Evolution of time-harmonic electromagnetic and acoustic waves along waveguides

We study time-harmonic electromagnetic and acoustic waveguides, modeled by an infinite cylinder with a non-smooth cross section. We introduce an infinitesimal generator for the wave evolution along the cylinder, and prove estimates of the functional calculi of these first order non-self adjoint differential operators with non-smooth coefficients. Applying our new functional calculus, we obtain a one-to-one correspondence between polynomially bounded time-harmonic waves and functions in appropriate spectral subspaces.


Introduction
A linear partial differential equation, PDE, or a system of PDEs, is often analyzed by studying the evolution of solutions u with respect to one of the variables, say t. In this way the PDE becomes a vector-valued ordinary differential equation, ODE, like (1) ∂ t u(t, x) + T u(t, x) = 0 in the homogeneous case. We assume here that our PDE is of first order. If it is of a higher order, we first rewrite it as a system of first order equations. Here T , an infinitesimal generator, is a first order differential operator acting in the remaining variables x only, for each fixed t. Formally solutions to (1) are given by (2) u(t, x) = (exp(−tT )u(0, ·))(x).
However, as T is an unbounded operator, we need to be careful in the definition and analysis of such a solution operator exp(−tT ). The heuristics are as follows. For a parabolic equation, say the heat equation, T is the positive Laplace operator and exp(−tT ) is a well defined bounded operator for any t ≥ 0 and any initial function. For a hyperbolic equation, say the wave equation as a first order system, T is skew symmetric and exp(−tT ) is unitary and well defined for any −∞ < t < ∞ and any initial function. For an elliptic equation, say the Cauchy-Riemann system, T is symmetric but with spectrum running from −∞ to +∞. In this case we need to split the function space for initial data as a direct sum of two Hardy subspaces. Then exp(−tT ) is well defined and bounded for t > 0 when the initial data is in one of the Hardy subspaces, and for t < 0 when initial data is in the other Hardy subspace.
The aim of the present paper is to study infinitesimal generators T arising as above in the elliptic case. Our motivation comes from the theory for waveguides, and our results yield a powerful mathematical representation of time-harmonic waves propagating along waveguides with general non-smooth materials. The waveguide is modeled by the unbounded region R × Ω, where Ω is a bounded domain in R 2 , or more generally in R n . Note that we study time-harmonic waves. Therefore the PDE is elliptic rather than hyperbolic, and t is not time but rather the spatial variable along the waveguide. For an acoustic waveguide, the PDE is of Helmholtz type, as in Section 2.1, with coefficients which we allow to vary non-smoothly over the cross section Ω, but they are homogeneous along the waveguide. For an electromagnetic waveguide, the system of PDEs is Maxwell's equations as we describe in Section 2.2.
We show in Section 2 that the infinitesimal generators T arising in this way when studying waveguide propagation are of the form where D 1 is a self-adjoint first-order differential operator, D 0 is a normal bounded multiplication operator and B is a bounded accretive operator depending on the material properties of the cross section of the waveguide. With such variable coefficients, the operator T will not be self-adjoint. Even in the static case D 0 = 0, T is only a bi-sectoral operator (see [3]) and L 2 (Ω) bounds of exp(−tT ), and more general functions f (T ) of T , is a non-trivial matter. However, in the general non-smooth case, this is well understood from the works of Axelsson, Keith and McIntosh [4] and Auscher, Axelsson and McIntosh [5]. In the present paper we extend these results to the case D 0 = 0 which occurs for example in general time-harmonic, but non-static, wave propagation in waveguides.
To have a theory for general frequencies of oscillation, encoded by the zero-order term D 0 , it is essential to require the cross section Ω to be bounded, which ensures that the spectrum is discrete. However, the compactness of resolvents and the discreteness of spectrum only holds for T in the range of D 1 + D 0 , which is invariant under T . Building on fundamental quadratic estimates (see [8]) for operators T in the static case, we are able to construct and prove L 2 (Ω) estimates of a generalised Riesz-Dunford functional calculus of T . To yield a well defined and bounded operator f (T ), the symbol f (z) is required to be uniformly bounded and holomorphic on an open neighbourhood of the spectrum of T except at ∞, where it is only required to be bounded and holomorphic on a bi-sector |y| ≤ tan ω|x|, ω < π/2, in a neighbourhood of ∞. Due to the deep quadratic estimates from harmonic analysis used in Proposition 3.16, this suffices to bound f (T ) at ∞.
Another novelty in estimating f (T ), due to the non-self adjointness of T , is that f (T ) may depend not only on |f (λ)|, but also on a finite number of derivatives f (k) (λ) at a given eigenvalue λ of T . In particular, an eigenvalue of T on the imaginary axis with index/algebraic multiplicity greater than 1, will result in propagating waves u t = exp(−tT )u 0 which grow polynomially.
Note that since the spectrum is discrete, a symbol like for t > 0, is admissible provided no eigenvalue lies on Rez = a, and will yield an operator bounded on L 2 (Ω). In this sense the functional calculus that we here construct is more general than that considered by Morris in [1].
In the final Section 4, we apply our new functional calculus for operators T to show how all polynomially bounded time-harmonic waves in the semi-or bi-infinite waveguide can be represented like (2), with u 0 in appropriate spectral subspace for T .
2. Partial differential equations expressed as vector-valued ordinary differential equation In this section we consider Helmholtz and Maxwell's equations and express them as vector-valued ordinary differential equations in terms of operator DB, which is introduced later.
Throughout this paper Ω = Ω + ⊂ R n denotes bounded open set, separated from the exterior domain Ω − = R n \ Ω by weakly Lipschitz interface Γ = ∂Ω, defined as follows.
Definition 2.1. The interface Γ is weakly Lipschitz if, for all y ∈ Γ, there exists a neighbourhood V y ∋ y and a global bilipschitz map ρ y : R n → R n such that were R n + = R n−1 × (0, +∞) and R n − = R n−1 × (−∞, 0). In this case Ω is called a weakly Lipschitz domain.

Helmholtz equation.
Let Ω ⊂ R n be a bounded weakly Lipschitz domain and A ∈ L ∞ (Ω; L (C n+2 )) be t-independent and pointwise strictly accretive in the sense that there exist α > 0 such that for all x ∈ R n and v ∈ C n+2 . For complex number k = 0, we consider an equation in Ω × R with u ∈ H 1 0 (Ω) for all t ∈ R. Let us set H div (Ω; C n ) := {f ∈ L 2 (Ω; C n ) : divf ∈ L 2 (Ω)} and define divergence and gradient operators, respect to an argument x, with domains H div (Ω) and H 1 0 (Ω) by div and ∇ 0 respectively. Splitting C n+2 in to C and C n+1 , we decompose the matrix A(x) in the following way Then we can write the equation (5) in form Next define f as Since A is pointwise strictly accretive, all diagonal blocks are pointwise strictly accretive, and consequently invertible. In particular, A ⊥⊥ is invertible. Hence, due to (7), we obtain . Therefore we can write equation (6) in On the other hand, from definition of f , we obtain which, together with (8), give us the system of equations In vector notation, we equivalently have together with the constraint f ∈ R(D) for each fixed t ∈ R. Since A is a pointwise strictly accretive operator, in [ [5], Proposition 3.2] it was noted that B is a strictly accretive multiplication operator just like A.
By the above arguments, the equation (5) for u implies that f , defined above, solves the equation (9). Moreover, the converse is also true, i.e. the following proposition holds. (9), then g solves equation (5).
Proof. Let (f, ∇ 0 g, kg) ∈ R(D) be a solution of equation (9), then (10) The first equation of (10) can be written in form From the second equation of the system (10), we see Setting (12) and (13) in to the formula (11), we get This shows that g solves equation (5).
Let us define operators with domains D(D) and L 2 (Ω; C n+2 ). Then Remark 2.3. Note that D 1 is a self-adjoint operator, see [ [6], theorem 6.2], and D 0 is a bounded operator. Therefore D is a closed operator and

Maxwell's equation.
Let Ω ⊂ R 2 be a bounded weakly Lipschitz domain. By Rademachers's theorem the surface ∂Ω has a tangent plane and an outward pointing unit normal n(x) at almost every x ∈ ∂Ω. We introduce the Sobolev spaces The last two spaces have the following geometric meaning. Assume that f ∈ H 0 div (Ω; C 2 ), then there exists a sequence {ψ k } ∞ k=1 ⊂ C ∞ 0 (R 2 ; C 2 ) such that ψ k → f and divψ k → divf , see [ [12], Definition 8.14, Lemma 8.18]. Hence for φ ∈ C ∞ 0 (R 2 ), we obtain Hence the Stokes' theorem implies formally Therefore we interpret f ∈ H 0 div (Ω; C 2 ) as saying, beside divf ∈ L 2 (Ω), that f is tangential on the boundary in a weak sense. Similarly, the condition f ∈ H 0 curl (Ω; C 2 ) means curlf ∈ L 2 (Ω) and that f is normal on the boundary in a weak sense.
Let µ(x), ε(x) ∈ L ∞ (R 2 ; L (C 3 )) be pointwise strictly accretive matrices, see (4). For complex number ω = 0, we consider the Maxwell's system of equations According to the splitting C 3 into C and C 2 , we write Since µ, ε are pointwise strictly accretive, we conclude that µ ⊥⊥ , ε ⊥⊥ are pointwise strictly accretive, and consequently µ, ε and A are invertible. Let i,j=1 be a 6 by 6 matrix such that I 1,1 ⊥ = I 4,4 ⊥ = 1 and other elements are zeros. We set I = I − I ⊥ . From the first and forth equations of (14), we get From the second and third equations of (14), we obtain Since I ⊥ AG = I ⊥ AG, I AG = I AG, G = I AG and I ⊥ G = I ⊥ AG, we can combine equations (15) and (16) in the following way To see that the system of equations (14) and the equation of (18) are equivalent, let us prove analogue of Proposition 2.2.
Since (f, g) is a solution for (18), we see By assumption, (f, g) ∈ R(D) for fixed t ∈ R, and hance Proposition 2.11 implies Therefore, in terms of H and E, we can write Combining (19) and (20), we conclude that H, E solve the system of equations (14).
Since µH = f and f ∈ D(DB) for each fixed t ∈ R, it follows that . Proposition 2.11 and relation between E and g lead to E ∈ H 0 curl (Ω; C 2 ), so that for any fixed t ∈ R, Let us define operators with domains D(D) and L 2 (Ω; C n+2 ). Then Remark 2.6. Note that D 1 is a self-adjoint operator, see [ [6], theorem 6.2], and D 0 is a bounded operator. Therefore D is a closed operator and

Properties of D.
Here we prove that the operators defined in Sections 2.1 and 2.2 have closed range and compact resolvent. We will use symbols σ(·) and ρ(·) to denote spectrum and resolvent set of an operator.
Let us start by considering operator D defined in Sections 2.1. First, we prove that the range R(D) is closed.
Let Ω ⊂ R n be a bounded weakly Lipschitz domain and D be the operator defined in Section 2.1. Then the range R(D) is a closed subspace of L 2 (Ω; C n+2 ).
Since k = 0, the rage R(D) is closed.
To prove Proposition 2.7 we use that k = 0, however, by applying Poincaré inequality, one can prove it also for k = 0.
Next, we find the exact expression for the range R(D).
Proof. By definition of operator D, we obtain that R(D) ⊂ H. Conversely, assume This, by Proposition 2.7, implies that f ∈ R(D).
Finally, we prove that the resolvent operators are compact, which implies that the spectrum σ( D| R(D) ) contains only the eigenvalues of D| R(D) and each eigenvalues has a finite geometric multiplicity.
Let Ω ⊂ R n be a bounded weakly Lipschitz domain and D be the operator defined in Section 2.1. Assume λ ∈ ρ( D| R(D) ), then is a compact operator.
Proof. Since R(D) ⊂ L 2 (Ω; C n ) is closed and the operator D(λ − D) −1 is closed and defined on whole L 2 (Ω; C n ), we see that is a bounded operator. Therefore, it is suffices to show that the embedding Since Ω ⊂ R n is bounded, the Sobolev Embedding Theorem gives that H 1 (Ω) ֒→ L 2 (Ω) is compact. Hence, the sequence {f l } ∞ l=1 contains a Cauchy subsequence in L 2 (Ω). The same conclusion can be drawn for {g l } ∞ l=1 . From estimate (22), we obtain div∇ 0 g l + k 2 g l + g l ≤ C, l=1 contains a Cauchy subsequence in L 2 (Ω; C n ), because the embedding H 0 curl (Ω; C n ) ∩ H div (Ω; C n ) ֒→ L 2 (Ω; C n ) is compact, see [2] or [11].
Further on, we prove similar results, but for the operator D defined in Section 2.2. Proof. Let h ∈ D(D * ) and D * h ∈ D(D). Then Thus Since ω = 0, the rage R(D) is closed.
There is also the following analogue of Proposition 2.9.
Let Ω ⊂ R 2 be a bounded weakly Lipschitz domain and D be the operator defined in section 2.2. Assume λ ∈ ρ( D| R(D) ), then is a compact operator.
Proof. As in Proposition 2.9, we see that is a bounded operator. Therefore it remains to verify that the embedding

Spectral projections and functional calculus for DB
In this section we modify the functional calculus designed by McIntosh in [7], for the operators described below.
Let Ω ⊂ R n be a bounded weakly Lipschitz domain. From now on we consider a pointwise accretive multiplication operator B ∈ L ∞ (Ω; C M × C M ) on L 2 (Ω; C M ) and a closed range operator satisfying the following conditions (1) There exist a bounded operator D 0 and a self-adjoint homogeneous first order differential operator D 1 with constant coefficients and local boundary conditions so that D = D 1 + D 0 . (2) The operator (λ − D| R(D) ) −1 is compact for some, and therefore for all λ belonging to the resolvent set ρ( D| R(D) ).
Remark 3.1. In both the Helmholtz and the Maxwell's cases, the operators B and D satisfy conditions above. Moreover, D 0 is a normal operator, and hence D is normal as well.

Preliminary for functional calculus.
Here we consider basic properties of the operator DB in order to construct a functional calculus in the next subsections. We begin with a well known result and give its prove for sake of completeness. Since B * is an accretive operator, for g ∈ R(D) and h ∈ N(D * ), we obtain for some constant C > 0. Therefore In particular, f ∈ N(D * ) and f ⊥ R(D). Since B is an accretive operator, we see that f = 0. Therefore Similarly, one can prove the second splitting.
Hence g − f ≤ ε C , that is the set D(DB) ∩ R(D) is dense in R(D).
Let P R(D) and P N(D * ) be orthogonal projections to R(D) and N(D * ) corresponding to the splitting   is a compact operator.
Proof. As in Propositions 2.9 and 2.12, it suffices to prove that the embedding and therefore P R(D) Bf l + DP R(D) Bf l ≤ C for some C > 0. Since (λ − D| R(D) ) −1 is a compact operator, we see that is a compact embedding. Hence, the sequence {P R(D) Bf l } ∞ l=1 contains a Cauchy subsequence, and therefore Lemma 3.5 implies that the sequence {f l } ∞ l=1 contains a Cauchy subsequence in L 2 (Ω; C M ) as well.
The Θ(Σ) functional calculus. In this subsection we introduce the following preliminary functional calculus.
A justification of this definition follows from the next proposition.
Proof. We give only the main ideas of the proof. For ψ ∈ Θ(Σ), Proposition 3.4 implies Therefore, the first statement follows from the convergence Next, let us prove that the integral is independent of the choice of ν. Assume ω < ν 1 < ν 2 < µ. For P > 0, we set where l(δ ± P ) is the length of δ ± P . Letting P → ∞, we obtain independence on the choice of ν.
Finally, suppose b 1 and b 2 satisfy appropriate assumptions of the proposition and b 1 < b 2 . Then, there is no spectrum point inside the region b 1 ≤ Reλ ≤ b 2 . This shows that integral is independent of choice of b.
The proofs of the next three propositions are standard and based on proofs for bisectorial operators, see for instance [8], [9]. First we prove that the map given by (36) is an algebra homomorphism. Proposition 3.9. If ψ 1 , ψ 2 ∈ Θ(Σ), then and Proof. For 0 < r 1 < r 2 < R, 0 < ν 1 < ν 2 < µ and b 1 > b 2 > a such that we define two curves γ 1 and γ 2 as in (35). Note that γ 1 belongs to the interior of γ 2 . Then Using the Cauchy formula, we see that the second term vanishes and (2πi) 2 ψ 1 (T )ψ 2 (T ) = 2πi Next we prove the convergence lemma for the Θ(Σ) functional calculus.
The following proposition together with Proposition 3.16 allow us to derive an H ∞ (Σ) functional calculus from the Θ(Σ) functional calculus.
be a sequence such that f j L ∞ (Σ) < C and f j (T ) < C for all j ∈ N and some C > 0. Assume f ∈ H ∞ (Σ) and f j → f uniformly on compact subsets of Σ. Then, for any u ∈ H, the sequence {f j (T )u} ∞ j=1 is convergent in H. Moreover, if f (z) = 1 on Σ, then f j (T )u → u in H.
Next, let u ∈ H. Since D(T ) is a dense set in H, there exists a sequence {u k } ∞ k=1 ⊂ D(T ) converging to u in H. Thus By choosing k large enough and then letting m, n → ∞, we conclude that {f j (T )u} ∞ j=1 is a Cauchy sequence.
Finally, if f (z) = 1 on Σ and u ∈ D(T ), then arguments above imply that f j (T )u → u in H. For u ∈ H, there exists a sequence {u k } ∞ k=1 ⊂ D(T ) converging to u in H. Thus By choosing k large enough and then letting j → ∞, we get f j (T )u → u in H.
Remark 3.12. Note that we do not use the uniform boundedness of the sequence {f k (T )} ∞ k=1 to prove the second part of Proposition 3.11. Definition 3.13. For an eigenvalue λ ∈ σ(T ), define the index of λ, as the smallest nonnegative integer m such that Next we prove that all purely imaginary eigenvalues of T have finite index.
Since p i ∈ Θ(Σ), we can define Π i := p i (T ) for i = 1, ...N. By Proposition 3.6, (λ − T ) −1 is a compact operator for all λ ∈ ρ(T ). Hence Π i is a compact operator as the Riemann sum of compact operators. Moreover, Proposition 3.9 implies that Π i is a projection. Therefore Π i is a finite rank operator and Finally, for any integer m > 0, we obtain N((λ 0 i − T ) m ) ⊂ R(Π i ). Therefore, the index of λ 0 i is a finite number. We conclude this subsection with the following inequality, which will be used in Section 4. Proposition 3.15. For fixed i = 1, ..., N, there exists a constant C > 0 such that for all h ∈ H ∞ (Σ) satisfying h(z) = 0 for z / ∈ {ζ ∈ C : |λ 0 i − ζ| < R}, the following estimate holds h(T ) ≤ C max Proof. From the assumption, h(T )u = h(T )Π 0 u = 0 for u ∈ N(Π i ). Therefore, due to (37), it suffices to prove for any v ∈ R(Π i ). This implies (38).

3.3.
The H ∞ (Σ) functional calculus. Here we prove that T has a bounded H ∞ (Σ) functional calculus. In order to do this, analogously to functional calculus for bisectorial operators, we need the following quadratic estimate.
Next we prove the following auxiliary lemma. Observe that f m → 1 pointwise on Σ. Actually, {f } ∞ m=1 converges uniformly on compact subsets of Σ. Indeed, assume there exist a compact subset K ⊂ Σ and {x k } ∞ k=1 ⊂ K such that |f m (x m ) − 1| > c for some c > 0. Since K is compact, without lost of generality we assume that x m → x for some x ∈ K. Then The first term tends to 0, because of pointwise convergence. To estimate the second term, let us note that dist(iτ, Σ) > 0, and hance there exists C > 0 such that 1 1 + (αz) 2 < C for any α ∈ [0, 1 τ ], z ∈ Σ. Therefore, straightforward calculations give This contradicts with our assumption, that is f m → 1 uniformly on compact subsets of Σ. Therefore Proposition 3.11 and Remark 3.12 imply that On the other hand, Proposition 3.9 yields for each u ∈ H, and therefore f m (T )u → P u + Qu.
Now we prove that T has a bounded H ∞ (Σ) functional calculus . The main idea is contained in [10], [9]. Proof. Let ψ t (z) = tz 1+t 2 z 2 ∈ Θ(Σ) and P , Q be the operators as in Lemma 3.17. Then, for v, u ∈ H, We estimate each summand separately. For the first two terms, using Proposition 3.9, we obtain To estimate the last term, we note for t, s ∈ (0, 1 τ ) and some α > 0. Denote η(x) = min (x α , x −α ) (1 + |log x|). Then The Cauchy-Schwartz inequality yields Finally, using the quadratic estimate from Proposition 3.16, we get Now we are on a position to introduce the following H ∞ (Σ) functional calculus for the operator T .
. Also observe that the sequence { im im+z f (z)} ∞ m=1 ⊂ Θ(Σ) converges to f uniformly on compact subsets of Σ for f ∈ H ∞ (Σ). Therefore Proposition 3.18 implies that we have a well defined bounded operator f (T ) on H for any f ∈ H ∞ (Σ).
Proposition 3.11 also shows that Definition 3.19 agrees with Definition 3.7 for functions in Θ(Σ).
Let us consider the basic properties of the H ∞ (Σ) functional calculus. First we prove that the map given by Definition 3.19 is an algebra homomorphism. Proof. Let f , g ∈ H ∞ (Σ) and {f j } ∞ j=1 , {g j } ∞ j=1 ⊂ Θ(Σ) be the corresponding sequences, see Definition 3.19. Then {f g j } ∞ j=1 ⊂ Θ(Σ) is uniformly bounded and f g j → f g on compact subsets of Σ. Therefore Next we show the convergence lemma for the H ∞ (Σ) functional calculus.
According to the above proposition, we have a topological splitting For given u ∈ R(Π 0 ) ⊕ R(Π ± ), we define for ±t > 0, where e −tT is the operator obtained from the function by the functional calculus.
for ±t > 0 and u t → u as t → 0.

Application to waveguide propagation
In this section, we return to the Helmholtz equation and Maxwell's system of equations and use our new functional calculus for the operator T := DB| R(D) to investigate acoustic and electromagnetic waves along the waveguide. More precisely, in Theorems 4.1 and 4.2 we prove that all polynomially bounded time-harmonic waves in the semi-or bi-infinite waveguide have representation in R(Π 0 ) or R(Π 0 ) ⊕ R(Π + ) respectively. 4.1. The bi-infinite waveguide. We start by considering the bi-infinite waveguide, that is we consider the ordinary differential equation  for t < s. By integrating over (P, s) and letting P → 0, one can prove f s = f s , so that f t = h t (T )f 0 .
Integrating over (s, P ) for some P > s, gives Theorem 3.18 and estimate (52) imply now that g P −s (T )Π − f P ≤ Ce −aP e εP .
Hence f t = h t (T )f 0 . Since h t → π 0 + π + uniformly on compact subsets of Σ, we conclude f t → f 0 strongly in H.