Remark on Justification of Asymptotics of Spectra of Cylindrical Waveguides with Periodic Singular Perturbations of Boundary and Coefficients

To perform an asymptotic analysis of spectra of singularly perturbed periodic waveguides, it is required to estimate remainders of asymptotic expansions of eigenvalues of a model problem on the periodicity cell uniformly with respect to the Floquet parameter. We propose two approaches to this problem. The first is based on the max–min principle and is sufficiently easily realized, but has a restricted application area. The second is more universal, but technically complex since it is required to prove the unique solvability of the problem on the cell for some value of the spectral parameter and the Floquet parameter in a nonempty closed segment, which is verified by constructing an almost inverse operator of the operator of an inhomogeneous model problem in variational setting. We consider boundary value problems on the simplest periodicity cell: a rectangle with a row of fine holes.

UDC 519.958:531.33:517.956.8 To perform an asymptotic analysis of spectra of singularly perturbed periodic waveguides, it is required to estimate remainders of asymptotic expansions of eigenvalues of a model problem on the periodicity cell uniformly with respect to the Floquet parameter. We propose two approaches to this problem. The first is based on the max-min principle and is sufficiently easily realized, but has a restricted application area. The second is more universal, but technically complex since it is required to prove the unique solvability of the problem on the cell for some value of the spectral parameter and the Floquet parameter in a nonempty closed segment, which is verified by constructing an almost inverse operator of the operator of an inhomogeneous model problem in variational setting. We consider boundary value problems on the simplest periodicity cell: a rectangle with a row of fine holes. Bibliography: 33 titles. Illustrations: 3 figures.

Motivation.
In this paper, we study the model problem on a periodicity cell coming from the Floquet-Bloch theory [1]- [4] in the analysis of the spectrum of a periodically perforated waveguide (cf. Figure 1). Eigenvalues of the model problem depending on the Floquet parameter η ∈ [−π, π] (the dual variable of the Gelfand transform [5]) determine the location and size of spectral segments and gap opening between them, i.e., generate the band-gap structure of the waveguide spectrum. The main (and essentially new) problem appearing in the justification of asymptotic expansions of eigenvalues of model problems on the periodicity cell is to derive η-uniform estimates for remainders of asymptotic expansions since only such estimates provide a competent information about spectral segments. We note that estimates of such a quality are not necessary in the usual situation of a single spectral problem, but they are required if we deal with a family of problems parametrized by the Floquet parameter. The known justification schemes involve elements that, after adaptation to the class of problems under consideration, do not guarantee the required uniformity. In this paper, we discuss approaches to compensate this lack.
We study several boundary value problems on the same cell of the simplest shape as in Figure 1 (b) for which we have different formulations and different proofs of the corresponding results. In Sections 2 and 3, calculations and arguments are rather simple since they are based on variational methods, whereas a new rather complicated and laborious analysis including new ideas and tools is required to study some problems in Section 4. The approaches we propose can be also applied to other methods of periodic singular perturbations of cylindrical and even originally periodic waveguides.
(1. 7) In what follows, we deal with a composite rectangle. Together with Equation (1.3), we also consider the differential equation in the holes (1.1) and the transmission conditions on the boundaries of the inclusions connecting the restrictions u ε T • on ω ε and u ε T on Ω ε of the function u ε T defined in Ω and the normal derivatives of these restrictions. Moreover, T ∈ (0, +∞) is also a parameter of the problem. We emphasize that, in the particular case T = 1, the singular perturbation vanishes since the transmission conditions (1.9) are transformed to the continuity conditions, whereas Equations (1.3) and (1.8) remain unchanged (cf. Remark 1.2). For the sake of brevity we sometimes omit the subscripts T , η and •, .
Remark 1.1. The integral identity of Problem J ε N (η) can be obtained from Problem J ε T (η) as T → +0, but the limit passage to "absolutely rigid" inclusions as T → +∞ does not lead to Problem J ε D (η). This situation is discussed in Subsection 5.

Two scenarios of obtaining uniform estimates for remainders.
Usually, a scheme for justifying asymptotic expansions of eigenvalues and eigenfunction (eigenvectors) of singularly perturbed spectral problems consists of three steps. First, formal asymptotic expansion of eigenpairs is constructed. Second, the global asymptotic approximation to the eigenfunction is obtained when we glue together the external and internal expansions obtained by the method of matched asymptotic expansions (cf., for example, [7,8]) or summarize smooth type solutions with boundary layer type solutions within the framework of the method of composite asymptotic expansions (cf., for example, [9,10]). Then it is possible to calculate the residuals generated by the approximate eigenpair in the singularly perturbed initial problem and, finally, to apply the lemma about almost eigenvalues and eigenvectors [11] provided by the spectral representation of the resolvent (cf., for example, [6,Chapter 6]). Thus, at the second step, in a small neighborhood of the κ k (η)-multiple eigenvalue λ 0 Mk (η) of the limit problem, we can find the terms of the subsequence of eigenvalues of the initial problem P ε M (η). It is important that the factor c(η) in the final estimates can be made independent of the Floquet parameter η ∈ [η 0 − δ 0 , η 0 + δ 0 ] for some α 0 ∈ (0, 1] and η 0 ∈ (−π, π], δ 0 > 0 (we refer to Subsection 5.2 for details). It remains to verify that P ε k (η) = k in the list (1.19), i.e., to realize the third step of the scheme. Traditionally, for this purpose one uses the so-called convergence theorems establishing the limit passage as ε → +0 (usually, the spaces in (1.21) should be specified, but it is not necessary for our goals): u ε Mj (·; η) → u 0 MJ(j) (·; η) weakly in H 1 and strongly in L 2 . (1.21) The number J(j) ∈ N of the eigenpair {λ 0 MJ(j) (η), u 0 MJ(j) (·; η)} of the limit problem is not specified at this step, but it becomes j after completing proofs of all necessary results. Thus, owing to formula (1.21) for any j ∈ N and η ∈ [−π, π], it is possible to prove by contradiction that, in the case ε ∈ (0, ε j (η)], there are no "superflous" eigenvalues satisfying (1.20). Therefore, it is easy to prove that P ε k (η) = k in the list (1.19) for sufficiently small ε. We note the brittleness of the third step: we do not know the character of dependence of ε j (η) on j and η. Consequently, it is not clear whether the estimates (1.20) with P ε k (η) = k are uniform with respect to the Floquet parameter.
In this paper, we propose two approaches to overcoming the above-mentioned difficulty arising in the justification of asymptotic expansions of spectral segments generating the spectrum of an infinite waveguide with the periodicity cell (1.2) (cf. Figure 1 (a)). We emphasize that if estimates for the remainder in the asymptotic expansions of the eigenvalues λ ε Mk (η) are not uniform with respect to η, then we cannot make any conclusion about geometric characteristics of the connected compact set υ ε Mk for ε ∈ (0, ε Mk ] even if ε Mk > 0 is small. The first approach (cf. Sections 2 and 3) is to somehow use the classical max-min principle (cf., for example, [6,Theorem 10.2.2]) to prove the inequalities where θ > 0 and c(η) c θ , η ∈ [−π, π]. (1.24) By (1.23) and (1.24), it is easy to show that it is not necessary to use convergence theorems in the justification scheme and we can verify that P ε k (η) = k in the list (1.19) for every η (we refer the reader for details to [12], where this approach is realized for a singularly perturbed cell of some other shape in the space R d , d = 3).
According to the second approach, it is not necessary to exclude the above-mentioned convergence theorems from the scheme, but we can ignore the fact that we do not know whether the convergences (1.21) are uniform by verifying the following assertion: if λ • is not eigenvalue of the limit problem P 0 . Thus, it becomes indifferent for which value of the parameter the multiplicity is calculated.
In Subsection 5.2, we present another way to use this fact. Namely, since the eigenvalues , the dispersion curves cannot intersect the horizontal segment (1.25), and, consequently, on the segment [0, λ • ], the multiplicity #σ(η) of the spectrum σ(η) of Problem P(η) = P ε T (η) T =0 on the whole cell Ω coincides with the multiplicity #σ ε N (η) of the spectrum σ ε N (η) of Problem P ε N (η) = P ε T (η) T =0 on the cell with the Neumann perforation. Thus, the asymptotics of the spectral segments (1.22) is justified without convergence theorems since the lemma about almost eigenvalues and eigenvectors asserts that there are at least #σ(η) eigenvalues (1.19) By the aforesaid, the number of eigenvalues is equal to #σ(η).

Structure of the paper.
In Section 2, we use the max-min principle to obtain the right inequality in (1.23) for the eigenvalues of Problem P ε T (η); here, we use rather elementary calculations. This method does not provide the required result completely, but we can slightly modify Equation (1.8) (cf. Subsection 2.2), and then use the same principle to obtain the left inequality in (1.23) for T > 1. Unfortunately, the modified problem is useless to study Problem P ε N (η) (cf. Remark 5.1).
In Section 3, we study Problem P ε D (η). Owing to the Dirichlet conditions on the hole boundaries, we can derive weighted estimates (Lemma 3.1 and Proposition 3.1). Based on these estimates, we use the max-min principle to prove (1.23) completely. In a sense, the problem with the conditions (1.6) is simpler than other problems. In Subsection 5.3, we show that the second approach is also applicable to obtain uniform estimates for remainders of asymptotic expansions.
Section 4 represents a technically difficult result. We show how the second approach is realized by considering Problem P ε N (η). To establish the unique solvability of Problem P ε T (η) with parameters λ = λ • and η ∈ [η • − δ • , η • + δ • ], we construct almost inverse operators for the operators A ε T (λ • ; η) of the family of problems. We first use the trick [13] of smoothing the right-hand side of the singularly perturbed problem and then, in fact, repeat the procedure for constructing asymptotics for the solutions and estimate the appeared small residuals.
The proposed method for verifying whether there are eigenvalues on the segment (1.25) can be also used in other approaches to the study of singular perturbations of periodicity cells. Therefore, we mention some works dealing with differential equations with strongly contrast coefficients [14]- [17], the case where the periodicity cells split in limit [18]- [21], the case of thin domains [22]- [25], and the case of regular and singular perturbations of boundaries [26]- [28].
In Section 5, we describe the second approach by considering Problems P ε T (η), T ∈ [0, 1], and P ε D (η). This section also contains auxiliary results used in this paper.
The case T = 0 corresponding to Problem P ε N (η) with the Neumann condition (cf. Remark 1.2 and Subsection 5.1) is covered by Proposition 2.1 because no essential modifications are required to verify formula (2.7) in this case. As shown in Subsection 3.3, the eigenvalues of Problem P ε D (η) with the Dirichlet conditions satisfy an equality similar to (2.7).

2.2.
The modified problem P ε T (η). Unfortunately, the max-min principle does not yield immediately the left inequality in (1.23) for the eigenvalues of Problem P ε T (η). We discuss one problem to which this approach is applicable by using elementary calculations. Namely, for T > 1 we replace Equation (1.8) with the following: In other words, we eliminate T from the right-hand sides of the differential equations (1.8), and thereby the "material density" of the cell Ω becomes constant. We assign the symbol denoting this operation to ingredients of the relations (1.3)-(1.5), (1.9) generating Problem P ε T (η). The eigenvalues of this problem are denoted by λ ε T j (η). We assume that the corresponding eigenfunctions u ε T j (·; η) satisfy the orthonormality conditions (1.18). We apply the max-min principle to the operator A (η) of Problem P(η) = P ε T (η) T =1 , consisting of Equations (1.16), (1.4), (1.5) without the parameter ε and make the required changes in formula (2.1). The linear span in the max-min principle, whereas the intersection of these sets contains the following linear combination similar to (2.2): Since U (·; η); L 2 (Ω) = 1 in view of (1.18) and the second identity in (2.2), we have Proof. It suffices to note that the transformations providing Proposition 2.1 can be also used for the Rayleigh fraction in the max-min principle for the eigenvalues of the operator A ε T (η), i.e., in fact, the second inequality in (2.10) is not different from (2.7). The first inequality in (2.10) is contained in (2.9).

Perforations with Dirichlet Conditions
3.1. The limit problem P 0 D . By the conditions (1.6) on the boundaries of the densely located fine holes (1.1), Problem P ε D (η) significantly differs from other problems considered in the paper because of the Poincaré-Friedrichs inequality The inequality (3.1) is verified by stretching variables x → ξ j (cf. formula (1.1)). Summarizing the inequality (3.1) with respect to j = −N, . . . , N, we get the estimate where Ω ε = {x ∈ Ω ε : |x 1 | < }. Since the factor on the left-hand side of (3.2) is large, we can conclude that the limit problem of P ε D (η) is Problem P 0 D (η) consisting of the differential equation   It is remarkable that the eigenvalues (3.5) are independent of the Floquet parameter. Owing to this fact, the asymptotic analysis in [29] shows that there are narrow spectral segments (cf. Figure 2) separated by wide gaps in the spectrum of the periodic waveguide with Dirichlet perforation (cf. Figure 1 (b)). We emphasize that the dependence of the eigenfunctions (3.6) on η is largely fictitious; the periodicity cell in the waveguide is taken in an arbitrary way and, by the quasiperiodicity conditions (1.5), the eigenfunctions in the shifted cell Ω → = (0, 1)×(−H, H) take the form sin(πqx 1 ) cos(πp(2H) −1 (x 2 + H)) without the Floquet parameter. We omit the argument η even in the notation of the functions (3.6) and denote by P D the limit problem (3.3), (3.4), (1.4), (1.5). We enumerate the eigenvalues (3.5) in nondescending order and re-numerate terms of the subsequence {λ 0 j } j∈N with one index. We assume that the corresponding eigenfunctions u 0 j satisfy the orthonormality conditions (1.18).

Weighted estimates.
We first apply the one-dimensional Hardy inequality which is valid for any R > 0 and V ∈ C 1 c (0, R] vanishing at t = 0. Indeed,

8)
where c ω is independent of ε and u ε .
In the following assertion, we use the trick proposed in [30] and used, in particular, in [12].
In what follows, we omit η in the notation. We set U ε k = R −θ ε u ε k . By (3.8), we have Two terms on the right-hand side cancel, and the remaining two terms satisfy the relations Thus, for θ (2c ω ) −1 the right-hand side of (3.12) is estimated from below by 3 4 Remark 3.1. By construction, the exponent θ in Proposition 3.1 cannot be large. Thus, in the limit problem P D without small parameter ε, but with the Dirichlet condition (3.4), we have θ < 1/2 in view of smoothness of the eigenfunctions (3.6).

Application of the max-min principle.
In the following representation of the eigenvalues of the operator of Problem P ε D (η), similar to (2.1) and (2.11), We introduced the cut-off function (3.9) to satisfy the Dirichlet conditions (1.6) and guarantee that X ε u 0 Dj belong to the space H 1 η,D (Ω ε ). At the same time, for the eigenfunctions (3.6) (which are enumerated and satisfy (1.18)) we have the elementary estimates which imply that the smoothing functions U ε 1 , . . . , U ε k with small ε inherit the linear independence property of u 0 D1 , . . . , u 0 Dk . Furthermore, for the linear combination (3.16) For the sake of brevity we omit the argument η and do not indicate that the linear combination (3.15) belongs to E k D (η) in the notation. In (3.16), we used formulas (3.9), (3.14) and their consequence |u 0 D,j (x)| 2c 0 j hε, x ∈ supp |∇X ε |. From (3.13) and (3.16) it follows that A similar calculation for the max-min principle (3.13) at ε = 0, i.e., for the eigenvalues of the problem (3.4), (1.4), (1.5) without parameters η and ε, yields the estimate We explain necessary replacements. First of all, E k D (η) is the subspace of the space H 1 η,D (Ω) of functions vanishing on the mean line Υ of the rectangle Ω and some test functions in this subspace have the form where u ε j are the eigenfunctions of Problem P ε D (η) satisfying the orthonormality conditions (1.15), and, as above, the cut-off function (3.9) satisfies the Dirichlet conditions on Υ.
By the weighted estimate (3.11) in Proposition 3.1, Taking into account this estimate, we obtain the following assertion from (3.17).
As was shown in Subsection 1.3, Theorem 3.1 is sufficient to realize the first approach for obtaining uniform estimates for remainders of asymptotic expansions. This fact was verified in [29] by other arguments.
We note that the max-min principle was also used in [12] to prove analogues of (1.23). This max-min principle is based on a priori weighted estimates for eigenfunctions of the model problem on a three-dimensional periodicity cell with the Neumann condition on the boundary of a single small cavity. The multidimensional (d > 3) case can be treated in the same way, whereas the plane problem has not been studied yet.

Almost inverse operator.
If the spectral parameter λ • is fixed, we can associate the variational statement of the inhomogeneous problem P ε T (η) where H 1 η,T (Ω) is the Sobolev space H 1 η (Ω) equipped with the norm depending on the parameter T ∈ (0, 1] Here, a ε T and b ε T denote the bilinear forms (1.11) and f ε T ∈ H 1 η,T (Ω) * is a linear continuous functional on H 1 η,T (Ω). In this subsection, we assume that T > 0. The case T = 0 is considered in Subsection 4.6.
To prove Theorem 4.1, we construct the so-called almost inverse operator

is the exponent and Id is the identity mapping. Then the operator
) −1 is the usual inverse of the mapping (4.2). It is important that α • and c • in (4.5) are independent of the Floquet parameter.
The problem (4.6) is uniquely solvable for small and ε in view of the Riesz representation theorem and the following inequality obtained from Proposition 4.1: Moreover, for u ε T ∈ H 1 η,T (Ω ) we have To conclude the first step, we define the first term in the coming representation of the almost inverse operator The cut-off function χ 1 ∈ C ∞ c (R) satisfying χ 1 χ 0 = χ 0 was introduced to (4.16) since

Smooth solution in Ω.
Taking ψ ∈ H 1 η,T (Ω) and substituting the test function ψ = ψχ 1 into the integral identity (4.6), we get becomes infinitely differentiable, at least, in the rectangle Ω /3 . In particular, in view of local estimates for solutions to elliptic equations, we have Furthermore, Equation (1.3) holds in Ω /3 and Equation (1.8) holds in ω ε j . Therefore, the residuals left by the sum R ε T + R εΩ T in the representation (4.17) are concentrated on the boundaries of the inclusions (1.1). Indeed, To compensate such residuals, we construct a boundary layer.

Boundary layer problem.
As usual, near the row of fine holes we have the boundary layer phenomenon described by the solutions to the problem in the strip Π = (−H, H) × R: on the space H T,per (Π) obtained by completion of the linear set C ∞ c,per (Π) of infinitely differentiable compactly supported functions that are 2H-periodic with respect to ξ 2 in the energy norm By the Poincaré inequality on the sets {ξ ∈ Π : |ξ 1 | < 2h} and ω and the one-dimensional Hardy inequality (3.7) with R = +∞, the norm (4.31) is equivalent to the weight norm uniformly with respect to the parameter T ∈ (0, 1]. As known, the constants belong to the space H T,per (Π), in particular, the norms (4.32) are finite. Thus, the following assertion holds (cf., for example, [31, Section 3]). where c is independent of g and T ∈ (0, 1]. If the contour ∂ω and the right-hand side g of the second transmission condition in (4.28) are smooth, then the components w and w • of the solution w ∈ H T,per (Π) are also smooth on the sets Π \ ω and ω respectively. In the general case, the component w is smooth outside any neighborhood of the compact set ω, whereas the component w • is smooth inside the domain ω. These obvious facts are true because of local estimates for solutions to elliptic equations: based on the periodicity condition, it is possible to reduce the problem (4.26)- (4.29) to the case of the cylindrical surface R × S 2H , where S L is the circle of length L.
By the Fourier method, we have the representation and the following estimate for the remainder: Moreover, c w is a constant and χ ± ∈ C ∞ (R) are cut-off functions vanishing on ω, We emphasize that the function (4.35) is stabilized, generally speaking, to different constants c ± w as ξ 1 → ±∞, i.e., the representation (4.35) providing the zero sum of these constants c ± w = ±c w distinguishes the solution w ∈ H T,per (Π), i.e., it is a counterpart of the orthogonality conditions (4.34) in Proposition 4.2.

Component of boundary layer type.
Thus, in the construction (4.17) of an almost inverse operator, we set Furthermore, The moduli of the first two inner products on the right-hand side of (4.40) do not exceed since the estimate (4.36) applied to the solutions W q implies We consider each term I ε m (ψ ε ). Since the smooth function decays at infinity at a rate O(e −(2H) −1 π|x 1 |/ε ) (cf. (4.35), (4.36) and (4.41)), we find Note that the estimates for the L 2 (−1/2, 1/2)-norm of the test function ψ ε are obtained from the usual trace inequality for the extension (4.10) of the restriction ψ ε Ω ε on Ω in the Sobolev class H 1 satisfying the relation Furthermore, since W q is harmonic in Π \ ω and ω, we get Taking into account formulas (4.35) and (4.36), we derive the estimate We represent the remaining terms I 2 q (ψ ε ) in (4.42) as By the definition of the special solutions W q (cf. Subsection 4.4), the sum coincides with the expression (4.25) taken with the opposite sign, i.e., the residual generated by the smooth solution u εΩ T are eliminated. We deal with another pair of terms on the right-hand side of (4.45) as follows: We summarize the above calculation. First, the estimates (4.18), (4.23), and (4.39) show that the norm of the mapping (4.4) is uniformly bounded. Second, the sum (4.17) of the three terms (4.16), (4.37), and (4.38) generates the residual f ε T (ψ ε ) in the problem (4.1) satisfying the inequality (4.19), (4.20), (4.25), (4.46), and also (4.21), (4.24), and (4.43)-(4.44), (4.47). It is this inequality (4.5) with exponent α • = 1/2 that was required to construct the almost inverse operator (4.4) and, consequently, the true inverse operator (4.2) of the problem (4.1). Thus, Theorem 4.1 is proved for T ∈ (0, 1]. 4.6. Case T = 0. All calculations and arguments of the previous subsections can be easily adapted to Problem P ε N (η). In fact, it suffices to set T = 0 in all formulas and remove the first identities from the transmission conditions (1.9) and (4.28), i.e., transform them to the Neumann conditions (1.7) and ∂ n w(ξ) = g(ξ) for ξ ∈ ∂ω respectively. It is obvious that the obtained estimates are uniform with respect to the Floquet parameter η ∈ [η • − δ • , η • + δ • ]. The fact that ε • > 0 can be different in the situations T ∈ (0, 1] and T = 0 does not affect the final formulation of Theorem 4.1 which thereby is valid for all T ∈ [0, 1].

Comments
5.1. Limit as T → +0. We fix ε > 0, i.e., the inclusions (1.1) are not assumed to be small, and construct the formal asymptotic expansions of the eigenvalues (1.13) T of Problem P ε T (η) as T → +0. We refer, for example, to [32] for asymptotic procedures including the proof of estimates for remainders of asymptotic expansions (the uniformity in the Floquet parameter is not necessary).
The transmission conditions (1.9) split into two boundary conditions as T → +0, so that the Neumann conditions are imposed at the more rigid inclusion, whereas the Dirichlet conditions are related to the softer one. Thus, the asymptotic ansätze for eigenpairs {λ ε T k (η), u ε T k (·; η)} of the problem (1.3)-(1.5), (1.8), (1.9) contain eigenpairs of the limit problem consisting of the equation where Λ ω > 0 is the first (least) eigenvalue of the Dirichlet problem in the domain ω. Conse-Moreover, the constants c ε j are not a priori fixed, but are found when we solve the problem. The variational statement (1.12) of the problem is considered in the subspace . . , N}.

Justification of the asymptotics the spectral segments with the Neumann perforations.
A simple result presented in [32] and rewritten in Subsection 5.1 leads to the following important conclusion: Theorem 4.1 shows that the multiplicities of the spectra on [0, λ • ] coincide for Problem P ε N (η) with the Neumann conditions on the boundaries of holes (1.1) and Problem P(η) on the whole cell. We consider an example to show how to apply this result.
Formula (1.17) for the eigenvalues of the limit problem P(η) involving Equation (1.16) in the rectangle Ω = (−1/2, 1/2) × (−H, H) and the conditions (1.4), (1.5) on the sides of Ω shows that the corresponding dispersion curves form a truss of a rather complex structure. The lower part of the truss is shown in Figure 3 (b), (c) in the case 1/6 < H < 1/4, where two gaps γ ε N 1 and γ ε N 2 (the projections of tinted rectangles on the ordinate axis in Figure 3 (a)) can appear in the spectrum of an infinite periodic waveguide with the Neumann perforation (cf. Figure 1 (a)). These gaps are located in cε-neighborhoods of the points λ ε N 1 = π 2 and λ ε N 2 = (2H) −2 π 2 . To identify the gaps γ ε N 1 , we need asymptotic formulas for the upper and lower bounds of the spectral segments υ ε N 1 and υ ε N 2 respectively. Due to the interwining of dispersion curves, the justification of asymptotics for λ ε N 1 (η) and λ ε N 2 (η) is performed in two steps. First, Theorem 4.1 applied to the segment Λ  Figure 3 (a) contains arcs of two dispersion curves. We emphasize that the rectangle is not necessarily symmetrically located because the function η → λ ε Mk (η) is even. Since we can choose λ •1 < λ •2 and δ 1 > π − 2δ 2 (cf. Figure  3 (b), where the segments Λ •1 and Λ •2± are marked with dash-dotted lines ending with the symbol •), we can derive the required uniform estimates with respect to the Floquet parameter for the remainders in the asymptotic representations of the eigenvalues and obtain exhaustive information about the spectral segments (1.22) N with k = 1 and k = 2.
(a) (b) (c) −π π −π π −π π Figure 3. The dispersion curves in the model problem with the Neumann perforation (a) and on the whole cell (b) and (c). The dash-dotted line ended with the symbol • represents the segment (1.25) which is free from the spectrum of Problem P ε T (η). To study the opening of the gap γ ε N 1 , we use Figure 3 (c). The upper and lower bounds of the segments υ ε N 2 and υ ε N 3 are determined by the eigenvalues λ ε N 2 (η) and λ ε N 3 (η), but, in view of the structure of the truss of dispersion curves, we need to justify the asymptotic expansions of four eigenvalues λ ε N 1 (η), . . . , λ ε N 4 (η) on η ∈ [−δ 3 , δ 3 ]. The segments Λ •3 and Λ •4± are presented in Figure 3 (c) as above. It is natural to expect that similar coverings of the range interval of the Floquet parameter can be also constructed for more complicated trusses.

5.3.
Another approach to justifying asymptotics for spectral segments with the Dirichlet perforation. The asymptotics of the eigenvalues of the problem (1.3)-(1.6) is constructed in [29]. The asymptotics of narrowed spectral segments justified in (1.22) (cf. Figure  2) can be done by different methods, in particular, by using the inequality (3.18) in Theorem 3.1.
The second above-discussed approach is also applicable. Namely, the construction of an almost inverse operator for mapping of the inhomogeneous problem P ε D (η) in the variational setting repeats (with some simplifications) the arguments of Section 4. Moreover, because of the simplicity of dispersion curves, in Problem P 0 D we deal with horizontal segments of level λ = λ Dk . Therefore, we choose points λ • ∈ {λ Dk } k∈N for which the operator (5.13) realizes an isomorphism for all η ∈ [−π, π], ε ∈ (0, ε k• ] and some ε k• > 0. As a result, for such values of ε the multiplicities of the discrete spectra of Problems P ε D (η) and P 0 D on [0, λ • ] coincide. Since the asymptotic expansions of the eigenvalues λ ε Dk (η) are obtained in [29], it is obvious that the estimates for the remainders are uniform in η ∈ [−π, π].