface structures allow surface waves that propagate along the structure without decaying. These waves do physically arise at certain exceptional values of the Bloch parameter, and mathematically they are eigenfunctions of a certain periodic eigenvalue problem involving the periodic structure. The corresponding eigenvalue determines the surface wave's frequency, and the surface wave itself is the periodically extended eigenfunction.
Since the eigenfunction is a non-zero solution to a corresponding periodic scattering problem from the periodic structure, the latter scattering problem cannot be uniquely solvable at any of these eigenfrequencies. For this reason, such frequencies are often excluded from the analysis (see, e.g. [4]) by proper assumptions on the refractive index. In the past decade, however, the study of surface waves, also known as resonant scattering, has attracted a lot of attention. For an overview we suggest the interesting paper [24] by S. Shipman. From the mathematical point of view the formulation of a correct radiation condition is challenging which ensures both, uniqueness and the existence of surface waves.
In this paper we extend the paper [15] to the scattering problem in R 3 by an infinite inhomogeneous cylinder involving a periodic (with respect to the axis of the cylinder) refractive index. We believe that the so-called limiting absorption principle is the natural approach for finding the physically correct solution of the scattering problem. By construction, this solution is, in a certain topology, limit of the unique solutions to a family of coercive problems with artificial complex-valued wave numbers.
This limiting absorption solution consists of two parts that we determine via the Floquet-Bloch transform: The first part belongs to H 1 in any cylinder of finite radius and the second part is made up of surface waves or propagative modes. This second part vanishes if no propagative mode exists. However, if such modes exist then the direction of propagation is determined through a finite-dimensional eigenvalue problem in the finite dimensional space of propagating modes. This paper seems to be a first instance of such a limiting absorption principle for a scattering problem by an infinite tube. There exist, however, several contributions concerning problems in R 3 which are periodic with respect to two or all three variables, see, e.g. [24]. Our problems serves as a simple model how tubes in R 3 are scattered by, e.g. point sources.
The structure of the spectrum, the limiting absorption principle, and the construction of radiation conditions for frequency scattering problems in free space, in closed waveguides, and in stratified media has a long history. We refer to [2, 3, 7-9, 13, 17, 22, 23, 25-27] for a few references. Further, in [10] (see also [14]) a limiting absorption principle for scattering in a closed waveguide has recently been shown that relies fundamentally on the Floquet-Bloch transform and has substantially motivated our first paper [15]. In [10], the authors decompose fields via the eigenfunctions of the generalized quasi-periodic Laplacian in the unit cell. This technique cannot be applied in our case-not even in the two-dimensional case-, as such decompositions cannot be directly transferred to structures that form open instead of closed waveguides. Our analysis is indeed rather different compared to the one in [10], and also compared to the independent study in [12].
Precisely, we consider the scattering of an incident field u inc by an infinite cylinder T R := B 2 (0, R) × R ⊂ R 3 where B 2 (0, R) ⊂ R 2 denotes the disc centered at the origin with radius R > 0. We assume that the index of refraction n ∈ L ∞ (R 3 ) is positive and 2π-periodic with respect to x 3 and equals to one for x ∈ T R and construct a (weak) limiting absorption solution u s ∈ H 2 loc (R 3 ) such that the total field u t = u inc + u s solves ∆u t +k 2 n u t = 0 in R 3 .
Here, k > 0 denotes the wave number. The limiting absorption principle leads to a special decomposition of the solution into a field u (1) which decays along the axis of the tube and a second field u (2) which consists of a finite combination of propagative modes. Outside of the cylinder T R the field u (1) is a solution of the exterior boundary value problem ∆u (1) +k 2 u (1) = −h for some h ∈ L 2 (R 3 \ T R ) with H 1/2 -boundary data on the boundary Γ R = ∂T R . This limiting absorption solution u (1) can be explicitly expressed by the (generalized) Fourier transform in terms of Hankel functions of the first kind.
Both parts, the decomposition of the field u into a decaying field u (1) along the axis and a propagating field u (2) and the particular form of u (1) outside of the cylinder allows the formulation of a radiation condition which we carry out in the second part of this paper. There, we will prove uniqueness under this radiation condition and also existence by a direct method; that is, without using the limiting absorption principle.
The methods we apply are all well-known and in principle simple enough to extend our analysis to more involved scattering problems in linear elasticity or electromagnetics. This, however, has to be done and is planed for the future.
To briefly comment on this paper's structure, the following section 2 discusses the scattering problem in more detail and transforms it into a family of periodic problems with the help of the Floquet-Bloch transform. We reduce the problems to a bounded cell by introducing the Dirichlet-Neumann operator for the exterior of the cylinder T R . In section 3 we prove the limiting absorption principle and exhibit the particular form of u (1) in the exterior of T R . Finally, in the appendix we prove several properties of Hankel functions with complex arguments and solve an exterior boundary value problem for the Helmholtz equation with the use of the Fourier transform.

Formulation of the scattering problem and the Floquet-Bloch transform
We begin by setting up some notations (see figure 1). Let k ∈ C with Rek > 0 and Imk 0 be the wave number, B N (y, R) = {x ∈ R N : |x − y| < R} the ball in R N with center y and radius R > 0, and T R = B 2 (0, R) × R ⊂ R 3 the tube (or infinite cylinder) in x 3 -direction with boundary Γ R := ∂T R = ∂B 2 (0, R) × R. Furthermore, we define the finite cylinder by C R := B 2 (0, R) × (0, 2π) ⊂ R 3 and C ∞ := R 2 × (0, 2π) ⊂ R 3 and the vertical part of the boundary by γ R := Γ R ∩ C R . We consider in the following the case that a point source at some point y ∈ R 3 is scattered by a tube T R0 ⊂ R 3 of radius R 0 which is filled by some medium with index of refraction n ∈ L ∞ (R 3 ) which is assumed to be 2π-periodic with respect to the variable x 3 and equals to one outside of T R0 . The incident field u inc is given by the fundamental solution Φ k of the Helmholtz equation in R 3 ; that is, for some fixed y ∈ R 3 . The scattering problem is to determine the total field u t ∈ H 1 loc R 3 \ {y} with ∆u t + k 2 n u t = 0 in R 3 \ {y}, (2) and such that the scattered field u s = u t − u inc is more regular than the incident field, in particular u s ∈ H 1 loc (R 3 ). In other words, we wish to determine the Green's function of the differential operator ∆ + k 2 n. The solution is not uniquely determined by (2) because some kind of radiation condition for the scattered field is required. It is the purpose of this paper to derive a correct form of a radiation condition from the limiting absorption principle. First, we transform this problem into an inhomogeneous equation in H 1 loc (R 3 ) with a source term of bounded support. Indeed, choose > 0 and a function χ ∈ C ∞ (R 3 ) with χ(x) = 0 for |x| /2 and χ(x) = 1 for |x| and set u(x) = u s (x) + χ(x − y) u inc (x). Then u coincides with u s for |x − y| /2 and coincides with u t for |x − y| . Setting χ y (x) = χ(x − y) we observe that u solves has support in the ball B 3 (y, ) ⊂ R 3 and depends analytically on k. From now on we treat f = f k ∈ L 2 (R 3 ) as an arbitrary function with compact support in the disc We enlarge the radius R 0 of the tube to include the support of the source function f . In the case of the scattering problem the scattered field is then given by u s = u − χ y u inc and the total field by u t = u + (1 − χ y )u inc . The solution of (3) is understood in the variational sense: for all ψ ∈ H 1 (R 3 ) with compact support.
By choosing ψ ∈ H 1 (R 3 ) in (4) with compact support in R 3 \ T R0 we note that u is a classical solution of the Helmholtz equation ∆u + k 2 u = 0 for x 2 1 + x 2 2 > R 2 0 . The solution u is therefore analytic in the exterior of the tube T R0 .
In the following we will consider the source problem (3) for arbitrary functions f ∈ L 2 (R 3 ) with compact support and make the following assumption on the data.

Assumption 2.2. Let k ∈ C with
Rek > 0 and Imk 0 and let n ∈ L ∞ (R 3 ) which is assumed to be 2π-periodic with respect to the variable x 3 and equals to one outside of T R0 . Furthermore, we assume that there exists n 0 > 0 with n(x) n 0 for all x ∈ R 3 . Finally, we assume that f ∈ L 2 (R 3 ) has compact support which is also contained in T R0 and depends analytically on k in the neighborhood of some k > 0.
As mentioned above, a further condition is needed to assure uniqueness. It is one of the main goals of this paper to develop a proper radiation condition for real wave numbers. For wave numbers with positive imaginary part we simply require that u ∈ H 1 (R 3 ). Theorem 2.3. Let assumption 2.2 hold. If Imk > 0 then there exists a unique variational solution u = u k ∈ H 1 (R 3 ) of problem (4).
Proof. This is a standard application of the Lax-Milgram theorem. □ We will show in the first part of this paper that the solution u = u k converges (in some topology specified later) to some solution uk of the Helmholtz equation ∆uk +k 2 n uk = −fk in R 3 when k tends to some real valued k > 0; that is, the limiting absorption principle holds.
We use the (periodic) Floquet-Bloch transform to reformulate the problem as a family of 2π-periodic problems. Recall that the periodic Floquet-Bloch transform F : This formula directly shows that for smooth functions f and fixed α the transformed function The inverse transform is given by where we extended h(·, α) to a 2π-periodic function in R. In view of our scattering problem, we apply the Floquet-Bloch transforms to the variable x 3 and consider (x 1 , x 2 ) as a parameter. We use the same symbol F for this extension. Then one can show that F is an isometry from L 2 (T R ) onto L 2 C R × (−1/2, 1/2) ; that is, Further, the restriction of F to H 1 (T R ) is an isomorphism from H 1 (T R ) onto the space [20, section 6]).
We transform (3) using the Floquet-Bloch transform where e (3) = (0, 0, 1) denotes the third coordinate unit vector. We note that f α depends analytically on α because the series reduces to a finite sum. We arrive at the problem to determine for every Here, H 1 per,loc (C ∞ ) is just the local space corresponding to H 1 per (C ∞ ); that is, where g ,m = 1 2π 2π 0 2π 0 g(ϕ, x 3 ) e −i[mϕ+ x3] dϕ dx 3 , , m ∈ Z, are the Fourier coefficients of g ∈ L 2 (γ R ) (see [5] (6). The solutions for Imk > 0 necessarily satisfy a Rayleigh expansion. From now on we fix R > R 0 .
for r R 1 , ϕ, x 3 ∈ (0, 2π). Here, H (1) m (z) denote the Hankel functions of the first kind and order m ∈ Z. The branch of the square root √ z for z ∈ C with Imz 0 is chosen such that Rez 0 and Imz 0. The series converges in H 1 (C R2 \ C R1 ) for every R 2 > R 1 .
(c) For every R 1 > R and k ∈ C with Imk 0 the operator S k,α : H (a) Existence and uniqueness of a solution ũ ∈ H 1 per (C ∞ \ C R ) follows again by the Lax-Milgram theorem. The uniqueness and the proof of part (b) imply that ũ has to be of the form (10). To show the exponential decay we set k = k 2 − ( + α) 2 and use part (a) of lemma A.2 which yields that, for any R 1 > R, where σ = min{Imk : ∈ Z} which is positive because Imk > 0. This yields the estimate because ,m∈Z b ,m < ∞. The estimate for the derivative is obtained by the same argument. (b), (c) Let Imk 0 and ũ given by (10). We define By lemma A.4 of the appendix there exists c > 0 such that This proves that the function given by (10) provides a solution of (9). The solution is also unique. Indeed, let g = 0 and ũ ∈ H 1 per,loc (C ∞ \ C R ) be a solution satisfying the Rayleigh expansion (8). The Fourier coefficients u ,m (r) = 1 2π 2π 0 2π 0ũ (r, ϕ, x 3 ) e −imϕ−i x3 dϕ dx 3 satisfy Bessel's differential equation and the initial condition u ,m (R) = 0. By the Rayleigh expansion, u ,m is given by and some a ,m ∈ C. By analyticity, u ,m (r) is given by this formula for all r > R. The initial condition yields a ,m = 0. □ Now we consider the source problem (6) also for the case of real wave numbers k =k > 0 and include the Rayleigh expansion.
per,loc (C ∞ ) as a solution of (6) for k =k > 0 which satisfies also the Rayleigh expansion (8) for k =k > 0; that is, for some R 1 > R and a ,m ∈ C.
As the example of a constant index n shows for given k > 0 there might exist certain values of α such that the Problem (P α ) does not have a unique solution.

We define the set
of cut-off values and note that A consists of one or two elements. We make the following assumption for the rest of the paper.
Assumption 2.8. The cut-off values α ∈ A are not exceptional values; that is, for α ∈ A the only solution of (11) and (12) The name 'propagating mode' is justified by the following.

Proof.
(a) Let φ ∈ H 1 per,loc (C ∞ ) be a non-trivial solution of (P α ) with f α = 0. Green's formula in C R1 and the Rayleigh expansion (12) yields and are real valued (see [1]). The left hand side and the second series on the right hand side are real valued. Furthermore, for s > 0 by the Wronskian relationship. This implies that a ,m = 0 for all ( , m) ∈ Z 2 with | + α| <k. Finally, we use part (a) of lemma A.2 which yields as in the proof of lemma 2.6 that, for any R 2 > R 1 , where σ = min{| + α| −k : | + α| >k, ∈ Z}. This proves part (a) because the series over |b ,m | converges. (b) This is clear from the definition. □ Next, we reduce the problem to a boundary problem in the bounded cell C R using the Dirichlet-Neumann operator.

Definition 2.10. Let
Rek > 0 and Imk 0. The periodic Dirichlet-Neumann operator Λ k,α : H where Here we cut the complex plane along the negative imaginary axis so that the square root is holomorphic in C \ (iR 0 ). Furthermore, are the Fourier coefficients of g ∈ H 1/2 per (γ R ). The operator is well defined and bounded by part (b) of lemma A.2.
We can even extend this operator Λ k,α to (k, α) ∈ C × C in a neighborhood of real values (k,α) ∈ R >0 × [−1/2, 1/2] ⊂ R × R. This and other properties are shown in the following theorem.

Theorem 2.11.
(a) The operators Λ k,α are well defined and bounded from H Then the mapping (k, α) →Λ k,α is well defined and strongly holomorphic from Here, ·, · denotes the dual form; that is, the extension of the L 2 -inner product to H

Proof.
(a) This follows directly from the estimate of part (b) of lemma A.2 of the appendix. (b) Boundedness of Λ k,α for every (k, α) ∈ P follows again by the estimate of lemma A.2 of the appendix. For the analyticity of the mapping (k, α) →Λ k,α it is sufficient to show that this mapping is weakly holomorphic; that is, the mapping The fact that every weakly holomorphic function is strongly holomorphic is shown, e.g. in Chapter 8 of [6] for operator valued functions of one complex variable. Since the proof uses only Cauchy's integral formula -which is valid also in C 2 -this property holds also for functions of two complex variables. The series converges uniformly by lemma A.2 which proves analyticity of this function. (c) Let Λ k,α g = ∂u/∂r| γR and Λ i,0 g = ∂v/∂r| γR where u and v solve

A Kirsch Inverse Problems 35 (2019) 104004
We define the shell S := {x ∈ R 3 : and vanishes on the boundary of S. Here, We notice that h ∈ L 2 (S) and the mapping g → h is bounded from H 1/2 per (γ R ) into L 2 (S) (by the continuous dependence of u and v on g). Standard regularity results for elliptic partial differential equations imply continuity of h → w from L 2 (S) into H 2 (S). This shows continuity of g → w from H 1/2 per (γ R ) into H 2 (S) and thus compactness as a mapping into H 1 (S). The trace theorem yields because v j and ∂v j /∂r decay exponentially by part (a) of lemma 2.6. This shows that Λ i,0 is selfadjoint. Furthermore, for g = g 1 = g 2 we observe that Λ i,0 is negative because Λ i,0 g, g = 0 holds only for v = 0 which implies g = 0.
We recall the bounded cell C R = B 2 (0, R) × (0, 2π) and formulate the source problem for ũ k,α ∈ H 1 per (C R ) as the variational equation (14) for all ψ ∈ H 1 per (C R ). The proof of the following lemma is simple and left to the reader. Lemma 2.12. Let k ∈ C with Rek > 0 and Imk 0.
(a) If ũ k,α ∈ H 1 per,loc (C ∞ ) is a solution of the scattering problem (6) and Rayleigh expansion (8) then the restriction ũ k,α | CR ∈ H 1 per (C R ) solves (14). (14) then the extensioñ with the operator S k,α introduced in lemma 2.6 is the solution of the scattering problem (6) and Rayleigh expansion (8).
We show that the equation (14) is of Fredholm type. Indeed, we first decompose defines an inner product in H 1 per (C R ) which is equivalent to the ordinary norm in H 1 per (C R ). Therefore, (14) is equivalent to We note that the source term f α in (14) can also depend on k (as in the original scattering problem (3)), and we write f k,α from now on. Let again k ∈ C with Rek > 0 and Imk 0. By the theorem of Riesz, the compact imbedding of H 1 per (C R ) in L 2 (C R ), and the compactness of Λ i,0 − Λ k,α there exists a compact opera- Then we can rewrite the variational equation (14) as an operator equation in the form The operator equation is well defined for all k ∈ C with Rek > 0 and Imk 0. For Imk > 0 we have uniqueness and existence by theorem 2.4.
For real values k =k > 0, however, we expect non-uniqueness at certain values of α that we called exceptional values (see definition 2.7) . In other words, we expect that for some α ∈ [−1/2, 1/2] there is an eigenvalue λ = 1 of the non-selfadjoint operator Kk ,α . We note that by lemma 2.12 the corresponding eigenfunctions are exactly the propagating modes of definition 2.7. Proof. Assume on the contrary that there exists an (infinite) sequence (α j ) j in [−1/2, 1/2] and a sequence (w j ) j in H 1 per (C R ) of corresponding normalized functions such that (I − Kk ,αj )w j = 0 for all j . Let again A = {α ∈ [−1/2, 1/2] : | + α| =k for some ∈ Z}. We can assume that the sequence belongs to one of the at most three intervals of [−1/2, 1/2] \ A, say to I = [−1/2, τ ) where | + τ | =k for some ∈ Z. By theorem 2.11 there exists an open set U such that I ⊂ U and the mapping α → Kk ,α is analytic from U into L(H 1 0,per (C R )). From [11, theorem 5.1] it follows that the equation (I − Kk ,α )w = 0 has the same number of linearly independent solutions at every parameter α ∈ I except for finitely many. Since for the infinite sequence α j this number is at least one, it has to be at least one for all α ∈ I except for finitely many. From the continuity of α →Kk ,α and the injectivity of Kk ,τ by assumption 2.8 the operators Kk ,α have to be injective for all α in a neighborhood of τ . This is a contradiction. The other cases of I are treated in the same way. □

The limiting absorption principle
In this section we fix an arbitrary wave number k ∈ R >0 and investigate the operator equation (17) in a neighborhood of the exceptional values α j for j ∈ J. (Of course, such exceptional values do not need to exist for every k > 0.) The following lemma is obvious by the Fredholm property of the operator K k,α and the definition of an exceptional value.
Lemma 3.1. For any fixed δ > 0 the solutions ũ k,α ∈ H 1 per (C R ) of (17) for Imk > 0 converge to ũk ,α in H 1 per (C R ) as k →k uniformly with respect to α ∈ [−1/2, 1/2] : |α −α j | δ ∀j ∈ J . It remains to study the convergence of ũ k,α in neighborhoods of the exceptional values α j . To this end, we formulate the following result from abstract functional analysis.

be families of compact operators and elements, respectively, such that
is selfadjoint and positive definite and B := P ∂ ∂α K(0, 0)| N : N → N is selfadjoint and one-to-one.
Here, u (1) (ε, α) H is uniformly bounded with respect to (ε, α), and λ , φ : = 1, . . . , m is an orthonormal eigensystem of the following generalized eigenvalue problem in the finite dimensional space N (where m = dim N ): For the proof we refer to [16]. We want to apply this theorem to the equation (17) and set K(ε, α) = Kk +iε,αj+α and f (ε, α) = rk +iε,αj+α where k > 0 is fixed and α j , j ∈ J, is one of the exceptional values. In the following we assume always that assumptions 2.2 and 2.8 are satisfied. We have to show the assumptions of the previous theorem. Let X j = N (I − Kk ,αj ) denote the kernel of I − Kk ,αj . By lemma 2.12 it is given by is again equipped with the inner product (·, ·) * from (15). We set m j = dimX j .
is orthogonal with respect to (·, ·) * . Therefore, the projection operator P j : Here, φ ,j ∈X j is identified with its extension in C ∞ .
Proof. Since j ∈ J is fixed we drop j from the notation.
(a) First we note that for v ∈X and ψ ∈ H 1 per (C R ) the dual form takes the form (see proof of lemma 2.9) = v, Λk ,α ψ because the ratio is real valued. Again, K m are the modified Bessel functions and k = k (k,α) = k2 − ( +α) 2 . Therefore, for v ∈X and ψ ∈ H 1 per (C R ) we have by partial integration From this we conclude for v ∈ N (I − Kk ,α ) and ψ ∈ H 1 per (C R ) that We note that ak ,α has the following form on X ×X because v and ψ decay exponentially as x 2 1 + x 2 2 tends to infinity. On the right hand side v, ψ ∈X are again identified with their extensions in C ∞ . Therefore, This proves part (c) because X is finite dimensional. For a more convenient formulation we translate the spaces X j into the (isomorphic) spaces X j of α j -quasi-periodic solutions of the homogeneous Helmholtz equation; that is, we replace the periodic function φ j by φ j (x) = e iαjx3φ (x). Then φ j ∈X j if, and only if, φ j ∈X j wherê Here, H 1 αj (C ∞ ) denotes the space of α j -quasi-periodic functions (wrt x 3 ); that is, the subspace of H 1 (C ∞ ) consisting of functions φ such that φ (x 1 , x 2 , 2π) = e iαj2πφ (x 1 , x 2 , 0) for all x 1 , x 2 . The eigenvalue problem (19) is equivalent to Again, φ ,j are normalized by 2k C∞ nφ ,jφ ,j dx = δ , .
Therefore, all of the assumptions of theorem 3.2 are satisfied if k > 0 is regular in the following sense.
. . , m j and j ∈ J where λ ,j ∈ R, = 1, . . . , m j , are the eigenvalues of the selfadjoint eigenvalue problem (21) in the finite dimensional space X j .
Proof. We compute In the first integral we substitute α = tε/|λ| and in the second integral t = αx 3 . This yields For ε → 0 the expression on the right converges to uniformly with respect to |x 3 | a, for arbitrary a > 0. The derivative of the integral with respect to x 3 converges uniformly for |x 3 | a for every a > 0 as well. □ Remark 3.6. As lim a→∞ a 0 sin(t)/t dt = π/2 we observe that ψ ± ∈ C ∞ (R), defined by tends to 1 as x 3 → ±∞ while it converges to 0 for x 3 → ∓∞. Thus, as ε tends to zero, u where This separates u (2) 0 into groups of modes propagating to the left and the right. (27) yields the following main result. Theorem 3.7 (The limiting absorption principle). Let assumptions 2.2 and 2.8 hold and let k > 0 be regular in the sense of definition 3.4. Then the solution uk +iε of (3) for k =k + iε has a decomposition in the form uk +iε = u and u (2) ε ∈ W 2,∞ (R 3 ) is given by (23). Furthermore, for every R > R 0 we have that u for a ,j ∈ C given by Here, the functions ψ ± are defined in (25), and φ ,j are the α j -quasi-periodic solutions of ∆φ ,j + k 2 nφ ,j = 0 in R 3 , given by the eigenvalue problem (21). The func- Proof. Only the second equality in (29) has to be shown. But this follows directly from □ This result holds for any R > R 0 . Therefore, the solution u = u (1) + u (2) is defined in all of R 3 and a solution of the differential equation (3) for k =k.

Remarks 3.8.
(a) We note that we can replace the functions ψ ± by any functions with ψ + ( 3 → ±∞ (and ψ − analogously) because the difference of this choice of ψ ± and the one of (25) differ only by a H 1 -function. In particular, one can choose ψ + such that ψ + (x 3 ) = 0 for x 3 −τ and ψ + (x 3 ) = 1 for x 3 τ for some τ > 0 (and ψ − analogously) or The representation (28) (with (29)) corresponds to the asymptotic formulas for closed waveguides T R ; that is, with boundary condition u = 0 on ∂T R see, e.g. theorem 7 in [10]. (c) From part (b) of lemma 2.9 and the eigenvalue problem [21) we note that we can assume that φ ,−j =φ ,j for all and j ∈ J and thus λ ,−j = −λ ,j . Therefore, there exist as many propagating modes propagating upwards as ones propagating downwards.
(d) In the special case that X j is one-dimensional with basis {φ j } such that 2k C∞ n|φ j | 2 dx = 1 the wave number is regular if λ j = −2i C∞ ∂φj ∂x3φ j dx = 0. In particular λj 2 = Im C∞ ∂φj ∂x3φ j dx corresponds to the group velocity of this eigenfunction and measures its energy flux in vertical direction: In the the case λ j > 0 the energy of the wave is traveling upwards, in the case λ j < 0 the energy of the wave is traveling downwards (see [18]).
As a corollary we apply this result to the special case of the scattering of a point source u inc (x) = Φ k (x, y) by the periodic waveguide. In this case the total field u t = u inc + u s is the Green's function of the differential operator ∆ + k 2 n and is given by u t = u + (1 − χ y )u inc where χ y (x) = χ(x − y) and χ ∈ C ∞ (R 3 ) is such that χ(x) = 0 for |x| /2 and χ(x) = 1 for |x| for some > 0 and u solves (3) with f := (∆ + k 2 n) (1 − χ y ) u inc . In this case we can compute the coefficients a ,j explicitly.

Proof.
We have to compute a ,j from (29) for the special form y) . By Green's theorem we have for any δ ∈ (0, ε/2) (note that f has compact support and χ y ( and this converges to φ ,j (y) as δ tends to zero. Therefore, a ,j = 2πi |λ ,j |φ ,j (y). The assertion follows from the decomposition and the form of u (2) . □

The radiation condition
It is obvious that u (1) from theorem 3.7 is a solution of the following boundary value problem with h 0 := ∆u (2) +k 2 u (2) in R 3 \ T R and g 0 := u (1) = u − u (2) on Γ R . The solution of this boundary value problem is not unique without a condition away from the waveguide. We will derive such a radiation condition below. First we note that h 0 ∈ L 2 (R 3 \ T R ). Indeed, from (26) we observe that for x / ∈ T R the right hand side h 0 is given by where u ± j are linear combinations of the evanescent modes φ ,j , = 1, . . . , m j , see (27). From this we observe that not . In order to formulate a correct radiation condition for u (1) normal to the axis of the cylinder we need the cylindrical Fourier transform Fg : Z × R → C of g which is given by Here, ϕ ∈ (0, 2π) and y 3 ∈ R are the cylindrical coordinates of y ∈ Γ R . Then F is well defined and bounded from L 2 (Γ R ) into The inverse transform is then Also, Parseval's identity holds in the form (33) The one-dimensional Fourier tranform F 1 is related to the Floquet-Bloch transform F by for ∈ Z and ξ ∈ (−1/2, 1/2]. This translates to the cylindrical Fourier transform F as for r > 0, , m ∈ Z, ξ ∈ (−1/2, 1/2] where r, ϕ, x 3 are the cylindrical coordinates of x ∈ R 3 . Analogously, the inverse of F is expressed as where r, ϕ, x 3 are the cylindrical coordinates of x ∈ R 3 . Therefore, by (5), The Fourier transform is, in particular, useful to define the trace space H 1/2 (Γ R ) by Then the trace theorem holds for u ∈ H 1 (T R ) or u ∈ H 1 (TR \ T R ) for R > R (see [5]). We will now study the following exterior boundary value problem for the Helmholtz equation in the exterior of the cylinder T R such that the Fourier transform satisfies the family of one-dimensional radiation conditions for all m ∈ Z and almost all ξ ∈ R where k(ξ) = k 2 − ξ 2 .
for some σ > 1 and g ∈ H 1/2 (Γ R ) and k ∈ C with Rek > 0 and Imk 0. The function v(r, θ, for r > R, θ ∈ [0, 2π], and x 3 ∈ R, is the unique solution of the boundary value problem (36), (37). Again, k(ξ) = k 2 − ξ 2 , and G is given by for r, ρ R, m ∈ Z, and ξ ∈ R where r + = max{r, ρ} and r − = min{r, ρ}. Furthermore, the solution depends continuously on h, g, and k, in the sense that for any R > R and σ > 1 the mapping Proof. In this-and only this-proof we abbreviate the (cylindrical) Fourier transform by writing f instead of Ff . Let k 0 > 0 be arbitrary, but fixed, and k ∈ C with Rek > 0 and Imk 0 and |k| k 0 . The proof is divided into five parts.
(a) We show that v is well defined and depends continuously on h and g. The function v consists of two parts; i.e.
and thus by Parseval's identity where c depends only on k 0 , R, and R . This shows the estimate v 1 H 1 (TR\TR) c h L 2 σ (R 3 \TR) . Now we discuss v 2 as in [5]. Part (a) of lemma A.2 yields v 2 (r, m, ξ) ĝ(m, ξ) . Furthermore, where we used the elementary estimate Imk(ξ) |ξ|/2 for |ξ| 2|k| (see lemma A.5). Furthermore, with lemma A.4, Altogether we have shown that Taking the inverse Fourier transform proves that v 2 ∈ H 1 (TR \ T R ) and v 2 H 1 (TR\TR) c g H 1/2 (ΓR) where c depends only on k 0 , R, and R . (b) Now we show that v from (38) satisfies the differential equation.
and g ∈ C ∞ (Γ R ) have compact supports. Then ĥ = Fh and ĝ = Fg are smooth with respect to ρ and ξ and ĥ vanishes for large values of ρ. The partial sections (r, m, ξ) e imθ+iξx3 dξ and as it is shown by using (A.7). Let ψ ∈ C ∞ (R 3 \ T R ) have compact support in some TR \ T R . Then Both sides converge as N tends to infinity which shows that that is, w satisfies ∆w + k 2 w = −h in R 3 \ T R . For arbitrary h ∈ L 2 σ (R 3 \ T R ) and g ∈ H 1/2 (Γ R ) we approximate h and g by smooth functions with compact support and repeat the argument. This ends the second part of the proof.
(c) Now we study the dependence of the solution on k. Here we use Lebesgue's theorem on dominated convergence. Concerning v 1 estimate (40) provides an integrable bound uniformly with respect to k. Furthermore, the integrand (left hand side of (40)) is continuous with respect k. Lebesgue's theorem, applied to the integration with respect to r and ξ and to the summation with respect to m yields the continuous dependence of v 1 on k. The same arguments applies to v 2 using estimate (41). This ends the proof of part (c). (d) Next we show that v = Fv satisfies (37). For v 2 this follows from H v ∈ H 1 (R 3 \ T R ) be a solution of (36) satisfying (37). We take the Fourier transform v(r) = (Fv)(r, m, ξ) of v. Then v solves the Bessel differential equation

Now we use again that H
and v(R, m, ξ) = 0 for all parameters m ∈ Z and almost all ξ ∈ R. The general solution is given by v(r) = a H (1) m (k(ξ)r) + b H (2) m (k(ξ)r) for r > R and some a, b ∈ C. The boundary condition and the one-dimensional radiation condition v (r) − ik(ξ)v(r) = o(1/ √ r) yields (Fv)(r, m, ξ) =v(r) = 0 for all r > R, m ∈ Z and almost all ξ ∈ R. This yields v = 0 and ends the proof. □ Theorem 4.2. Let u = u (1) + u (2) be the solution of the source problem ∆u +k 2 nu = −f in R 3 derived by the limiting absorption principle of theorem 3.7. Then the Fourier transform (Fu (1) )(r, m, ξ) of u (1) satisfies the one-dimensional radiation condition (37) for all m ∈ Z and almost all ξ ∈ R.
We note that the coefficients a ,j are given by (29). Again, the representation of (a) corresponds to the well known radiation condition for closed waveguides as in definition 5 of [10]. Part (b) is needed for open waveguides to describe the behavior normal to the axis of the cylinder.
We will show in Part (B) that this radiation condition assures uniqueness and existence directly; that is, without using the limiting absorption property. for all ξ ∈ R and m ∈ Z and k ∈ C with |k| k 0 and Rek > 0 and Imk 0.