Spectral analysis of one-dimensional high-contrast elliptic problems with periodic coefficients

We study the behaviour of the spectrum of a family of one-dimensional operators with periodic high-contrast coefficients as the period goes to zero, which may represent e.g. the elastic or electromagnetic response of a two-component composite medium. Compared to the standard operators with moderate contrast, they exhibit a number of new effects due to the underlying non-uniform ellipticity of the family. The effective behaviour of such media in the vanishing period limit also differs notably from that of multi-dimensional models investigated thus far by other authors, due to the fact that neither component of the composite forms a connected set. We then discuss a modified problem, where the equation coefficient is set to a positive constant on an interval that is independent of the period. Formal asymptotic analysis and numerical tests with finite elements suggest the existence of localised eigenfunctions ("defect modes"), whose eigenvalues situated in the gaps of the limit spectrum for the unperturbed problem.


Introduction.
1.1. The general context for the problem in hand. The description of the effective behavior of high-contrast composites ("high-contrast homogenization") has been of particular interest in the analysis and applied mathematics communities over the last decade. The analytical part of the related literature starts with the work [18], which developed in detail some earlier ideas of [1] concerning the use of "two-scale convergence" for the analysis of the limit behavior of the boundary-value problem −div(A ε (x/ε)∇u) = f, f ∈ L 2 (Ω), u ∈ H 1 0 (Ω), A ε = ε 2 χ 0 I + χ 1 I, ε > 0, where Ω ⊂ R n is a bounded domain, and χ 0 , χ 1 are the indicator functions of [0, 1) nperiodic sets in R n such that χ 0 + χ 1 = 1. Several contributions to the high-contrast homogenization followed-in the linear and nonlinear scalar and vector contexts, with various sets of assumptions about the underlying geometry of the composite. With applications mainly in solid mechanics and electromagnetism, high-contrast media have served as a theoretical ground for a number of effects observed in physics experiments, in particular those related to photonic band-gap materials and cloaking metamaterials [15]. The range of techniques developed in these contexts and their applications continue their rapid expansion, and the present paper is aimed at addressing some aspects that have thus far been left out of the scope of the related research.
More specifically, we approach the question of the analysis of the spectral behavior of high-contrast composites when the component represented by the function χ 1 (the "matrix" of the composite) is disconnected in R n . Clearly, this is always so in one dimension (n = 1). In the present article we study this particular case.

Problem set-up.
We consider solutions u to the following family of elliptic problems on an interval (a, b) ⊂ R: where the operators A ε are given by the closed sesquilinear form Here p = p(y) > 0 is a 1-periodic function in R such that p, p −1 ∈ L ∞ (0, 1), the functions χ 0 and χ 1 are the indicator functions of 1-periodic open sets F 0 and F 1 such that F 0 ∪ F 1 = R, and H denotes a closed linear subspace of H 1 (a, b) that contains C ∞ 0 (a, b). We make no assumptions regarding boundedness of the interval (a, b); in particular, it may coincide with the whole space R.
In applied contexts the problem (1.1) corresponds to, e.g., the study of wave propagation in a layered two-dimensional or three-dimensional composite structure where f = 0, λ > 0. In what follows we study the spectrum S ε of the problem (1.1), i.e., the set of values of λ for which A ε − λI does not have a bounded inverse in L 2 (a, b). Throughout the article we employ the notation σ(A) for the spectrum of an operator A and the notation Q for the "unit cell" [0, 1) whenever we describe the behavior with respect to the "physical" variables x, y. We continue writing [0, 1) for the "Floquet-Bloch dual" cell when we refer to the domain of the quasimomentum θ.
1.3. Our strategy for the analysis of (1.1). It has been well understood in the existing literature on the subject (see [2], [18], [20]) that in the analysis of convergence of spectra of families of differential operators with periodic rapidly oscillating coefficients one has to deal with two distinct issues: the lower semicontinuity of the spectra in the sense of Hausdorff convergence of sets and the possibility of spectral pollution, the lack of which is often referred to as "spectral completeness." The former issue, which in the wider spectral analysis context has been looked at from a more general perspective (see, e.g., [4]), is usually dealt with by first proving a variant of the strong resolvent convergence. In the case of periodic operators involving multiple scales, one typically makes use of two-scale convergence (see, e.g., [14], [1], [18]). In the present paper we follow this general approach in proving the related lower semicontinuity statements both for the whole-space problem and for the problem in a bounded interval. This part of the analysis of spectral convergence does not eliminate the need for a study of spectral completeness: unless some assumptions are made concerning the geometry of the periodic composite in question (see, e.g., [18]), one may not get the best possible "lower bound" for the limit spectrum. It has been noticed that in order to capture the behavior with respect to all Bloch components in the limit as ε → 0, it is preferable to use an advanced, "multicell," version of the standard two-scale convergence; see, e.g., [2], [6,Chapter 5], where this more refined c 2015 SIAM. Published by SIAM under the terms of the Creative Commons 4.0 license approach is adopted. It is a version of this last, more detailed, procedure that we develop in the present article.
In the proof of spectral completeness, a natural strategy seems to be to analyze the relative strength of different Bloch components in a given (convergent) sequence of eigenfunctions. This idea has been elaborated in [2] in the specific context of "high-frequency" homogenization, with the use of what the authors refer to as the "Bloch measures." A combination of a compactness argument in the related space of measures and a special "slow-variable modulation" construction then yields the simultaneous convergence of the given sequence to a limit eigenfunction and of the associated eigenvalues. In the present work we deal with a situation where such a compactness argument does not suffice, since the limit of an eigenfunction sequence may have a nontrivial part in the orthogonal complement (in the two-scale version of the L 2 space) to functions that are constant in the matrix component. (In the set-up studied in [2] this orthogonal complement is zero.) Once we have suitable control of this part, we can prove spectral convergence for some operator families not amenable to the approach of [2], including those considered in [17], [18], and [6,Chapter 4].
The key element in our analysis, which allows us to implement the above idea, is Lemma 3.2 below (see section 3.1). This statement establishes a uniform version of the Poincaré-type inequality between the projection of a given function onto the "poorly behaving" subspace of H 1 and the L 2 -norm of its derivative on the part of the domain where solutions of the eigenvalue problem can be shown to be a priori small as ε → 0. Different versions of the same idea have appeared in a number of other contexts, serving a similar purpose of somehow compensating for the apparent loss of compactness in the problem, for example, in the form of the Korn inequality in elasticity (see, e.g., [7] and also [19] for its multiscale versions), in the form of the energy method in classical homogenization (see [13]), and, more recently, in the form of a generalized Weyl decomposition for problems with degeneracies (see [11]). For nonlinear variants of the same idea, the reader is referred to the geometric rigidity (see [8]) and A-quasiconvexity (see [9]).
For an easier introduction to the problem, in what follows we start with the analysis of the problem (1.1) in the whole-space case, (a, b) = R; see section 2. While a version of the compactness argument developed in the bounded-interval setting (see section 3) applies here as well (once complemented by a suitable Weyl-sequence argument), we present a different argument, based on some ideas of [6,Chapter 5], where the spectral analysis is carried out in a more challenging setting of the Maxwell system.
Throughout this paper we assume for simplicity that the restriction of χ 0 to the periodicity cell [0, 1) is the indicator function of an open interval (α, β), which we also denote by Q 0 . We use the notation Q 1 for the interior of the complement of Q 0 to the interval (0, 1).

2.
Limit analysis for the whole space: (a, b) = R. Let us consider for a moment the case p ≡ 1. One well-known procedure for calculating S ε , the spectrum of the problem (1.1), is the Floquet-Bloch decomposition (see, e.g., [3]) following the rescaling y = x/ε. For θ ∈ [0, 1), let λ = λ(θ) be the eigenvalues corresponding to θ-quasiperiodic eigenfunctions of the differential expression ((χ 0 + ε −2 χ 1 )u ) on the interval (0, 1). Such eigenvalues are obtained by solving the dispersion equation Passing to the limit in the above equation as ε → 0 yields By varying θ as indicated, we obtain (for α = 1/4, β = 3/4) the set shown in Figure  1. The following statement is a particular case of our main result, Theorem 2.2 below.
In what follows we consider the case of an arbitrary p while adopting the above Floquet-Bloch approach to the description of the set lim ε→0 S ε of limit points of S ε as ε → 0. For θ ∈ [0, 1), we denote by H 1 θ (Q) the space of functions u ∈ H 1 (Q) that are θ-quasiperiodic, i.e., such that v(y) = exp(2πiθy)u(y), y ∈ Q, for some 1 u ∈ H 1 # (Q). We also denote Note that V (θ) is a closed subspace of H 1 (Q) and is therefore a Hilbert space when inheriting the standard H 1 -norm. The sesquilinear form pu v is clearly closed and nonnegative on V (θ). Therefore, it defines a self-adjoint operator A(θ) (see, e.g., [12]) such that where the domain of A(θ), denoted by dom(A(θ)), is a dense subset of V (θ) with respect to the L 2 -norm. Henceforth (·, ·) denotes the usual inner product in L 2 (Q). Under the adopted conditions on the coefficient p, the operator A(θ) is self-adjoint and has a compact inverse (except for the case θ = 0, when it has a compact inverse as an operator on V (θ) C). Therefore, the spectrum σ(A(θ)) is discrete and unbounded; i.e., it consists of eigenvalues 0 ≤ λ 1 (θ) ≤ λ 2 (θ) ≤ . . . of finite multiplicity with eigenfunctions v k (θ) = v k (θ, y). The eigenfunctions corresponding to different eigenvalues are automatically orthogonal in L 2 (Q). We also carry out the orthogonalization process on those eigenfunctions that correspond to the same eigenvalue, and we normalize each eigenfunction so that v k (θ) L 2 (Q) = 1 for all θ ∈ [0, 1), k ∈ N.
Our aim within this section is to show that the set lim ε→0 S ε coincides with the union of the spectra of the operators A(θ), θ ∈ [0, 1), i.e., More precisely, we establish the following theorem.
In order to demonstrate property 1 we use an appropriate modification of the strong two-scale resolvent convergence, introduced in [18]. By showing, for each N ∈ N, that a subsequence of A ε strongly two-scale resolvent converges to an "intermediate" operator A N on the space of L 2 loc -functions that are N Q-periodic, we establish property 1 for θ = j/N, 0 ≤ j ≤ N − 1. The details of this argument, which rely on a procedure that we refer to as "N Q-periodic homogenization," are given in Appendix A.
An essential ingredient in extending property 1 to hold for θ ∈ [0, 1) and in proving Property 2 is the following continuity property of the set V (θ) with respect to θ.
Proof. Property 1: We shall outline the proof here and refer the reader to Appendix A for the full details. Note that property 1 holds for rational θ, the set lim ε→0 S ε is closed, and the rationals are dense in [0, 1). Therefore, it suffices to show that the eigenvalues λ(θ) of A(θ) are continuous with respect to θ. Indeed, in Appendix B we show this to follow from Lemma 2.3, i.e., from the continuity of V (θ).
Remark 2.1. The above "limit spectrum" lim ε→0 S ε is strictly larger than the set obtained by the two-scale analysis of the operator A ε of the paper [18]. In particular, the spectrum of the one-dimensional version of the homogenized operator obtained in [18] coincides with {λ k (0)} ∞ k=1 , using our notation.
Our analysis above shows that the set lim ε→0 S ε has, in fact, a band-gap structure, with infinitely many gaps opening in the interval [0, ∞), as ε → 0. This fact suggests possible applications of the above composite structures to the design of optical or acoustic band-gap materials, which we discuss in section 5. The above effect also raises a mathematical question regarding the analysis of the limit behavior of the operators A ε in the case when (a, b) is a bounded interval, which we study in the next section.

Spectral behavior on a bounded interval.
It is known that the classical, "moderate-contrast," analogue of the problem (1.1)-(1.2) leads to limit spectra of different kinds for problems on bounded and unbounded intervals (a, b): the limit set in the case of the problem in the whole space is purely absolutely continuous, while in the case −∞ < a < b < ∞ it is purely discrete; i.e., it consists of eigenvalues with finite multiplicities; see, e.g., [3]. A similar situation occurs in multidimensional highcontrast problems where the inclusion F 0 ∩ Q has a nonzero distance to the boundary of Q; see [18], where, in addition, some eigenvalues of infinite multiplicity are present. As we shall see below, this is not the case for the problem (1.1)-(1.2), when the spectrum of the operator A ε defined by (1.2) contains the right-hand side of (2.4), i.e., the spectrum of the operator defined by the form (1.2) with (a, b) replaced by R and H replaced by H 1 (R). This makes the limit spectrum in question considerably richer than that described in [18].
In this section we employ, for convenience, the following notation: The main convergence result. The following theorem holds. Theorem 3.1. Consider an operator A ε from the class described in section 1.2. The limit set lim ε→0 S ε is given by the union σ Bloch ∪ σ boundary , where An essential element to proving Theorem 3.1 is validating the following statement. Lemma 3.2. There exists a constant C > 0, which depends on α and β only, such that for all θ ∈ [0, 1) one has Here Proof. It suffices to show that the statement of the lemma holds for the equivalent H 1 -norm Indeed, assuming that the statement of the lemma holds for the norm ||| · |||, with a constant C = C in the analogue of (3.2) for ||| · |||, andĉ : e.g., the claim of the lemma with C =ĉ C. In what follows we keep the notation V ⊥ (θ) for the orthogonal complement to V (θ) with respect to the inner product induced by ||| · |||. We first note some properties of functions that belong to the space V ⊥ (θ), which follow immediately from the characterization of the space V (θ) given in the proof of Lemma 2.3. Proposition 3.3. Let w ∈ V ⊥ (θ); then the following hold.
(i) The equation w (y) = 0 holds for y ∈ Q 0 . In particular, the function w is linear on the Q 0 -component of the unit cell:

holds.
We now return to the proof of Lemma 3.2. We consider three different cases, depending on the location of the quasimomentum θ within the Floquet-Bloch cell [0, 1).
Case I: θ = 0. Using Proposition 3.3, we find that Since w(1) = w(0), we obtain the estimate Case II: θ ∈ (0, δ) ∪ (1 − δ, 1), where 0 < δ < 1/2 is to be chosen appropriately. Again, Proposition 3.3 implies that and therefore In view of this observation and in order to have a bound below on |d θ |, we require that δ < 1/4. Further, by continuity of the embedding of H 1 (Q) in C(Q), there exists a constantĉ, which is independent of θ, such that and thus We and hence |d θ | −2 is bounded above by a constant independent of θ in the intervals considered. The inequality (3.3) now immediately implies the required estimate.
In particular, substituting x = 1, y = 0 and using the fact that w(1) = exp(2πiθ)w(0), we obtain Similarly, we write which upon integration over (0, α) yields Combining (3.4) and (3.5), we obtain Squaring both sides and using the Cauchy-Schwarz inequality yields A direct calculation shows that the coefficient on the left-hand side of the last inequality is separated from zero in the range of θ considered. Similarly, the coefficient on the right-hand side is bounded. Finally, we argue that The required inequality follows since by Proposition 3.3(i) one has This completes the proof of Lemma 3.2. We now prove Theorem 3.1.
Proof. The inclusion θ∈[0,1) σ A(θ) ⊂ lim ε→0 S ε is proved in the same way as in the case of the whole-space problem; see the proof of Theorem 2.2. In what follows we therefore discuss the converse inclusion; i.e., for λ ε ∈ S ε , λ ε → λ such that λ / ∈ σ boundary one has λ ∈ θ∈[0,1) σ A(θ) . Let us consider such a sequence λ ε . Notice first that for each λ ε ∈ S ε , there exist u ε ∈ H, u ε L 2 (a,b) = 1, such that for any sequence δ ε ε→0 −→ 0, there exists a constant c > 0 uniformly bounding the L 2 -norms of u ε away from the boundary of Ω, holds for all ϕ ∈ H. Indeed, if this were not the case, then λ would be an element of σ boundary ; see (3.1). Rescaling the above statement with y = x/ε yields the existence of functions for all ϕ ∈ H := {v | v(y) = u(εy) for some u ∈ H}. Choosing ϕ = v ε in (3.7) yields the estimates (a priori bounds) where C B > 0 is independent of ε.
For every N ∈ N we introduce the function space (cf. (2.2)) and denote V ⊥ N to be the orthogonal complement of V N in the space H 1 # (N Q) equipped with the usual inner product. For each ε we set the χ ε v ε to take zero values on N ε Q \ Ω ε , and we consider the functions U ε : Nε denote the orthogonal projections in the space H 1 # (N ε Q) onto its subspaces V Nε and V ⊥ Nε , respectively. Note that, due to the normalization of v ε and (3.6), the bounds hold for all ε. Further, we show that there exists a constant C ⊥ , independent of ε, such that The inequality (3.11) is a consequence of (3.9) and the following proposition. Proposition 3.4. There exists a constant C ⊥ > 0 such that . Indeed, if Proposition 3.4 holds, then (3.11) follows from the a priori bounds (3.8). Note that Lemma 3.2 is one of the key ingredients in the proof of Proposition 3.4, which we give next.
Proof. Consider a function u ∈ V ⊥ N for some positive integer N . Notice that for each j = 0, 1, . . . , N − 1 the function belongs to the space V (θ j ) ⊥ , θ j := j/N. Indeed, for any v ∈ V (θ j ) ⊂ H 1 θj (Q) it is clear that v belongs to the space V N when extended in a quasiperiodic fashion. Further, for any j = 0, 1, . . . , N − 1, one has Combining the above two identities yields as required. Now using the Parseval identity and Lemma 3.2 we obtain where the positive constant C does not depend on N .
Step 2. There exists a constant C independent of ε such that U ε L 2 (NεQ) ≤ C. (3.12) Indeed, by the a priori bounds (3.8), as well as (3.9) and (3.11), we find that Further, the identity χ ε v ε = U ε + V ε and the bounds (3.10) imply As the functions U ε belong to the spaces V Nε , by the discrete Floquet-Bloch transform the following decomposition holds: Recalling, from section 2, that for fixed ε the eigenfunctions v k (j/N ε ) form a complete sequence in the L 2 (Q)-closure of the set V (j/N ε ), we can decompose U j ε with respect to this sequence, e.g., and (3.13) imply (3.15) Denoting by δ(· − θ) the Dirac mass at θ, the inequalities (3.15) can be rewritten as Clearly, for each k the sequence {μ k ε } ε is bounded in the space of Radon measures on [0, 1]. Therefore, up to a subsequence, μ k ε weakly converges as ε → 0 to some measure μ k , e.g., The above result follows from recalling that the space of finite Radon measures on [0, 1] coincides with the dual space C[0, 1] , and hence bounded sets of Radon measures are relatively compact with respect to weak star convergence in this space.
Step 3. Here we show that which implies in particular that there exists at least one nonzero measure μ k0 for some integer k 0 . The bounds (3.18) follow from (3.16), (3.17), and the following result. Proof. Notice that In the three equalities of (3.19) we use (3.14), (3.16), and the fact that v k (θ) are orthogonal eigenfunctions of A(θ) with eigenvalues λ k (θ); see section 2. Combining this observation with (3.12), we find that where we use the fact that U ε = 0 on F 1 ∩ N ε Q. For any integer K ≥ 2 the inequality (3.20) immediately implies This concludes the proof of the proposition.
4. An example of a family with lim ε→0 S ε = σ Bloch . Here we consider the case when the endpoints of the material domain belong to the "stiff" component. We also assume for simplicity that (a, b) = (0, 1) and restrict our analysis to values ε = N −1 , where N ∈ N; hence (ε −1 a, ε −1 b) = (0, N). Recall that F 0 denotes the 1-periodic extension of Q 0 = (α, β) to the whole of R and F 1 = R\F 0 . For all N ∈ N we denote by W N , W ⊥ N the subspace and its orthogonal complement in H 1 (0, N), respectively. We also denote by P W ⊥ N the orthogonal projection of H 1 (0, N) onto W ⊥ N . Lemma 4.1. There exists a constant C > 0 independent of N such that Proof. Recall (see Proposition 3.4) that there exists C ⊥ > 0 independent of N such that where P VN is the orthogonal projection of H 1 # (0, N) onto In what follows we use this fact to prove the desired estimate (4.1). Take χ to be a smooth cut-off function such that χ = 1 on (0, α/2) ∪ (N − (1 − β)/2, N), χ = 0 on [α, N − 1 + β], and the inequality |χ | ≤ max {4/α, 4/(1 − β)} holds. For fixed u, the function w := u−χu is seen to belong to H 1 # (0, N) and satisfy u−P VN w = P V ⊥ N w+χu. Therefore, since V N ⊂ W N , one has It remains to prove the required bound for χu H 1 (0,N ) . By noting the formulae The result now follows from the formulae u(x) = We now consider a sequence of λ ε such that λ ε → λ as ε → 0, ε −1 ∈ N, for some λ ∈ R, and the corresponding eigenfunction sequence v ε ∈ H satisfies the identity for all values of ε from the indicated set. Here, as before, H is a subspace of H 1 (0, N) and χ 0 , χ 1 are the characteristic functions of the sets F 0 , F 1 .
Notice first that (4.2) in combination with Lemma 4.1 implies the existence of C > 0 such that Further, the function u ε = P WN v ε is not periodic in general; however, it is piecewise constant on (0, N) ∩ F 1 . We introduce an extension u ε by the formula ∈ (N, 2N ).
Clearly, the function u ε is 2N -periodic, but it is not necessarily an element of the space V 2N . However, in cases when u ε does belong to V 2N , we can proceed with the Bloch measure argument in the proof of Theorem 3.1 to show that λ ∈ σ Bloch .

5.
A modified problem with a compact perturbation, and the associated defect modes.

Analytical set-up.
In this section we discuss a modified version of the set-up of section 1.2, as follows. Consider the operator A ε defined by the sesquilinear form (5.1) Here the space H is as before (see section 1.2), and χ d is the indicator function of a "defect" interval I d whose closure is assumed to be contained in (a, b), and p d is the corresponding "defect" coefficient (or "defect strength"). We denote by S ε the spectrum of the operator A ε .
A formal two-scale asymptotic procedure carried out on the equation (cf. (1.1)) A ε u = λu suggests the following.
(1) The set lim ε→0 S ε is given by the union of the "Bloch spectrum" given by σ Bloch = θ∈[0,1) σ A(θ) , the "boundary spectrum" σ boundary (see (3.1)), and a sequence of "defect eigenvalues" {p d π 2 j 2 /|I d | 2 } ∞ j=1 . (2) Those defect eigenfunctions u j , j ∈ N, that correspond to the defect eigenvalues situated in the gaps of σ Bloch decay exponentially away from the boundary of the defect: Here c 2 depends on the distance of the defect eigenvalue to σ Bloch . In section 5.2 we present numerical evidence that supports these claims. Note that the rate of decay in the estimate (5.2) increases as ε → 0. This complements the recent results of [5], where the decay of eigenfunctions in multidimensional highcontrast problems is analyzed. It was shown in [5] that under the assumption that the matrix of the composite forms a connected set in the whole space the defect eigenfunctions corresponding to eigenvalues in the gaps of the limit spectrum satisfy the estimate u j (x) ≤ c 1 exp(−c 2 dist{x, ∂D} , x ∈ (a, b) \ D, with c 1 , c 2 > 0, where D is the defect set, and c 2 may depend on j. The higher rate of decay in (5.2) can be interpreted as a stronger localization effect in the case when the matrix of the composite is disconnected.

Numerical results for the modified problem.
We consider a defect of length |I d | = 1/2 and strength p d = 2 in the middle of the interval (a, b) = (0, 1), e.g., I d = (1/4, 3/4). For each value of ε such that N := (4ε) −1 is a positive integer, we describe the intervals (0, 1/4) and (3/4, 1) on either side of the defect according to (5.1): each of them consists of N cells of the same length ε = 1/(4N ), and in one half of each cell the coefficient in the form (the "modulus") takes the value 1/(4N ) 2 , while in the other half it is equal to unity. Note that the described set-up satisfies the conditions of section 4, so in this case we expect lim ε→0 S ε to coincide with the union of σ Bloch and the sequence of defect eigenvalues.
The results of solving the above problems with finite elements are given in Tables  In addition, the profiles obtained for such trapped modes (see Figure 2 for the case of periodic boundary conditions) suggest that the number of half-oscillations in a trapped mode is equal to the number of the mode in the sequence, which resembles the behavior of the usual Neumann eigenfunctions on the defect. We also note that the decay of the trapped modes appears to be exponential, as can be seen in Figure  3: the larger the contrast (and hence the number of subdivisions of the string), the more localized the mode, irrespective of the boundary conditions at the endpoints of the string.

Photonic band gaps and trapped modes in high-contrast multilayered dielectric structures.
The string problem emerges, among other contexts, in the study of wave propagation in one-dimensional photonic crystals, e.g., multilayered dielectric structures invariant along two directions. In what follows, we set these directions to be x 1 and x 3 in the usual Euclidean representation x = (x 1 , x 2 , x 3 ).
We consider solutions (E, H) to the classical system of Maxwell's equations [10] that have the form where t is time, ω is the angular frequency, and κ ≥ 0 is a "propagation constant." We write the Maxwell's equations for the field variable (E, H): Here μ is the magnetic permeability, and ε is the electric permittivity at each point of the dielectric. We rearrange the above six equations into two groups of equations for (E 1 , H 2 , H 3 ) (transverse magnetic polarization), and (H 1 , E 2 , E 3 ) (transverse electric polarization). We choose E 1 and H 1 as the unknown functions within the respective groups and notice that the remaining unknowns are expressed in terms of these two scalar functions only. The equations satisfied by E 1 , H 1 are Note that (5.4) coincides with (1.1) when κ = 0 by setting where we use η rather than ε to denote the structure period in order to avoid confusion with the standard notation for electric permittivity. Our analysis in sections 2 and 3 carries over to the case κ > 0. In particular, for p ≡ 1 we get a κ-dependent version of the dispersion relation (2.1), as follows: Assuming infinitely conducting walls on either side of the dielectric (see, e.g., [21] for further details), we supply (5.3) and (5.4) with homogeneous Dirichlet and Neumann boundary conditions, respectively.
In the numerical solution of the above problem we employ finite elements with perfectly matched layers, e.g., anisotropic absorptive reflectionless layers (see, e.g., [21]), on the top and bottom of the computational domain. Our results are shown in Figure 4 for κ = 0.1, N = 16, and for the transverse electric mode with frequency λ = 78.34 inside the third stop band. The latter corresponds to the first trapped mode shown in Figure 2(a), in view of the fact that for κ = 0.1 there is an additional zero-frequency stop band. The magnetic component of this mode (Figures 4(a) and (d)) clearly shares the same features as the string mode in Figure 2(a). where " 2 " is weak two-scale convergence (see [1]). The function u ∈ L 2 (a, b); V 1 is the unique solution to the problem b a Q p(y)χ 0 (y) ∂u ∂y (x, y) ∂ϕ ∂y Proof.
Step 1. Let u ε be a solution to problem (1.1), e.g., for all ϕ ∈ H. It is clear, by choosing ϕ = u ε , that the sequences u ε and εu ε are bounded in L 2 (a, b). Therefore, by [1, Proposition 1.14] there exists u(x, y) ∈ L 2 (a, b); H 1 # (Q) such that up to a subsequence, u ε 2 u and εu ε 2 ∂u ∂y . We now show that, in fact, u(x, y) ∈ L 2 (a, b); V 1 . In view of the convergence εu ε 2 ∂u ∂y , we have Furthermore, since the sequence χ 1 u ε is bounded in L 2 (a, b), we have This observation implies, in particular, that the spectra corresponding to (A.1) and (A.4) coincide. Therefore, denoting by A 1 the operator defined by the sesquilinear form A.2. "NQ-periodic" homogenization. In the argument above we found the two-scale limit operator A 1 by choosing the unit cell Q to be the periodic reference cell and by passing to the two-scale limit in (1.1) as ε → 0. Replacing now Q with N Q, N ∈ N, we obtain an analogue of Lemma A.1, as follows.

Appendix B.
Here we show that the eigenvalues of the operators A(θ) defined by (2.3) are continuous in θ. The statement of continuity of λ k = λ k (θ), k ∈ N, is not a simple consequence of the continuity of eigenvalues for the usual Floquet-Bloch decomposition. An important distinct feature of our case is the dependence on θ of the operator domain V (θ). The continuity of λ k (θ) in θ is therefore shown to be a consequence of the continuity of the spaces V (θ) with respect to θ; see Lemma 2.3. In fact, we show that the continuity of V (θ) leads to the operators A(θ) being continuous with respect to θ in the norm-resolvent sense, which implies as one particular consequence the continuity of λ k (θ).
Proof. Let u n be as in the statement of the lemma; then (B.1) Q pu n ϕ + Q u n ϕ = Q f n ϕ ∀ϕ ∈ V (θ n ).
As the sequence f n is weakly convergent, it is bounded, and we find by choosing ϕ = u n in (B.1) that u n H 1 (Q) ≤ f n L 2 (Q) ; e.g., the sequence u n is bounded in H 1 (Q). In particular, up to a subsequence, u n converges weakly in H 1 (Q) (hence strongly in L 2 (Q)) to some u ∈ H 1 (Q). It it readily shown that u ∈ V (θ). Furthermore, for a fixed ϕ ∈ V (θ) there exist, by Lemma 2.3, ϕ n ∈ V (θ n ) such that ϕ n → ϕ strongly in H 1 (Q). Choosing ϕ n as the test function in (B.1) and passing to the limit n → ∞ shows that u is a solution to Q pu ϕ + Q uϕ = Q f ϕ ∀ϕ ∈ V (θ).
By virtue of the fact that the solution u is unique, the above argument applies to any subsequence of u n . Therefore, the claim holds for the whole sequence u n .
We shall now proceed with the proof of Theorem B.1.