Periodic wave-guides revisited: Radiation conditions, limiting absorption principles, and the space of bounded solutions

: We study the Helmholtz equation with periodic coeﬃcients in a closed wave-guide. A functional analytic approach is used to formulate and to solve the radiation problem in a self-contained exposition. In this context, we simplify the non-degeneracy assumption on the frequency. Limiting absorption principles (LAPs) are studied and the radiation condition corresponding to the chosen LAP is derived; we include an example to show diﬀerent LAPs lead, in general, to diﬀerent solutions of the radiation problem. Finally, we characterize the set of all bounded solutions to the homogeneous problem.


Introduction
This paper is devoted to the study of equations modelling waves in a periodic wave-guide.We consider (1.1) The domain is an unbounded cylinder with a cross-section S, we assume that S ⊂ R d−1 is a bounded Lipschitz domain, d ≥ 2 is the dimension of the wave-guide.The wave-number k ∈ C is prescribed and satisfies Im k ≥ 0, the coefficient function n : Ω → R is assumed to be 2π-periodic in x 1 , the right hand side f ∈ L 2 (Ω) is assumed to have compact support or, more general, decay properties, see (1.3).We treat the homogeneous Dirichlet boundary condition u = 0 on R × ∂S.We are interested in solutions u to (1.1) that satisfy, additionally, a radiation condition.
In this article, we show existence and uniqueness results for (1.1), we investigate different Limiting Absorption Principles (LAPs), and we characterize function spaces that are related to (1.1).Regarding the LAPs, we show that a vanishing absorption can yield, indeed, a (radiating) solution to the original problem; we additionally show that different damping mechanisms in the LAP can lead to different radiation conditions and, hence, select different solutions to (1.1).
In this work, we treat only the case of a strictly periodic coefficient n = n(x).Nonetheless, we mention that our work has implications for the case that the medium is only periodic outside a compact set.Such a case is treated in [20] under a nondegeneracy assumption on the frequency.It is one aim of the article at hand to relate that non-degeneracy assumption to a more standard formulation.
We always use a weak solution concept.Solutions to (1.1) are functions u ∈ H 1 loc ( Ω) := {u : Ω → C | u| (−R,R)×S ∈ H 1 (−R, R) × S for every R > 0}, and (1.1) is interpreted in the weak sense: We demand that (1.2) holds for all ϕ ∈ H 1 0 (Ω), and u = 0 on R × ∂S in the sense of traces.We assume that the right hand side is in the space In a first step, we construct solutions u ∈ H 1 (Ω).For wave numbers k ∈ C with Im k > 0 the existence of such solutions follows for all f from the Lax-Milgram theorem.We show that, for real values of k, the existence of H 1 (Ω) solutions can be obtained with the Floquet-Bloch transform for right hand sides f that satisfy an orthogonality condition (we will write g instead of f for sources with this property).This construction of solutions u ∈ H 1 (Ω) can be used to show existence and uniqueness of a radiating solution u ∈ H 1 loc ( Ω) for a general right hand side f .In Section 4 we turn to Limiting Absorption Principles.In a first result we consider a real number k > 0 and use the wave-number k + iη in the equation.We find that, as η > 0 tends to zero, solutions u η tend to solutions of the limit problem with η = 0.It is interesting to compare this result with other mechanisms of a small absorption: We show that different absorption terms lead, in general, to different limit solutions.We can characterize the radiation condition for different absorption mechanisms.
The starting point for all these results is the Floquet-Bloch transform.It allows to transform the original equation (1.1) to a family of problems on the bounded domain W := (0, 2π) × S. The family of problems is parametrized by a parameter α ∈ I := [−1/2, 1/2].Equation (1.1) has then to be solved on W for all α ∈ I, demanding the α-quasi-periodicity of solutions on the lateral boundaries {0} × S and {2π} × S. To obtain an equivalent formulation of the problem, it is important to impose, additionally, certain boundedness properties of solutions with respect to the parameter α.
When a fixed parameter k ∈ R is considered, we obtain a one-parameter family of problems (α is the only parameter).For a wave number of the form k + iη, we will deal with a two-parameter family of problems, where η > 0 is a second parameter.
1.1.Known results and literature.The Helmholtz equation is an old and intensively treated research subject.Classical contributions concern homogeneous media and treat the appropriate radiation conditions in different (unbounded) geometries, the development of appropriate numerical schemes and the field of inverse scattering.Here, we refrain from citing any of the corresponding results.
The two simplest cases for heterogeneous media are (a) periodic media and (b) compact perturbations of periodic media.The methods for the two cases are closely related.In particular, in both cases, one can exploit the tool of the Floquet-Bloch transform [17,19].Within this setting, the simplest geometry is that of a closed wave-guide.An important contribution is [9], where the appropriate radiation condition was specified and an existence and uniqueness proof was presented.A related work is [12], where a limiting absorption principle for the periodic wave-guide was shown.In [7], the focus is on equivalent descriptions with Dirichlet-to-Neumann maps, which are useful also in numerical approaches.Such an approach was also used to study, e.g., wave-guides with different periodic geometries in the two directions.We refer to [10] for a typical result and further references.
All of the above articles are based on complex integrals to invert operators or operator families.Another route to existence and uniqueness results was developed with [15,16] based on an idea taken from [5].Essentially, after a Floquet-Bloch transform of the equations, one has to deal with a family of operators that are, except for a discrete set of exceptional points, invertible.With an application of the implicit function theorem, one can construct bounded famlilies of solutions.These provide solutions in periodic wave-guides without advanced operator theory.In the paper at hand, we will use this method.
While all of the above approaches are based, in one way of the other, on the Floquet-Bloch transform, [20] is not using it; an existence result is shown and, in a more general geometry, a Fredholm alternative, the proofs use only energy methods.The only assumption that is made in [20] is that of a non-degeneracy of ω (which is essentially k in the article at hand).With Section 6 we show that our quite classical Assumption 3.5 implies the non-degeneracy that was assumed in [20].We note that similar ideas allow to introduce a different radiation condition, see [18] and, for a numerical scheme, [6].
Let us close this overview by mentioning some results beyond closed wave-guides.Perturbed periodic geometries in two dimensions are considered in [1,2,8,11,13,14].For open wave-guides, by which we mean here a domain that is unbounded in more than one direction, one has to introduce radiation conditions also in the additional direction; we refer to [3,4] for formulations of such conditions.
The emphasis of the present work is the following: A further simplification of the direct approach to the Helmholtz equation in periodic media.This comes with a simplification of the non-degeneracy assumption on frequencies, see Assumption 3.5.The non-degeneracy implies also a characterization of bounded solutions, Y = B; essentially, every bounded homogeneous solution must be a linear combination of α-periodic solutions, see Theorem 6.2.We derive limiting absorption principles with a characterization of the resulting radiation conditions and show that different solutions can emerge depending on the absorption mechanism.

Floquet-Bloch transform of the equation
This section is devoted to the application of the Floquet-Bloch transform to (1.1).We emphasize that, here, we only study coefficients that are x 1 -periodic in all of Ω.Only homogeneous Dirichlet conditions are treated here, but we note that, e.g., homogeneous Neumann conditions can be treated with only notational changes in the proofs.Also operators of the form u → −∇ • (a∇u) − k 2 u with strictly positive and 2π-periodic a ∈ L ∞ (Ω, R n×n ) can be treated with only notational changes.
2.1.The Floquet-Bloch transform.We perform the Floquet-Bloch transformation only in the x 1 -variable.We recall that the interval for the parameter α is I = [−1/2, 1/2] and that the periodicity cell is W = (0, 2π) × S. The transformation is a bounded linear map (2.1) For smooth functions u with compact support, writing x = (x 1 , x) for the argument, the transformation is defined by for every x 1 ∈ (0, 2π), x ∈ S, α ∈ R. The map F FB of (2.1) is defined as the continuous extension of this map.Proofs regarding properties of the map F FB are given in Appendix A.
We say that a function u ∈ H 1 (W ) is α-quasiperiodic when u(2π, •) = e 2πiα u(0, •) holds in the sense of traces.We define the space H 1 α (W ) as the subspace of H 1 (W ) that consists α-quasiperiodic functions.From the definition of F FB in (2.2) it is clear that, for almost every α, the function û(•, α) is α-quasiperiodic.
A direct consequence of definition (2.2) is that the transformation respects derivatives in the sense that This fact implies that we can interpret F FB also as a map from H 1 (Ω) onto Two remarks should be made at this point.One regards a notational difficulty: The target space H 1 α (W ) depends on the parameter α, hence and carries the topology of that ambient space.Our second remark is that H 1 α (W ) does not include a boundary condition on R × ∂S, but a boundary condition can and will be included later on.
With the above space, the transformation map F FB has a bounded inverse FB with its easy formula (A.4) is provided in Appendix A for convenience of the reader.The method is quite standard, for generalized approaches we refer to [17,19].

2.2.
A family of operators.We exploit the Floquet-Bloch transform to analyze equation (1.1).In this subsection, the wave-number can also be complex, we treat an arbitrary k ∈ C. The right hand side is denoted by g ∈ L 2 * (Ω) and not by f ; the reason is that, in this first step, we construct H 1 (Ω)-solutions u for right hand sides g with a particular structure.Later on, we treat general right hand sides f ∈ L 2 * (Ω).We consider, as in (1.1), (2.4) with the weak form as in (1.2), (2.5) for all ϕ ∈ H 1 0 (Ω).We always impose Dirichlet boundary conditions without further mentioning: u(•) = 0 on R × ∂S for (2.4) and, later on, û(•, α) = 0 on (0, 2π) × ∂S for almost every α ∈ I.
for almost every α ∈ I.The functions û(•, α) are weak solutions of (2.6). ( ) and û(•, α) is a weak solution of (2.6) with homogeneous Dirichlet conditions for almost all α ∈ I, then the inverse Floquet-Bloch transform and it is a weak solution of (2.4).Proof.(1) Let u ∈ H 1 0 (Ω) be a weak solution of (2.4).Our aim is to derive (2.7) for û := F FB (u).To this end, let φ ∈ H 1 α (W ) be a test-function, we write φ in the form φ(x) = ψ(x)e iαx 1 with a function ψ ∈ H 1 (W ) that is periodic with respect to x 1 .Additionally, we choose a number m ∈ Z.
We can now construct a test-function for u: We define ϕ ∈ H 1 0 (Ω) as the inverse Floquet-Bloch transform of the function φ(α, x) := φ(x)e iα2πm = ψ(x)e iα(2πm+x 1 ) .By the unitarity of the Floquet-Bloch transform, see (A.2), in the integral equation (2.5), the Ω-integrals transform into I × W -integrals, and we obtain Since m was arbitrary, all Fourier coefficients of the two terms in squared brackets coincide.This implies that the squared brackets coincide for almost every α ∈ I.Because of φ(x) = ψ(x)e iαx 1 , this is (2.7).
For the further development of the theory, it is useful to have a target space that is independent of parameters.We introduce (2.9) We denote the canonical inner product in X = H 1 per (W ) by •, • X .Note that we included the Dirichlet boundary condition into the space H 1 per (W ).We exploit the following equivalence for U ∈ H 1 (W ): It allows to map H 1 α (W )-functions to H 1 per (W )-functions and vice versa.Replacing û(x, α) by v(x, α)e iαx 1 and φ(x) by ϕ(x)e iαx 1 , we can re-write the problem described in (2.7) as a family of problems in the space X = H 1 per (W ): We seek for v ∈ L 2 (I, X) such that for almost every α ∈ I.
For fixed α ∈ I, we can consider the right hand side of (2.11) as a function of ϕ, defining a functional on X.We can also, for fixed v, consider the left hand side of (2.11) as a functional on X.By the Riesz representation theorem there exist y α ∈ X and for every ϕ ∈ X.With these representations, using Lemma 2.1 (b), the original problem (2.4) is solved when we find, for almost every α ∈ I, a solution v(•, α) ∈ X = H 1 per (W ) of (2.14) and if this family of solution satisfies v ∈ L 2 (I, X).
It is not obvious how to solve (2.14).Indeed, structural assumptions on g will be necessary in order to solve the equation.The reason for this restriction is that we are looking for solutions u of the original problem in the space H 1 (Ω), i.e., for solutions with decay properties.
On the other hand, some structural properties of L α follow immediately from the definition.For fixed α, the operator L α is a linear bounded operator from X = H 1 per (W ) into itself.The form of L α shows that L α is self-adjoint and that we can write L α = id + K α , where K α is a compact linear operator.Accordingly, every operator L α is a Fredholm operator with index 0. Additionally, the definition of L α extends, for ε > 0, to the increased interval I ε := (−1/2 − ε, 1/2 + ε).The operators depend continuously differentiable and even analytically on α.
A remark on notation.When k is replaced by k + iη with k > 0 and η ≥ 0 then we write L η α to indicate the dependence on the second parameter η.

Existence and uniqueness
In this section, we consider the case of a real wave number k > 0 and equation (1.1) for arbitrary f ∈ L 2 * (Ω), see (1.3) for the function space.Our approach will be the following.In a first step we search for solutions u ∈ H 1 (Ω); we can find such solutions only when the right hand side f = g has certain orthogonality properties.
Roughly speaking, g must be orthogonal to the space of quasi-periodic solutions of the homogeneous equation.For such g, we will show the existence of a solution u by a functional analytic singular perturbation theorem which we learned from [5].In a second step, we allow general f ∈ L 2 * (Ω), but we search for a solution in a larger class of functions u satisfying a radiation condition.
We say that (L α ) α is a regular C 1 -family of operators when additionally the following two conditions are satisfied for every α ∈ I for which L α is not invertible: (i) The operator L α has Riesz number 1, i.e., N := ker(L α ) = ker(L 2 α ).(ii) With the range R := L α (X) ⊂ X and the projection P onto N corresponding to X = N ⊕ R, the operator Remarks. 1.We demand that every operator L α is a Fredholm operator with index 0.This implies that, for every α and L = L α , the subspace N := ker(L) has finite dimension and the subspace R := L(X) is closed and has finite co-dimension; the latter agrees with the dimension of N since the index is 0. Together with the requirement ker(L) = ker(L 2 ), we conclude that the space possesses the decomposition X = N ⊕ R and corresponding continuous projections P : X → X onto N and Q = (id − P) onto R. We recall the easy argument why the intersection is trivial: u ∈ N ∩ R implies u = Lx and Lu = 0, hence L 2 x = 0 and thus Lx = 0, we find x ∈ N and u = Lx = 0. 2. When X is a Hilbert space with inner product •, • X and L is self-adjoint, it has Riesz number 1. Indeed, L 2 x = 0 implies Lx, Lx X = L 2 x, x X = 0 and thus x ∈ N .Theorem 3.2 (Functional analysis I).Let (L α ) α be a regular C 1 -family of operators.There holds: (1) The set of critical numbers is finite: For a number J ∈ N (we allow J = 0 for an empty set A) and values {α j | j = 1, . . ., J} ⊂ I holds (2) Let I ε α → y α be a C 1 -family of right hand sides such that y α j ∈ L α j (X) holds for every j = 1, . . ., J. Then the family of solutions can be continued to a C 0 -family on I ε .With C independent of the family (y α ) α , there holds Proof.
Step 1: An equivalent form of the system.We start the proof by investigating a point α 0 ∈ I with ker(L α 0 ) = {0}.It is no loss of generality to assume α 0 = 0.The critical (non-invertible) operator is L := L 0 .We use X = N × R with projections P and Q.We emphasize that these subspaces and projections are chosen for L and independent of α in the following.For α close to α 0 , we write the operator L α as Relation (3.5) defines linear operators Lα : X → X for α = 0. We want to extend this family of operators to the point We claim that the new operator family (−ε, ε) α → Lα ∈ L(X, X) is continuous.This is clear by definition in all points α ∈ (−ε, ε) \ {0}.Regarding α = 0 we note that the operators of (3.5) can be written as difference quotients: Because of L| N ≡ 0 there holds Since we extended with P L | N = M for α = 0, the resulting family is continuous in α.The same argument can be performed for the second entry of the matrix: Because of P L| R ≡ 0 we can write The limit operator is given by the derivative that is used in (3.6).Finally, regarding the third entry, we note that QL| N = 0 by the definition of N .We obtain that the family Lα is continuous in α.
We next observe that the operator L0 is invertible: The operator M : N → N is invertible by the definition of a regular family.The operator QL| R : R → R is invertible by definition of R and Q.As a triagonal matrix, L0 is invertible.
For α ∈ A, the claim holds, since small perturbations of invertible operators are invertible.
Step 3: Assertion (2).We have to consider again the situation of Step 2, with u α solving L α u α = y α (or, equivalently, (3.5)) for α ∈ (−ε, ε) \ {0}.Regarding the right hand side y α , we have imposed the property y α j ∈ L α j (X) for all j.In the local situation and with our assumption that the critical point is α 0 = 0, we have y 0 ∈ L(X) = R.This implies P y 0 = 0 and we can write the first entry of right hand side of (3.5) as 1 α P y α = 1 α P (y α − y 0 ), which can be extended continuously with P y | 0 for α = 0.
The fact that the family Lα is a continuous family of invertible operators on α ∈ (−ε, ε) together with the fact that the right hand sides of (3.5) can be extended continuously to (−ε, ε) shows that the family u α can be extended continuously.The proof also provides (3.3).Remark 3.3 (Functional analysis with two parameters).Definition 3.1 can be adapted to define C 1 -families of operators L η α depending on two parameters, α ∈ [−1/2, 1/2] and η ≥ 0. Regarding the definition of a regular C 1 -family, requirement (ii) of Definition 3.1 has to be replaced by the requirement that, for every α for which L 0 α is not invertible and any direction vector 0 = ξ ∈ R 2 with ξ 2 ≥ 0, the operator (3.7) α denotes the directional derivative.With these adaptions, the assertion of Theorem 3.2 holds in a slightly weaker form: For some ε > 0, the family of solutions To show this result, one considers, for fixed direction ξ, parameters along a semiray: (α, η) = τ ξ with τ > 0. The arguments of Theorem 3.2 can be repeated upon replacing the parameter α with the new parameter τ .

3.2.
Regularity of the C 1 -family of operators L α .We now consider the oneparameter family L α of (2.12).This family is a C 1 -family because of the smooth dependence of L α on α.Using the equivalence (2.10), the kernel Since each L α is a Fredholm operator, the kernel N α is finite dimensional and hence also Y α is finite dimensional.We are interested in the set of critical points (3.9) Without further assumptions, the set A can be finite or infinite.Theorem 3.2 yields that A is finite when we can show that L α is a regular C 1 -family.This is what we will obtain under a certain assumption.We define a sesqui-linear form E by setting, for u, v ∈ H 1 (W ), We emphasize that, typically, the arguments of E are α-periodic functions, but not necessarily elements of X = H 1 per (W ).We observe that E is hermitean, thus E(u, u) ∈ R for all u.
The form E is related to energy fluxes through sections of the form Γ t := {t}×S ⊂ Ω for t ∈ R. Indeed, when u and v are two solutions, (∆ + k where we denoted with ∂ ν the normal derivatives into the exterior of W s,t .The calculation shows that the flux quantity We obtain easily that, for different values of α ∈ (−1/2, 1/2], the spaces Y α are orthogonal with respect to the above sesqui-linear form: Proof.For quasiperiodic u and v as in the lemma, the expression of (3.11) satisfies, by its definition, F u,v,t+2π = e 2πiα e −2πiβ F u,v,t .On the other hand, as noted above, F u,v,t is independent of t.Because of |α − β| < 1 we conclude F u,v,t = 0 and thus E(u, v) = 0.
We can show that L α is a regular C 1 -family under the following assumption.Assumption 3.5 (Non-degeneracy assumption).For every α ∈ A, the sesqui-linear form E is non-degenerate on Y α in the following sense: For every Proof.We fix α ∈ A and consider the operator L := L α with kernel N := ker(L) and derivative L := ∂ α L α .We have to verify that M := P L | N : N → N is invertible, where P is the projection onto N .In the subsequent calculation, the definition of L α in (2.12) yields the first equality; we use here that e iαx 1 e −iαx 1 = 1 is independent of α.In the second equality we use that, when the derivatives are applied to u(x) e iαx 1 and to ϕ(x) e iαx 1 , but not on x 1 , the terms from the first term and from the second term cancel.
From this calculation we can conclude that P L | N is invertible.Indeed, let u ∈ N satisfy P L u = 0. Since L = L α is self-adjoint, N and R are orthogonal.In this situation, P L u = 0 implies that ϕ → L u, ϕ X is the trivial form on N .The above calculation, together with the fact that E is non-degenerate, implies that this is possible only for u e iαx 1 = 0, and thus for u = 0. We obtain that the kernel of P L | N is trivial and hence that M of (3.1) is invertible.
Corollary 3.7 (The spaces Y j and basis functions).We consider a Helmholtz equation for which Assumption 3.5 holds.In this situation, the family L α constructed in (2.12) is a regular C 1 -family of operators.There is a finite (possibly empty) set of values A = {α j |j = 1, . . ., J} such that Every space Y j has a finite dimension m j ∈ N and the spaces Y j are orthogonal with respect to E. We introduce the direct sum We choose, for every space Y j , an inner product •, • Y j , and solve the self-adjoint eigenvalue problem for λ ∈ R and φ ∈ Y j .This provides an orthogonal basis of Y j consisting of eigenfunctions φ ,j , = 1, . . ., m j .The value λ = 0 is not an eigenvalue.
Proof.Lemma 3.6 provides that L α is a regular family.The functional analysis Theorem 3.2 provides that the set of critical α-values is finite.Because of Lemma 3.4, the spaces Y j are also orthogonal to each other (with respect to E). Assumption 3.5 guarantees that λ = 0 is not an eigenvalue.The other assertions repeat the definitions and follow from the Fredholm assumption on the family of operators.
The solutions φ of (3.14) are orthogonal to each other in Y j by construction.
3.3.H 1 (Ω)-solutions.We turn to our first existence result for the Helmholtz equation.We characterize the right hand sides g such that equation (2.4) has a solution in H 1 (Ω).
Proof.Existence.Using the Floquet-Bloch transform, we have shown that equation (2.4) is equivalent to the family of equations In particular, it is sufficient to find a family v(•, α) of solutions to (2.14) and to verify that v ∈ L 2 (I, H 1 per (W )).By definition of the critical values A = {α j |j = 1, . . ., J}, a unique solution v(•, α) exists for every α ∈ I \ A. We claim that this family of solutions extends continuously to all of I.
We consider one of the critical values, α = α j ∈ A, and a small interval Ĩ = [α j − ε, α j + ε] that contains no other critical value.We want to use the functional analysis result of Theorem 3.2.We use the space X of (2.9), the family of operators L α of (2.12), and the family of right hand sides y α of (2.13).
We have to check the assumptions of Theorem 3.2.The operators L α depend smoothly on α and they are invertible for all α ∈ Ĩ \ A. We turn to the condition y α j ∈ R = L α j (X).For an arbitrary element ϕ ∈ N := ker(L α j ) ⊂ X, we note that there holds φ(x) := ϕ(x)e iα j x 1 ∈ Y j , and, by definition of y α , by the orthogonality assumption (3.15).This shows that y α j is orthogonal to N .Since L α j is self-adjoint, the subspaces N and R are orthogonal.Since L α j is also Fredholm with index 0, the space X is the orthogonal direct sum N ⊕ R. Since we have shown that y α j is orthogonal to N , we have found y α j ∈ R. Lemma 3.6 provides that L α is a regular family of operators in the sense of Definition 3.1, hence Theorem 3.2 can be applied.We find that . This provides a H 1 0 (Ω)-solution of (2.4).
We turn to the estimate for the solution.The right hand side is an element g ∈ L 2 * (Ω).With the functions g : W → C, g (x 1 , x) := g(x 1 + 2π , x), we can estimate the corresponding norm as g 2 . This allows to calculate, for arbitrary m < M , the norm of a finite sum, which is related to the derivative ∂ α ĝ(•, α) of the Floquet-Bloch transform of g with respect to α, compare (2.2): where C m,M is independent of g and tends to zero as m → ∞.The Cauchy argument shows that ∂ α ĝ(•, α) is well-defined in L 2 (W ) and is bounded by for some C > 0. We conclude that, for some C > 0, there holds ĝ(•, α) (Ω) for all α.Theorem 3.2 provides estimate (3.3) for solutions, which is a bound for û ∈ C 0 (I, H 1 (W )), hence, in particular, for û ∈ L 2 (I, H 1 (W )).This yields the bound for u ∈ H 1 (Ω), namely u H 1 (Ω) ≤ C g L 2 * (Ω) .Uniqueness.In order to show unique solvability of (2.4), it is sufficient to show the unique solvability of (2.6) for almost every α.For every α ∈ A, equation (2.6) can be solved uniquely by definition of the critical α-values.This already shows the uniqueness of the solution.
Let u ∈ H 1 (Ω) be a solution of (2.4).We have to show that the orthogonality (3.15) holds.We use equation (2.7), which is a consequence of (2.4): where we exploited that, for every α, integration by parts holds without boundary terms for two functions in the space H 1 α (W ).This concludes the proof of the theorem.Lemma 3.9 (Orthogonality criterion).The orthogonality condition (3.15) is formulated in terms of the Floquet-Bloch transform of g.With the original function g and the space Y of (3.13), an equivalent condition is Proof.We fix j ∈ {1, . . ., J}, set α := α j and choose a function φ in Y j .We identify φ with its α-quasiperiodic extension, which satisfies φ(x + 2π e 1 ) = φ(x)e i 2πα with the unit vector e 1 in x 1 -direction.We calculate for ĝ = F FB (g) We note that the series in the definition of the Floquet-Bloch transform is welldefined because of g ∈ L 2 * (Ω).
3.4.The radiation problem.In the previous subsection, we have obtained a solution u to (2.4) where g satisfies the orthogonality condition (3.17).This is not the kind of solution that is typically observed.In the physical problem, we have to consider the equation with a general right hand side f and obtain solutions that are, approximately, far away from the origin, linear combinations of outgoing waves.Such solutions are not in the space H 1 (Ω).We recall that we impose f ∈ L 2 * (Ω).In order to define the radiation condition, we use two cut-off functions.Definition 3.10 (Cut-off functions ρ ± ).We say that ρ + , ρ − ∈ C 2 (R, R) are admissible cut-off functions when they satisfy ρ ± (x 1 ) ∈ [0, 1] for every x 1 ∈ R and when the limiting behavior is given by ρ We additionally demand, for some C > 0, the decay properties Remark on cut-off functions.Formally, the radiation condition formulated below depends on the choice of ρ ± .But we will show later on that the solution u of the radiation problem does not depend on the choice of ρ ± .
The requirement ρ ± ∈ C 2 (R, R) can be replaced by ρ ± ∈ C 0,1 (R, R), i.e., Lipschitz continuity of the cut-off functions (we keep the property of the rate of decay).One can argue as follows: The existence result below is performed for cut-off functions of class C 2 .Remark 2 after Theorem 3.12 can provide that the constructed solutions are also solutions for arbitrary cut-off functions of class C 0,1 .Formally, our proof does not cover this case since we demand g ∈ L 2 (Ω) in the uniqueness statement below and therefore need w ∈ H 2 (Ω).In order to resolve this obstacle, one has to use weak solution concepts in all equations to conclude that ρ ± ∈ C 0,1 (R, R) is sufficient.
From now on, we use the spaces Y j and the basis functions φ ,j as chosen in Corollary 3.7.We slightly change notation at this point: We now collect all basis functions φ ,j as a new family with only one index and write (φ ) , where now 1 ≤ ≤ L := J j=1 m j .We recall that we have orthogonality with respect to the hermitean sesqui-linear form E, that is: E(φ , φ ) = δ , E(φ , φ ) for all , .Definition 3.11 (Propagating part and radiation condition).We fix admissible cut-off functions ρ ± as in Definition 3.10.For every ≤ L the mode φ is called right-going when E(φ , φ ) > 0, it is called left-going when E(φ , φ ) < 0. Note that, when E is non-degenerate, these are the only possible cases.For every such that φ is right-going, we set ρ := ρ + , for every for which φ is left-going, we set ρ := ρ − .
(i) Propagating part.For complex coefficients (a ) 1≤ ≤L , we say that is the propagating wave function corresponding to a ∈ C L .
(ii) Radiation condition.We say that a solution u ∈ H 1 loc (Ω) of (1.1) satisfies the radiation condition, when there exists a ∈ C L such that, with the corresponding propagating wave function w, there holds We note that the expression ∆w + k 2 nw has bounded support.This implies g ∈ L 2 * (Ω).The function g depends on the vector of coefficients a ∈ C L .We will construct a ∈ C L such that (3.20) has a solution v ∈ H 1 0 (Ω).We note that, by definition of the radiation condition in Definition 3.11, there is an equivalence of the solution concepts.Existence: When we find a ∈ C L such that (3.20) has a solution v ∈ H 1 0 (Ω), then u = w + v ∈ H 1 loc ( Ω) is a solution of (1.1) with radiation condition.Uniqueness: When u ∈ H 1 loc ( Ω) is a nontrivial solution of (1.1) with radiation condition, then there exists a ∈ C L and a solution v ∈ H 1 0 (Ω) of (3.20) such that a or v are non-trivial.Theorem 3.12 (Existence of radiating solutions).Let S, k, n, and f be as above and let ρ ± be fixed.We demand that Assumption 3.5 is satisfied.Then (1.1) has a unique solution u ∈ H 1 loc ( Ω) satisfying the radiation condition.With w, v, and a from the radiation condition, there holds with C = C(S, k, n, ρ ± ).The coefficients a for ∈ {1, . . ., L} are given by Proof.Existence.We want to determine a ∈ C L in the definition of w such that g of (3.20) satisfies the orthogonality condition (3.17).Using a basis function φ ∈ Y j for some j and extending this basis function to an α j -quasiperiodic function on Ω, we can calculate, using (3.17) in the first equality, We evaluate The scalar product can therefore be evaluated with an integration by parts, with the flux quantity F φ ,φ ,t of (3.11).The flux is independent of t and coincides with 1 2π E(φ , φ ) = 1 2π E(φ , φ ) δ , .We evaluate the right hand side for a rightgoing wave φ , i.e., for ρ with ρ (−∞) = 0 and ρ (+∞) = 1: For a left-going wave φ , ρ (+∞) − ρ (−∞) = −1 introduces a negative pre-factor.We find that the orthogonality condition (3.23) is This condition is identical to (3.22).The above calculation also shows that, choosing (a ) according to (3.22), the orthogonality condition (3.15) is satisfied for g.We can therefore solve for v with Theorem 3.8.With C depending on ρ ± , we have the estimate . This implies (3.21).
Uniqueness.Let u be a solution of the radiation problem with f = 0. Our goal is to show that u vanishes.Theorem 3.8 implies that the right hand side g = −(∆ + k 2 n)w of the equation for v satisfies the orthogonality condition (3.17).The existence part of the proof implies that the coefficients a ∈ C L for which the orthogonality condition is satisfied, are uniquely determined, hence, by f = 0, we conclude a = 0. Together with the uniqueness statement of Theorem 3.8, we find a = 0 and v = 0.This shows that u vanishes.
Remarks. 1.We note that the decomposition of the propagating modes φ into left-going and right-going modes is not needed from the mathematical point of view.Indeed, the proof works also for the case that we decompose {1, . . ., L} into {1, . . ., L} = L + ∪ L − for disjoint sets L ± and set ρ = ρ + for ∈ L + and ρ = ρ − for ∈ L − .
A particular choice would be to use L + := {1, . . ., L} and L − := ∅.With this choice, we impose that no propagating modes (neither left-going nor right-going) can be used on the right, but all propagating modes (not only outgoing / left-going) can be used on the left.
2. Above, we have constructed, for given ρ ± , solutions u = v + w.In order to investigate well-posedness of the radiation condition, let us consider the consequences of choosing another set of admissible cut-off functions, we denote them as ρ± .
We denote the corresponding solutions as u = v + w and ũ = ṽ + w.We write We observe that v − ṽ + ã (ρ − ρ ) φ is in H 1 (Ω).We emphasize that, at this point, we exploited the decay rate of the cut-off functions that was demanded in Definition 3.10.Therefore, u − ũ satisfies not only the homogeneous Helmholtz equation, but also the radiation condition with coefficients (a −ã ) .The uniqueness result of Theorem 3.12 implies that u = ũ and a = ã for all .In this sense, the choice of the cut-off functions has no influence on the solution.
3. The radiation condition depends on the choice of the inner product chosen in Y α .Regarding this point, it is very illustrative to study a simple example.
Example 3.13 (The standard example).In the two-dimensional case, d = 2, we use the cross-section S = (0, π), and the coefficient n ≡ 1, considered as a 2π-periodic function with respect to x 1 .Since we are interested in eigenspaces with dimension larger than 1, we choose a specific wave number k in the following.
The fluxes of φ 1 and φ 2 are We have therefore found a right-going wave φ 1 and a left-going wave φ 2 .
We will continue the above analysis in Example 5.1 where we show that, indeed, different absorption mechanisms can lead to different inner products, hence to different basis functions, and hence to different radiation conditions.

Limiting absorption principles
The boundary condition u η = 0 on ∂Ω remains unchanged.It is well known that this equation is uniquely solvable in H 1 (Ω) for every η > 0. This can be shown with an application of the Lax-Milgram theorem, the positivity of η implies that the bilinear form corresponding to (4.1) is coercive.
We use again the space X = H 1 per (W ) of (2.9) and the equivalence (2.10); the operator L η α ∈ L(X, X) and the element y α ∈ X are defined by Since the operators L η α depend on two parameters, we need the partial derivatives with respect to both parameters.The α-derivative is calculated as in the case η = 0: Taking the derivative of (4.3) with respect to η provides We introduce two operators, essentially given by the two derivatives of L η α .For a given α j ∈ A we consider the kernel N = ker(L 0 α j ) = {φ e −iα j x 1 | φ ∈ Y j }, the operator M η := iP ∂ η L 0 α j | N : N → N and the operator M α := P ∂ α L 0 α j | N : N → N .We note that, by the above formulas, M η is selfadjoint and positive definite (it can be identified with a multiplication with 2kn) and M α := P ∂ α L 0 α j | N is self-adjoint and one-to-one provided E is non-degenerate on Y j .4.2.Functional analysis for two-parameter families.Our aim is now to extend the one-parameter theory of the last section to a theory for two-parameter families.Definition 4.1 (Two-parameter family of operators).We consider a Banach space X and the unit interval I = [−1/2, 1/2] ⊂ R. We say that (L η α ) is a two-parameter family of Fredholm operators when there exists ε > 0 and a C 2 -map α is a Fredholm operator with index 0 and, for every α ∈ I for which L 0 α is not invertible, the operator L := L 0 α has Riesz number 1, ker(L) = ker(L 2 ).
Remarks. 1.We actually need less than the C 2 property of the operator family.The proof works when in L(X, X) the following approximation property holds: Here, the norm is the operator norm in L(X, X). 2. An illustrative example is X = C and L η α = α − iη (this will actually be, for α j = 0, the essential action of L η α on the kernel of L 0 0 ).For the family of right hand sides y η α = 1, we find the solutions (4.6) We observe that u η α has a singularity in (α, η) = (0, 0).This singular behavior was somehow to be expected, since y η α is not vanishing in (α, η) = (0, 0).Let us therefore look at a right hand side that vanishes in the singular point, we investigate y η α = α with y 0 0 = 0.The solution for this right hand side is We observe that the solution is bounded.On the other hand: The solution family is not continuous at (0, 0).Indeed, along the two coordinate axes, we find: u 0 α = 1 for all α and u η 0 = 0 for all η.
The following theorem considers the local situation with only one critical value α.Once more, without loss of generality, we choose the critical point to be α = 0. Theorem 4.2 (Functional analysis II).Let X be a Hilbert space and L η α be a twoparameter family of Fredholm operators in the sense of Definition 4.1.Let I α → y α ∈ X be a family of right hand sides that depends Lipschitz continuously on α ∈ I. Let the following properties be satisfied: In this representation, v η α X is uniformly bounded with respect to (α, η).The family φ | = 1, . . ., m , m = dim N , is an orthonormal eigensystem with eigenvalues λ | = 1, . . ., m of the following generalized eigenvalue problem in the finite dimensional space N : but this operator vanishes because of P L 0 0 = 0. Therefore, there remains only the derivative of the first term:

which is invertible (as seen already in (4.11)).
A theorem like Theorem 3.2 with two parameters (see Remark 3.3) implies that the solution family u N (α, η) is bounded.We note that it cannot be expected that the solution family is continuous, see the example in (4.7).4.3.Application of the functional analysis result.We want to apply Theorem 4.2 to equation (4.2), which we write again in the form L η α u η α = y α for u η α (x) = ûη (x, α)e −iαx 1 .We consider a fixed parameter α j ∈ I for some j ∈ {1, . . ., J} and recall that ker(L 0 α j ) = {φ e −iα j x 1 | φ ∈ Y j } where Y j has been defined in (3.12).Shifting the critical value α = 0 in Theorem 4.2 to α = α j yields the following decomposition.
Proof.In the end of Subsection 4.1 we have obtained characterizations for M α and M η ; they show that the abstract eigenvalue problem (4.9) reduces to the problem to determine λ and φ ∈ ker(L 0 α j ) with E φ e iαx 1 , ϕ e iαx 1 = λ 2 k W n φ φ for all ϕ ∈ ker(L 0 α j ) which coincides with (4.17) when replacing φ e iαx 1 and ϕ e iαx 1 by φ ,j ∈ Y j and ψ ∈ Y j , respectively.Formula (4.8) of Theorem 4.2 (for singularity at α j instead of 0) yields the representation follows from the definition of y α for ϕ(x) = φ ,j (x) e −iα j x 1 .
The inverse Floquet-Bloch transform.With (4.16) we have found an expression for the Floquet-Bloch transform ûη of the solution u η .Using the inverse transform yields an expression for u η .For the subsequent theorem, let ρ ± be two admissible cut-off functions as described in Definition 3.10 Theorem 4.4 (Limiting Absorption Principle).We consider solutions u η ∈ H 1 0 (Ω) of (4.1) for a right hand side f ∈ L 2 * (Ω).Let Assumption 3.5 be satisfied.We use the eigenvalues and eigenfunctions λ ,j and φ ,j of Proposition 4.3.Then, as η tends to zero, u η ∈ H 1 0 (Ω) converge to a solution u ∈ H 1 loc (Ω) of (4.1) with η = 0. Denoting cut-off functions as ρ ,j := ρ sign(λ ,j ) , the limit u can be written as Remark.We will derive the result for a specific pair of cut-off functions, namely, for some suitably chosen ε > 0, We note that the integral term behaves like This implies that the two functions ρ ± have the required properties of cut-off functions of Definition 3.10.
By Remark 2 after Theorem 3.12, the solution u is independent of the choice of the cut-off functions.This implies the following: When we verify that the limit solution u satisfies (4.18) with the cut-off functions of (4.19), then u satisfies (4.18) for every choice of admissible cut-off functions.
We now consider η → 0 in the different terms.
For fixed j ∈ {1, . . ., J}, we next treat the integral α j +ε α j −ε v η j (x, α) dα.The integrand v η j tends to v 0 j in L 2 (α j − ε, α j + ε), H 1 (W ) by Lebesgue's theorem of dominated convergence because v η j (•, α) tends to v 0 j (•, α) in H 1 (W ) for every α = α j and is uniformly bounded with respect to α and η.Again, the boundedness of the inverse Floquet-Bloch transform yields convergence of α j +ε Finally, we consider the integral in the last term for fixed j and .With a parameter transformation, we write the integral as (4.20) In the appendix, see (B.1), we show that, for ρ ± from (4.19), this integral converges to 2πi |λ ,j | ρ sign(λ ,j ) (x 1 ), uniformly with respect to |x 1 | ≤ R for every R > 0. Altogether, we have shown the local convergence of u η to for some v ∈ H 1 (Ω).It remains to note that f (•, α j ), φ ,j L 2 (W ) = f, φ ,j L 2 (Ω) , which was stated and shown in the proof of Lemma 3.9, exploiting the quasiperiodicity of φ ,j .

Alternative damping approaches
With equation (4.1), we have analyzed the LAP for a specific absorption term: k was replaced by k + iη.Other damping mechanisms are also physically relevant, e.g., non-homogeneous damping in the k-part or damping in the elliptic-part.We investigate here the LAP for these alternative damping mechanisms.
Non-homogeneous damping in the k-part.We choose a non-negative real valued function p ∈ L ∞ (Ω) that is 2π-periodic with respect to x 1 and with a positive lower bound, p ≥ p 0 > 0 on Ω.We consider (5.1) with the usual boundary condition u η = 0 on ∂Ω.This is a modification of the homogeneous damping of (4.1).Once more, an application of the Lax-Milgram theorem yields that the equation is uniquely solvable in H 1 (Ω) for every η > 0.
The variational form of the Floquet-Bloch transformed equation is equivalent to L η α u η α = y α for u η α (x) = ûη (x, α)e −iαx 1 where y α is given by (4.4) and L η α by (4.3), with k + iη replaced by k and with the refractive index n replaced by n + iηp.
The operator M η is given by a partial derivative of L η α with respect to η.We calculate it to be Therefore, the eigenvalue problem (4.17) has to be replaced by (5.2) E(φ ,j , ψ) = λ ,j k 2 W p φ ,j ψ for all ψ ∈ Y j .
Non-homogeneous damping in the elliptic part.As a second form of damping we consider, for p ∈ L ∞ (Ω) as above, with the usual boundary condition u η = 0 on ∂Ω.The variational form is to find The theorem by Lax-Milgram yields existence and uniqueness.The periodic form u η α (x) = ûη (x, α)e −iαx 1 of the Floquet-Bloch transform satisfies L η α u η α = y α , where y α is again given by (4.4) and L η α by for u, ϕ ∈ H 1 per (W ).The operator M η is now Therefore, the eigenvalue problem (4.17) has to be replaced by (5.5) E(φ ,j , ψ) = λ ,j W p ∇φ ,j • ∇ψ for all ψ ∈ Y j .
Example 5.1 (The standard example, continued).We continue Example 3.13, where we have found two linearly independent eigenfunctions φ 1 and φ 2 spanning Y α for α = 1/4.The wave φ 1 is right-going and the wave φ 2 is left-going.We now investigate different eigenvalue problems that are generated by different limiting absorption principles.The abstract eigenvalue problem is stated in (4.9), it uses the positive definite operator M η := i P ∂ η L 0 0 | N : N → N , and the selfadjoint operator M α := P ∂ α L 0 0 | N : N → N .For the standard absorption mechanism of (4.1), M η and M α are given, loosely speaking, by a multiplication operator (factor 2kn) and by the form E, respectively.The eigenvalue problem was calculated to be (4.17).For our concrete example, φ 1 and φ 2 are indeed eigenfunctions for this problem.The eigenvalues are (5.6) For a solution u = v + w of the radiation problem, the propagating function w has the form w = a 1 ρ + φ 1 + a 2 ρ − φ 2 .In particular, when ρ ± has support in (−L, L), the function w coincides with a multiple of φ 1 for x 1 ≥ L and with a multiple of φ 2 for x 1 ≤ −L.
Let us now choose a different absorption principle.Referring to (5.2), we consider u, v p = k 2 W p u v dx with some positive function p ∈ L ∞ (W ).The eigenvalue problem (3.14) takes the form E( φ, φ j ) = λ φ, φ j p for j = 1, 2. Making the ansatz φ = a 1 φ 1 + a 2 φ 2 leads to the generalized eigenvalue problem Two normalized orthogonal solutions to this problem are given by two complex vectors (a 1 , a 2 ) and (b 1 , b 2 ).Accordingly, we find new eigenfunctions φ1 = a 1 φ 1 + a 2 φ 2 and φ2 = b 1 φ 1 + b 2 φ 2 .This means that the wave that is outgoing to the right is, e.g., φ1 = a 1 φ 1 + a 2 φ 2 .This function is, for a generic coefficient p, neither a multiple of φ 1 , nor a multiple of φ 2 .The limiting absorption process then provides a radiating solution of the limit problem that uses on the right the function ρ + (a 1 φ 1 + a 2 φ 2 ).It is hence different from the previously obtained limit solution.
We obtain that the radiation condition indeed depends on the choice of the inner product or, in other words, on the damping mechanism.

Two spaces of homogeneous solutions
Let us recall the spaces that were used in the above constructions: The space Y j of (3.12) consists of α j -quasiperiodic homogeneous solutions, We recall that {α j | j = 1, . . ., J} ⊂ [−1/2, 1/2] are the quasi-moments that correspond to nontrivial spaces Y α .In the above formula, we identified H 1 α j (W ) with H 1 α j (Ω); the canonical identification is given by the α j -quasiperiodic extension of a function in H 1 α j (W ) (and, vice versa, the restriction to a function on W ). We furthermore introduced in (3.13) the space (6.1) It has a basis {φ | = 1, . . ., L} with orthogonality E(φ , φ ) = 0 for = .
Let us consider another space, the space B of bounded solutions.That space was extensively used in [20] (where it was named X).In order to impose a boundedness property, we introduce the norm , where W := (2π , 2π + 2π) × S. The space of bounded homogeneous solutions is defined as (6.2) It is clear that every quasiperiodic homogeneous solutions is a bounded homogeneous solution, hence Y ⊂ B. Our aim is to show that the spaces Y and B actually coincide.
Before we formulate the corresponding result, we note that an equivalent norm is obtained when we measure the H 1 -norm in every cell.Lemma 6.1 (Equivalent norms).There exists a constant C > 0 such that Proof.The lemma follows from Caccioppoli's inequality for solutions of elliptic problems.
We can now give the characterization of B. Theorem 6.2 (Every bounded homogeneous solution is a linear combination of quasiperiodic homogeneous solutions).When Assumption 3.5 holds, then the spaces Y of (6.1) and B of (6.2) coincide, The proof is given in the next subsection.We provide the proof in a more abstract setting such that it covers, e.g., compact perturbations of periodic media.If the reader wants to see the proof of Theorem 6.2 immediately: It is possible to jump to the proof of Theorem 6.5 and to read it as a proof of Theorem 6.2.6.1.A generalized setting.We write A for the underlying selfadjoint differential operator of second order, defined on some domain Ω ⊂ R d = R × R d−1 .In the main part of this text, we treat A = −∆ − k 2 n.By contrast, the next result holds also for compact perturbations of this operator, for example A = −∆ − k 2 (n + q) where q has bounded support, or A = −∇ • ((I + Q)∇) − k 2 where I is the identity and Q has bounded support.We always assume that the operator is everywhere uniformly elliptic.The domain Ω is assumed to be cylindrical outside a compact set: For some bounded set S ⊂ R d−1 and some M > 0 there holds Ω We always assume that the coefficients are 2πperiodic in x 1 in the cylindrical parts, more precisely: We assume that there exists a selfadjoint operator Â of second order in R × S with 2π-periodic coefficients (in The space in which we look for solutions is H 1 loc ( Ω).We consider the space B corresponding to the elliptic operator A, here defined with the norm U sH := sup ∈Z U H 1 (W ) : We emphasize that, due to the equivalence of norms of Lemma 6.1, in the setting of the last subsection, the definition of B was not changed with respect to (6.2).
In the following, we assume that cut-off functions ρ ± ∈ C 2 (R) with ρ ± (x 1 ) = 1 for ±x 1 ≥ 1 and ρ ± (x 1 ) = 0 for ±x 1 ≤ −1 are chosen.Let {φ | = 1, . . ., L} be quasiperiodic homogeneous solutions to the unperturbed operator Â in R × S with homogeneous Dirichlet conditions on R × ∂S.For two disjoint sets L + and L − with L + ∪ L − = {1, . . ., L}, we set ρ = ρ + for ∈ L + and ρ = ρ − for ∈ L − .Assumption 6.3 (An abstract existence and uniqueness result).We assume the following on the operator A. For every right hand side f ∈ L 2 * (Ω), there exist uniquely determined functions v ∈ H 1 0 (Ω) and w We note that Assumption 6.3 is verified in the standard setting of this contribution: For A = −∆ − k 2 n on the domain Ω = R × S with S ⊂ R d−1 a bounded Lipschitz domain, Assumption 3.5 implies Assumption 6.3.This is shown in Theorem 3.12.
For cylindrical domains and periodic coefficients, the space Y is defined in (3.13).When we treat compact perturbations of this setting (as described above), we have to define the space Y in a different way.We construct as follows: Let θ ∈ C 2 (R) be any function with θ(x 1 ) = 1 for |x 1 | ≥ M + 1 and θ(x 1 ) = 0 for |x 1 | ≤ M .For fixed ∈ {1, . . ., L}, we define the incident field u inc (x) := θ(x 1 )φ (x) and seek for a solution φ t (total field) of Aφ t = 0 in the form φ t = u inc + φ s ; here φ s is the scattered field, which has to satisfy the radiation condition.Assumption 6.3 allows to solve for u = φ s , since Aφ s = f := −A(θφ ) has compact support.Performing the construction of φ t for every , we can define (6.5) Y := span φ t | = 1, . . ., L .
The following lemma provides that the dimension of Y is L. Lemma 6.4 (Dimension of Y in compactly perturbed setting).The total fields (φ t ) 1≤ ≤L are linearly independent, there holds dim Y = L.
Proof.Let c φ t ≡ 0 be a linear combination of the trivial function.We can consider the incident field u inc := c θ φ and solve for the corresponding total field u t : By linearity of the equation, we find u t := c φ t ≡ 0 with the scattered field u s := c φ s satisfying 0 = u t = u inc + u s .On this basis, the principle argument is simple: Up to a H 1 0 (Ω)-function v , each function φ s is a linear combination of the outgoing fields, φ s = v + a , ρ φ , hence also u s is essentially a linear combination of the outgoing fields.On the other hand, u inc = c θ φ contains each field with a factor c .Let us study ∈ L − and a large (positive) position x 1 : In the left hand side of −u inc = u s , the pre-factor of φ is c , in the right hand side, it is vanishing.This shows c = 0. Similarly, one argues for ∈ L + by considering positions x 1 < 0.
We  Since the φ are linearly independent, we obtain ∈L − c e iγ φ = 0 and hence c = 0 for ∈ L − .Analogously, for m → −∞ we conclude that c = 0 for ∈ L + .
Step 3: Conclusion.It remains to insert the representation (6.6) of a into (6.9).We find (6.10) Since f was arbitrary, we find for all x ∈ Ω.We have therefore represented an arbitrary element U ∈ B with the L functions ξ (x) (1 + |x 1 | 2 ) 2 .This implies dim B ≤ L and hence the theorem.
6.2.Finite dimension of B in other settings.We return here to the geometry of the main part of this paper, Ω = R × S with S bounded.We note that the space B can be defined for any (positive) refractive index n ∈ L ∞ (Ω) without the assumption of periodicity.We ask: Does B have a finite dimension?We do not know the answer in the general case.
One particular case can be treated with the above methods.When n ∈ L ∞ (Ω) coincides with a periodic function n + for x 1 ≥ M and with another periodic function n − for x 1 ≤ −M (for some M > 0), then B can be characterization much as in the previous subsection: B is spanned by the solutions of scattering problems with incident fields φ ± (the right-going modes for index n − ) and φ ∓ (the left-going modes for index n + ).In particular, in this case, B is finite dimensional.
Another case that allows to show finite dimensionality of B is the following: Let n ∈ L ∞ (Ω) be of the form n(x 1 , x) = n 1 (x 1 ) + n 2 (x) for x 1 ∈ R and x ∈ S. In this case, we can use separation of variables techniques.Let λ j ∈ R and φ j ∈ H 2 (S) be the eigenvalues and eigenfunctions, respectively, of the selfadjoint operator − ∆ − k 2 n 2 , that is, − ∆φ j (x) − k 2 n 2 (x) φ j (x) = λ j φ j (x) in S , φ j (x) = 0 on ∂S .and denote the -th Fourier coefficient of û(x, •) by c ∈ C such that, for almost every x, there holds û(x, α) = ∈Z c e −i 2πα .We consider such a point x ∈ W and evaluate F FB (u) for u given by (A.This shows, in particular, that F FB : L 2 (R) → L 2 (W × I) is surjective.We conclude that F FB is an isometry and that the inverse is given by (A.3).

Appendix B. Evaluation of a complex integral
This appendix deals with an integral that appears in an inverse Floquet-Bloch transformation, see (4.20).For the following calculations, ε > 0 is an arbitrary number.We calculate In the limit η → 0, we therefore find, for this term, The convergence is uniform with respect to |x 1 | ≤ R for every R > 0.

Lemma 3 . 6 (
Regularity of the Floquet-Bloch transformed equation).Let L α be the C 1 -family of operators constructed in (2.12) and let Assumption 3.5 hold.Then L α is a regular C 1 -family of operators in the sense of Definition 3.1.

Definition 3 .
11 allows to show an existence result with our previously developed methods: We solve the radiation problem (1.1) by constructing v = u − w ∈ H 1 0 (Ω) with Theorem 3.8.We can write the equation for v as (3.20) −∆v − k 2 nv = g := f + (∆w + k 2 nw) .

Theorem 6 . 5
(Y = B in the abstract setting).When the existence and uniqueness property of Assumption 6.3 holds, then Y = B.Proof.The inclusionY ⊂ B is clear.We know that Y has dimension dim Y = L.In order to show B ⊂ Y , it suffices to show dim B ≤ L.In this proof we use, for arbitrary R > M , the piecewise affine cut-off functionϑ R : R → [0, 1] with ϑ R (s) = 1 for every s ∈ [−R, R], ϑ R (s) = 0 for |s| ≥ R + 1, affine on [−R − 1, −R] and on [R, R + 1].We interpret ϑ R also as a function on Ω by setting ϑ R (x) := ϑ R (x 1 ).