Scattering by a periodic tube in : part ii. A radiation condition

This second part of a pair of papers complements the first part (see Kirsch 2018 ( 35 104004)) but can be read independently. 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. As in the first part we consider a medium described by a refractive index which is periodic along the axis of an infinite cylinder in R 3 and constant outside of the cylinder. We formulate a proper radiation condition which allows the existence of traveling modes (and is motivated by the limiting absorption principle proven in the first part) and prove uniqueness and existence. Kirsch by a in R 3 part ii. A radiation condition in 10.1088/1361-6420/ab2e27


Introduction
This part continues the first part (see [2]) but can be read independently of it. While in the first part the limiting absorption principle for the scattering by a tube in R 3 filled with a periodic refractive index was proven and the corresponding radiation condition was derived we now take a different point of view and assume the radiation condition as given. With this radiation condition, formulated for real and positive wave numbers k > 0, we prove uniqueness and existence of a solution. The limiting absorption principle is not used in this part but serves only as a motivation for the definition of the radiation condition. Indeed, from the purely mathematical point of view one can replace the radiation condition by several others which also (by essentially the same proof) yields uniqueness and existence of a solution, see remarks 3.4 and 4.6 below.

Formulation of the problem
We begin by setting up some notations (see figure 1). Let k ∈ R with k > 0 be the wave number which is kept fixed throughout the paper. Let B N (0, R) = {x ∈ R N : |x| < R} be the ball in R N with center 0 and radius R > 0, and T R = B 2 (0, R) × R ⊂ R 3 be the tube (or infinite cylinder) in x 3 − direction. 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 . Furthermore, we assume that T R is filled with 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 for some R 0 > 0. Finally, let f ∈ L 2 (R 3 ) be given with support contained in T R0 . The problem is to determine u ∈ H 1 loc (R 3 ) with ∆u + k 2 n u = −f in R 3 . (1) The solution is understood in the variational sense; that is, we search for u ∈ H 1 loc (R 3 ) such that Without a radiation condition the solution is not expected to be unique. In [2] we constructed the so called limiting absorption solution of the problem; that is, the limit of the solutions u ε ∈ H 1 (R 3 ) of the coercive problems ∆u ε + (k + iε) 2 n u ε = f when ε > 0 tends to zero. The structure of the limiting absorption solution motivates the radiation condition below (definition 2.5). Its formulation needs some preparation.
Therefore, the α−quasi-periodic source problems are to determine u α ∈ H 1 α,loc (C ∞ ) such that ∆u α + k 2 n u α = −f α in C ∞ (2) in the variational sense; that is, is some given function with compact support in C R0 . For the α−quasi-periodic problem (2) a natural radiation condition is the extension of the classical Rayleigh expansion to our case; that is, the requirement that u α has an expansion of the form (3) for some R 1 > R 0 and a ,j ∈ C. 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 . This condition can equivalently be replaced by a one-dimensional radiation condition for the Fourier coefficients.
Proof. It is obvious that (a) implies (b). Indeed, if u has a Rayleigh expansion of the form (3) then the Fourier coefficients are given by u ,m (r) = 2π a ,m which satisfy (4) by the asymptotic behaviour of the Hankel functions as r tends to infinity.
Let now u ,m (r) satisfy (4). The fact that u satisfies the Helmholtz equation implies that the Fourier coefficients satisfy Bessel's differential equation The general solution is given by u ,m (r) = c ,m H (1) m (k r) for some coefficients c ,m , d ,m . Condition (4) implies d ,m = 0 which proves the assertion. □ But even with this Rayleigh expansion the solutions of (2) are not always unique. As the case of constant n explicitly shows there might exist parameters α ∈ [−1/2, 1/2] for which non-trivial quasi-periodic solutions of (2) for f = 0 exist. These parameters are called exceptional values. We define the set A = α ∈ [−1/2, 1/2] : there exists ∈ Z with |α + | = k of cut-off values (note that A consists of one or two elements) and make the following assumption.

Assumption 2.2.
For every α ∈ A the only α−quasi-periodic solution u ∈ H 1 α (C ∞ ) of (2) for f = 0 which satisfies the Rayleigh expansion (3) has to be the trivial one. In other words, no α ∈ A is an exceptional value.
The following can be shown (see, e.g. lemma 2.9 of [2]). Lemma 2.3. Let assumption 2.2 hold. Then there exist only finitely many exceptional values α ∈ [−1/2, 1/2]. Furthermore, if α is an exceptional value then also −α. Therefore, the set of exceptional values can be described by α j : j ∈ J where J ⊂ Z is finite and symmetric with respect to the origin and α −j = −α j for j ∈ J. The corresponding eigenspaceŝ are finite dimensional. Furthermore, the expansion coefficients a ,m in (3) of any eigenfunction φ ∈ X j vanish for all | +α j | k. This implies that every eigenfunction φ ∈X j is evanescent; We now choose a special basis in X j which is justified by the limiting absorption principle (see part A, [2]). In every X j we consider the m j − dimensional self-adjoint eigenvalue problem to determine λ ∈ R and φ ∈X j with  We denote the eigenvalues and eigenfunctions by λ ,j and φ ,j , respectively; that is, and every j ∈ J. We normalize the eigenfunctions φ ,j : = 1, . . . , m j such that 2k C∞ nφ ,jφ ,j dx = δ , for all , .
We make a second assumption and assume that the wave number k is regular in the following sense.
Then, for every j ∈ J we can split the propagating modes φ ,j in those with λ ,j > 0 and those with λ ,j < 0. These describe the modes which travel upwards and downwards, respectively. The radiation condition, formulated below in definition 2.5, consists of two parts. The first part (see part (a) of definition 2.5) describes the behavior along the axis of the cylinder while the second part (part (b) of definition 2.5) describes the behavior orthogonal to the cylinder. The second part is formulated in terms of the Fourier transform Fg : Z × R → C of g with respect to cylindrical coordinates which is given by 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 (9) In the formulation of the radiation condition we separate the propagating modes which travel upwards or downwards. This separation is formulated by auxiliary functions ψ ± ∈ C ∞ (R) with the properties Here, the constant c > 0 is independent on t. In particular, ψ + (t) tends to zero as t tends to −∞ while it tends to 1 as t tends to +∞. The function ψ − behaves analogously.

Definition 2.5 (Radiation condition).
Let assumption 2.2 hold and let k > 0 be regular in the sense of definition 2.4 and let ψ ± ∈ C ∞ (R) be given with the properties (10). The solution u ∈ H 2 loc (R 3 ) of (1) has a decomposition in the form u = u (1) + u (2) where: for some a ,j ∈ C. Here, λ ,j ,φ ,j : = 1, . . . , m j are the eigenvalues and eigenfunctions, respectively, of the eigenvalue problem (7).

Remarks 2.6.
(a) From (11) we observe that for The asymptotic behaviour of u (2) is not changed by choosing different functions ψ ± because for any functions ψ ± 1 and ψ ± 2 with (10) it holds that ψ ± 1 − ψ ± 2 ∈ H 1 (R). Therefore, the difference is subsumed in u (1) . (c) In part A we have shown that the limiting absorption solution satisfies this radiation condition. The coefficients a ,j are explicitly given by

Uniqueness
If we interpret Im γr u ∂u ∂ν ds as an energy flow along the axis of the tube then the energies of the guided modes are constant and positive as the following lemma shows. In the case of a closed waveguide; that is, posing the boundary condition u = 0 or ∂u/∂r on ∂T R , the following lemma implies almost directly uniqueness of the solution. We were not able to adjust the proof to the open waveguide problem but prove uniqueness in a different way (see below). The result of this lemma is, however, interesting in itself.
Lemma 3.1. Let γ r = R 2 × {r} for r ∈ R and u ± = j∈J λ ,j ≷0 a ,jφ ,j for some a ,j ∈ C. Then, for every r ∈ R and σ ∈ {+, −}, Therefore, the last integral vanishes for j = j . Thus we have Setting which proves the equality. Furthermore, by the definiton of φ ,j . Taking the imaginary part yields the assertion. □ The relationship between the original source problem (1) and the α−quasi-periodic problems (2) is given by the Floquet-Bloch transform F which is defined as From the definition we directly observe that for smooth functions f and fixed where we extended h(·, α) to a α−quasiperiodic function in R. Furthermore, the restriction of 0, α) for all α and j = 0, 1 with respect to the norm be the one dimensional Fourier transform which can be expressed by the Floquet-Bloch transform as where (Fg) (α) are the Fourier coefficients of (Fg)(·, α), ∈ Z. Therefore, In the following we use the same symbol F also for the Floquet-Bloch transform with respect to the variable x 3 of functions on R 3 ; that is, where ê (3) = (0, 0, 1) . Then the analogous of (15) and (16) are given by for r > 0, m, ∈ Z, and α ∈ (−1/2, 1/2]. Here, r, ϕ, x 3 are the cylindrical coordinates of x ∈ R 3 and (Ff ) ,m (r, α) are the Fourier coefficients of (Ff )(r, ·, ·, α). From part (a) of the radiation condition we observe that u (1) satisfies the differential equa- From the properties (10) we observe that h decays as 1/|x 3 | as |x 3 | tends to infinity. Therefore, the Floquet-Bloch transform (Fh)(x, α) is well defined for all α ∈ R. The following lemma computes it for the terms in the sum.

Theorem 3.3. Let assumption 2.2 hold and let k > 0 be regular in the sense of definition 2.4. Then there exist at most one solution u ∈ H 2 loc (R 3 ) of the source problem (1) satisfying the radiation condition of definition 2.5.
Proof. Let u ∈ H 2 loc (R 3 ) be a solution of the source problem (1) corresponding to f = 0 which satisfies the radiation condition. We recall that u (1) satisfies the differential equation ∆u (1) + k 2 nu (1) = −h where h is given by (19). Taking the Floquet-Bloch transform and using the previous lemma yields ∆(Fu (1) )(x, α) + k 2 n(x)(Fu (1) and ρ from lemma 3.2. We note that w(·, α) is α−quasi-periodic. Now we set v(x, α) = (Fu (1) )(x, α) + w(x, α) for x ∈ C ∞ and almost all α ∈ R. Then we observe that v(·, α) is α−quasi-periodic and ∆v(·, α) + k 2 n v(·, α) = 0 in C ∞ for almost all α ∈ R. Next we show that v(·, α) satisfies a Rayleigh expansion for x / ∈ C R. By lemma 2.1 it is sufficient to show that the Fourier coefficients of v(·, α) satisfy the one dimensional radiation condition. This is clear for the Fourier coefficients of w because of the exponential decay of φ ,j (x) as r tends to infinity. The Fourier coefficients û ,m (r, α) of (Fu (1) )(·, α) are given by (18); that is, and this satisfies the radiation condition of definition 2.5, part (b), by assumption. By lemma 2.1 this is equivalent to the Rayleigh expansion. This holds for almost all α ∈ [−1/2, 1/2]. The trivial uniqueness result for the α−quasi-periodic scattering problem at non-exceptional wave numbers implies that v(·, α) vanishes in C ∞ for almost all α. Thus, (Fu (1) )(·, α) = −w(·, α) in C ∞ for almost all α ∈ (−1/2, 1/2]. Now fix any j 0 ∈ J and choose a small open interval I such that α j0 ∈ I and α j / ∈ I for j = j 0 . Then, for almost all α ∈ I ,

This equation has the form
for some g which is in L 2 CR × I for every R > R. Therefore, The left-hand side is integrable over I in contrast to the right hand side unless the sum vanishes identically in CR. From the linear independence of φ ,j0 : = 1, . . . , m j0 we conclude that all of the coefficients a ,j0 vanish. This holds for all j 0 ; that is, u (2) vanishes identically. It remains to show that u (1) vanishes. Then u = u (1) solves ∆u + k 2 nu = 0 in R 3 and u ∈ H 1 (TR) for all R > R. Taking the Floquet-Bloch transform yields that Fu satisfies the Rayleigh expansion and ∆(Fu) + k 2 n(Fu) = 0 in C ∞ for almost all α ∈ (−1/2, 1/2). Since the set of exceptional is finite by lemma 2.3 we conclude that Fu vanishes for almost all α and thus also u = 0 almost everywhere. □ Remark 3.4. In the radiation condition the signs of λ ,j determine whether the corresponding propagating mode travels to x 3 → +∞ or to x 3 → −∞. From the proof we note that this particular decomposition {1, . . . , m j } = { : λ ,j > 0} ∪ { : λ ,j < 0}-which is justified by the limiting absorption principle-is not necessary. Any prescribed decomposi- j and L (2) j would also provide uniqueness.

Existence
In the first part [2] we have shown existence indirectly by the limiting absorption principle. It is the aim to present a direct proof of existence which is solely based on the radiation condition.
The radiation condition suggests that we search for the solution u of ∆u + k 2 nu = −f in the form u = u (1) + u (2) where u (1) ∈ H 1 (TR) for every R > R and u (2) ∈ W 2,∞ (R 3 ) is given by for a ,j ∈ C given by (13). Here, λ ,j ,φ ,j : = 1, . . . , m j are the eigenvalues and eigenfunctions, respectively, of the eigenvalue problem (7) for every j ∈ J. Furthermore, we choose t −∞ e −s 2 /2 ds, t ∈ R, and ψ − = 1 − ψ + . We set again h = ∆u (2) + k 2 nu (2) in R 3 . Then u (1) has to solve ∆u (1) Furthermore, since f and also h are in L 2 (R 3 ) (for h this follows from the form (19) and the decay of dψ + /dt and d 2 ψ + /dt 2 ) we can take the Floquet-Bloch transforms. Because of the exponential decay of f and h as |x 3 | → ∞ we note that Ff and Fh are continuous with respect to α. Therefore, it is the aim to solve for every α ∈ [−1/2, 1/2]. Assume for the moment that there exists a solution u α ∈ H 1 α,loc (C ∞ ) of (23) for every α ∈ [−1/2, 1/2] which satisfies also the radiation condition (4) such that for every R > R and u = u (1) + u (2) satisfies ∆u + k 2 nu = −f in R 3 and the radiation condition (12) by (17). Therefore, we have to study (23) and (4) with respect to solvability and continuous dependence on α.
In the first part we reduce the problem (23) and (4) to an operator equation of the form with a compact operator K α and right hand side r α ∈ H 1 per (C R ) which depend continuously on α (see lemma 4.2 below). Here, H 1 per (C R ) denotes the subspace of H 1 (C R ) consisting of 2π−periodic (wrt x 3 ) functions. The reduction to this equation on the bounded domain C R is not quite standard because the part (Fh)(·, α) in (23) does not vanish outside of any CR-in contrast to (Ff )(·, α) which vanishes outside of C R .
The equation (24) is singular in the sense of Colton and Kress (section 1.4 of [1]) because it is uniquely solvable for all α which are not exceptional. For α ∈ {α j : j ∈ J}, however, the kernel of I − K α is not trivial. We will apply a theorem from [1] (see theorem A.1 of the appendix) which proves that the mapping α →ũ α can be continuously extended to the whole interval [−1/2, 1/2]. Therefore, the inverse Floquet-Bloch transform yields that u (1) ∈ H 1 (TR) for any R > R and provides the solution u = u (1) + u (2) .
For the reduction of (23) to an equation of the type (24) we need to investigate the α−quasi-periodic problem (2) for right hand sides f α which do not vanish for large values of r but decay sufficiently large as r → ∞. We recall that in our case f α = (Ff )(·, α) + (Fh)(·, α). We define the weighted space ( This space is equipped with the canonical norm f L 2 σ (C∞) = f σ L 2 (C∞) . The spaces L 2 σ (C ∞ \ C R ) for R > 0 are defined analogously.
To reduce this problem (25) and (4) to a boundary value problem on the bounded tube C R we consider first the analog of problem (35) of Part A and solve the boundary value problem in the exterior of C R explicitly.
satisfying the one-dimensional radiation condition (4) for all , m ∈ Z. Here, f ,m (ρ) and g ,m are the Fourier coefficients of f α (ρ, ·, ·) and g α , respectively, and G α is given by (see (38) for r, ρ R and m, ∈ Z where r + = max{r, ρ} and r − = min{r, ρ}. Set A = α ∈ [−1/2, 1/2] : there exists ∈ Z with |α + | = k . Then the mapping α → v is continuous on [−1/2, 1/2] and continuously differentiable on [−1/2, 1/2] \ A as a mapping into H 1 α (CR \ C R ) for every R > R. The proof uses the same arguments as the proof of theorem 4.1 of part A and is omitted. For f α = 0 this theorem provides the Dirichlet-to-Neumann map Λ α : (27) and (4) with g α = 0 on γ R . Existence of v α is again assured by theorem 4.1. Then (25) and (4) is equivalent to the following α− quasi-periodic boundary value problem (see lemma 4.2 below): The variational form of this boundary value problem is to find for all ψ ∈ H 1 α (C R ). Later we will study the dependence on α. Therefore, it is convenient to eliminate the dependence of the solution space on α by replacing the α−quasi-periodic function u α by ũ α (x) = exp(−iαx 3 )u α (x) and, analogously for f α , v α , and the test functions ψ. We indicate the periodic functions by using the tilde sign on top of the symbol. Therefore, we search for Here, denotes the corresponding periodic Dirichlet-Neumann operator which is bounded from H 2 and g ,m denote the Fourier coefficients of the periodic function g. As in part A we write this variational equation as defines an inner product in H 1 per (C R ) which is equivalent to the ordinary inner product and Here, Λ 0,i denotes the operator Λ α for α = 0 and k = i. (27), (4) with g α = 0 on γ R . Furthermore, by the representation theorem of Riesz there exists a unique operator K α from H 1 per (C R ) into itself and r α ∈ H 1 per (C R ) such that K α u, ψ * = a α (u, ψ) and r α , ψ * = α (ψ) for all u, ψ ∈ H 1 per (C R ). Therefore, (25) and (4) is equivalent to the equation (24). Indeed, as shown as for lemma 2.12 of part A (i.e. [2]) we have the following result.
Proof. Let first u α ∈ H 1 α,loc (C ∞ ) satisfy (25) and (4). We write u and f instead of u α and f α , respectively, for brevity. If α is not an exceptional value then condition (35) is trivially satisfied because there are no such nontrivial φ. Therefore, let α be an exceptional value and φ ∈ H 1 α (C ∞ ) be a solution of ∆φ + k 2 nφ = 0 in C ∞ which satisfies the Rayleigh expansion. By lemma 2.3 φ is evanescent. Let φ ,m (r) and u ,m (r), , m ∈ Z, be the Fourier coefficients of φ and u, respectively. We split the integral C∞ f φ dx into the sum with w from (21) for α =α j . Then, for any ∈ {1, . . . , m j }, C∞ (Ff )(·,α j ) + (Fh)(·,α j ) φ ,j dx = C∞ (Ff )(·,α j ) + (∆ + k 2 n)w(·,α j ) φ ,j dx + mj =1 sign(λ ,j ) a ,j 1 π C∞ ∂φ ,j ∂x 3φ ,j dx = C∞ (Ff )(·,α j )φ ,j dx + sign(λ ,j ) a ,j 1 2π f (y)φ ,j (y) dy + sign(λ ,j ) a ,j 1 2π i λ ,j = 0 by the definition of a ,j . Here we have used Green's second theorem in C ∞ and the normalization of the eigenfunctions φ ,j (see (7) and (8)). Therefore, the equation (23) has a radiating solution for all α. Therefore, we have shown that the operator equation (34) is solvable in H 1 per (C R ) for all α ∈ [−1/2, 1/2]. The smoothness properties of α → K α and α → r α are shown in lemma 4.3. In the same lemma it has been shown that the Riesz number of I − Kα j is one. Therefore, all of the assumptions of theorem A.1 of the appendix has been shown except of the injectivity of the projected operator PK αj N : N → N where N = N (I − Kα j ) and P : H 1 per (C R ) → N is the projection with respect to the direct sum H 1 per (C R ) = N ⊕ R(I − Kα j ). We fix α j for some j ∈ J and let α be in a small neighborhood of α j . Let u, ψ ∈ N I − Kα j . We recall that where k (α) = k 2 − | + α| 2 and ψ ,m and u ,m are the Fourier coefficients of the periodic functions ψ| γR and u| γR , respectively. We extend the α−quasi-periodic functions e iαx3 u(x) and e iαx3 ψ(x) as in lemma 4.2 into all of C ∞ . Then, by Green's formula,