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. Recall that if the PDE is of second or higher order, then we can rewrite it as a system of first order equations, so without loss of generality we can assume that the PDE only contains first order derivatives in t. In this way the PDE becomes a vector-valued ordinary differential equation, ODE, like ∂ t u(t, x) + T u(t, x) = 0 (1.1) However, since 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 the 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 timeharmonic 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 Sect. 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 Sect. 2.2.
We show in Sect. 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 , are non-trivial matters. However, in the general non-smooth case, this is well understood from the works of Axelsson et al. [5] and Auscher et al. [4]. In the present paper we extend these results to the case D 0 = 0 which occurs in general time-harmonic, but non-static, wave propagation in waveguides. In Sect. 3 we study functional calculi of operators of the form (1.3), which we show have L 2 (Ω) spectra contained in regions S ω,τ := {x + iy ∈ C : |y| < |x| tan ω + τ }.
To have a theory for general frequencies of oscillation, encoded by the zeroorder 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 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 [11].
In the final Sect. 4, we apply our new functional calculus for operators T to show how all polynomially bounded time-harmonic waves in the semior bi-infinite waveguide can be represented like (1.2), with u 0 in appropriate spectral subspace for T .

Partial Differential Equations Expressed as Vector-Valued Ordinary Differential Equations
In this section we consider the 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 a bounded open set, separated from the exterior domain, Ω − = R n \Ω, by a 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 where R n + = R n−1 × (0, +∞) and R n − = R n−1 × (−∞, 0). In this case Ω is called a weakly Lipschitz domain.
We will use the symbols D(·), N(·), and R(·) to denote the domain, null space, and range of an operator, respectively.

The 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 exists α > 0 such that for all x ∈ R n and v ∈ C n+2 . For a complex number k = 0, we consider the equation By div and ∇ 0 , we denote the divergence and gradient operators on H div (Ω) and H 1 0 (Ω) respectively. Splitting C n+2 into C and C n+1 , we decompose the matrix A(x) in the following way Then we can write Eq. (2.2) in the form Next, we 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 (2.4), we obtain On the other hand, from definition of f , we obtain which, together with (2.5), gives us the system of equations In vector notation, we equivalently have with domains D(B) = L 2 (Ω; C n+2 ) and respectively. Then the equation becomes together with the constraint that f ∈ R(D) for each fixed t ∈ R. Since A is a pointwise strictly accretive operator, B is a strictly accretive multiplication operator just like A, see [4, Proposition 3.2]. By the above arguments, equation (2.2) for u implies that f , defined above, solves (2.6). Moreover, the converse is also true, that is the following proposition holds. Proof. Let (f, ∇ 0 g, kg) ∈ R(D) be a solution of Eq. (2.6), then The first equation of (2.7) can be written in the form From the second equation of the system (2.7), we see Setting (2.9) and (2.10) into the formula (2.8), we get This shows that g solves Eq. (2.2).
Let us define operators Remark 2.3. Note that D 1 is a self-adjoint operator, see [9,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 Rademacher'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 Hence the Stokes' theorem implies formally Therefore we interpret f ∈ H 0 div (Ω; C 2 ) to mean that divf ∈ L 2 (Ω), and that f is tangential on the boundary in a weak sense. Similarly, the condition f ∈ H 0 curl (Ω; C 2 ) means that curlf ∈ L 2 (Ω), and f is normal on the boundary in a weak sense.
By ∇, ∇ 0 , div and div 0 , we define the gradient and divergence operators on H 1 (Ω), H 1 0 (Ω), H div (Ω; C 2 ) and H 0 div (Ω; C 2 ) respectively. Remark 2.4. For a bounded weakly Lipschitz domain Ω ⊂ R 2 and function f ∈ H div (Ω; C 2 ), we see This gives Let μ, ε ∈ L ∞ R 2 ; L C 3 be pointwise strictly accretive matrices, see (2.1). For a complex number ω = 0, we consider Maxwell's system of equa- According to the splitting of 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 all other elements are zero. We set I = I − I ⊥ . From the first and forth equations of (2.11), we get From the second and third equations of (2.11), we obtain Since I ⊥ AG = I ⊥ AG, I AG = I AG, G = I AG, and I ⊥ G = I ⊥ AG, we can combine Eqs. (2.12) and (2.13) in the following way (2.14) Define Let B := AA −1 , F := AG, so that Eq. (2.14) becomes together with the constraint that F ∈ R(D) for each fixed t ∈ R. To see that (2.11) and (2.15) are equivalent, we prove an analogue of Proposition 2.2.

Proposition 2.5. Let f (t, x) and g(t, x) be three dimensional vector-valued functions such that (f, g) solves Eq. (2.15), and (f, g) ∈ R(D) ∩ D(DB) for each fixed t ∈ R. Then the vector-valued functions
solve the system of equations (2.11), and for any fixed t ∈ R, Proof. Splitting C 3 into C and C 2 , we write Since (f, g) is a solution for (2.15), we see By the assumption, (f, g) ∈ R(D) for fixed t ∈ R, and hence Proposition 2.11 implies Therefore, in terms of H and E, we can write Combining (2.17) and (2.18), we conclude that H, E solve the system of equations (2.11).
Since μH = f and f ∈ D(DB) for each fixed t ∈ R, it follows that . Proposition 2.11 and (2.16) lead to E ∈ H 0 curl (Ω; C 2 ). Therefore, for any fixed t ∈ R, Remark 2.6. Note that D 1 is a self-adjoint operator, see [9, 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 Sects. 2.1 and 2.2 have closed range and compact resolvents. We will use the symbols σ(·) and ρ(·) to denote the spectrum and resolvent sets of an operator, respectively.
Let us start by considering the operator D defined in Sect. 2.1. First, we prove that R(D) is closed. Proof. According to [8,Theorem 5.2], it suffices to prove that γ(D) > 0, where γ(D) is the reduced minimum modulus of D, that is the greatest number γ such that This implies that γ(D) ≥ |k| > 0, and consequently that R(D) is closed.
To prove Proposition 2.7 we used that k = 0. However, by applying the Poincaré inequality, one can prove that Proposition 2.7 also holds for k = 0.
Next, we find the exact expression for R(D).  Proof. By definition of operator D, we obtain R(D) ⊂ H. Conversely, assume there exists a function h ∈ N(∇ 0 ) and sequence in the L 2 norm. This, by Proposition 2.7, implies that f ∈ R(D).
Finally, we prove that the resolvent operators are compact. This implies that the spectrum σ( D| R(D) ) contains only the eigenvalues of D| R(D) , and each eigenvalue has finite geometric multiplicity. In fact, we prove in Proposition 3.14 that the indexes/algebraic multiplicities are finite.

Proposition 2.9.
Let Ω ⊂ R n be a bounded, weakly Lipschitz domain, and D be the operator defined in Sect.
Finally, after passing to subsequences three times, we conclude that {(f l , ∇ 0 g l , kg l )} ∞ l=1 contains a Cauchy subsequence in (R(D), · L2 ). We next derive similar results for the operator D defined in Sect. 2.2.
The following proposition gives the exact expression for R(D).
Conversely, assume (f, g) ∈ H. Let us set Then, from Remark 2.4, we obtain Next, since Combining all relations between (f, g) and (F, G), we conclude that (F, G) ∈ D(D), and

This implies that H ⊂ R(D), hence that H = R(D).
There is also the following analogue of Proposition 2.9. for some constant C > 0. In particular, As in Proposition 2.9, (2.21) implies that {div 0 h l 2 } ∞ l=1 contains a Cauchy subsequence in L 2 (Ω). Similarly, this statement holds for {divh l . From the compact embedding (see [6] or [12]) contains a convergent subsequence in L 2 (Ω; C 2 ). From the arguments above, we conclude that {Dh l } ∞ l=1 contains a Cauchy subsequence in L 2 (Ω; C 6 ).

Spectral Projections and Functional Calculus for DB
In this section we modify the functional calculus designed by McIntosh in [10], 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 exists 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 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) ).

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 proof for the sake of completeness. Since B * is an accretive operator, for g ∈ R(D) and h ∈ N(D * ), we obtain for some constant C > 0. Similarly, for some constant C > 0. Therefore One can prove the second splitting similarly.
Therefore, for any λ / ∈ S ω,0 and u ∈ D(DB), Thus, for sufficiently large τ > 0 and any λ / ∈ S ω,τ , and therefore Hence λ−DB is an injective operator with closed range. Next, let us consider the adjoint operator Similarly, we see that λ − D * B * is injective. Consequently, (λ − DB) * is also injective. Hence λ − DB is a surjective operator. Thus, λ / ∈ S ω,τ is contained in the resolvent set, and 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.
We conclude this preliminary subsection by introducing the following setup. We fix a constant τ > 0 from Proposition 3.4 and define By summarizing Propositions 3.3, 3.4, and 3.6, we conclude that T is a closed densely defined operator with σ(T ) ⊂ S ω,τ . Moreover, for each λ / ∈ S ω,τ , the operator (λ − T ) −1 is compact, and hence there may be only a finite number of eigenvalues of T on the imaginary axis. We denote them by {λ 0 i } N i=1 . We fix positive constants a and R such that R < a and For b > a such that and ν ∈ (ω, μ), r < R, we define anti-clockwise oriented curves

The Θ(Σ) Functional Calculus
Here we introduce the following preliminary functional calculus.
A justification of this definition follows from the next proposition.
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 the desired independence of the choice of ν.
Finally, suppose b 1 and b 2 satisfy the assumptions of the proposition, and b 1 < b 2 . Then, there is no spectral point inside the region b 1 ≤ Reλ ≤ b 2 . This shows that the integral is independent of the choice of b.
The proofs of the next three propositions are standard and based on proofs for bisectorial operators, see for instance [1,2]. First we prove that the map given by (3.12) is an algebra homomorphism. 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 (3.11). Note that γ 1 belongs to the interior of γ 2 . Then Using the Cauchy formula, we see that the second term vanishes. Therefore Next we prove the convergence lemma for the Θ(Σ) functional calculus. Proof. Let us fix ε > 0. One can find an integer m 1 ∈ N such that for any n > m 1 , Let γ p,q := {ζ ∈ γ : p ≤ |ζ| < q}, then we can fix M > 0 such that 3 .
The following proposition, together with Proposition 3.16, allows 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 Proof. Let τ 1 > τ and u ∈ D(T ). Since iτ 1 / ∈ S ω,τ , there exists v ∈ H such that By Proposition 3.9, we see that f j (T )u = ψ j (T )v, and therefore Proposition 3.10 implies that 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 the arguments above imply By choosing k large enough and then letting j → ∞, we get f j (T )u → u in H. Next, we prove that each imaginary eigenvalue of T has finite index.
Since p i ∈ Θ(Σ), we can define Π i := p i (T ) for i = 1, . . . N. Proposition 3.6 implies that (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, by Proposition 3.9, Π i is a projection. Therefore Π i is a finite rank operator, and (3.13) 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 Sect. 4. 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

Proposition 3.15.
Proof. From the assumption, h(T )u = h(T )Π i u = 0 for u ∈ N(Π i ). Therefore, due to (3.13), it suffices to prove for all v ∈ R(Π i ) and some C > 0.

The H ∞ (Σ) Functional Calculus
Here we prove that T has a bounded H ∞ (Σ) functional calculus. In order to do this, analogous to the functional calculus for bisectorial operators, we need the following quadratic estimate.
Proposition 3.16. There exists a constant C > 0 such that for all u ∈ H.
Next we prove the following auxiliary lemma. Proof. Let us consider the functions Observe that f m → 1 pointwise on Σ. Actually, {f m } ∞ m=1 converges uniformly on compact subsets of Σ. Indeed, assume there exist a compact subset K ⊂ Σ and 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 zero because of pointwise convergence. To estimate the second term, let us note that dist(iτ, Σ) > 0, and hence 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 our assumption c > 0. Thus 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.
Hence, due to (3.17), we derive P u + Qu = u. Now we prove that T has a bounded H ∞ (Σ) functional calculus . The main idea is contained in [2], [7]. We estimate each summand separately. For the first two terms, by using Proposition 3.9, we obtain To estimate the last term, let us set ψ t (z) = tz 1+t 2 z 2 ∈ Θ(Σ), and note that Finally, using the quadratic estimate from Proposition 3.16, we get Now we are in 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 {fg j } ∞ j=1 ⊂ Θ(Σ) is uniformly bounded and fg j → fg on compact subsets of Σ. Therefore Next we show the convergence lemma for the H ∞ (Σ) functional calculus.