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Research article

Optimization of machining parameters in drilling hybrid sisal-cotton fiber reinforced polyester composites

  • Received: 23 September 2021 Revised: 15 November 2021 Accepted: 17 November 2021 Published: 13 January 2022
  • Machining natural fiber reinforced polymer composite materials is one of most challenging tasks due to the material's anisotropic property, non-homogeneous structure and abrasive nature of fibers. Commonly, conventional machining of composites leads to delamination, inter-laminar cracks, fiber pull out, poor surface finish and wear of cutting tool. However, these challenges can be significantly reduced by using proper machining conditions. Thus, this research aims at optimizing machining parameters in drilling hybrid sisal-cotton fibers reinforced polyester composite for better machining performance characteristics namely better hole roundness accuracy and surface finish using Taguchi method. The effect of machining parameters including spindle speed, feed rate and drill diameter on drill hole accuracy (roundness error) and surface-roughness of the hybrid composite are evaluated. Series of experiments based on Taguchi's L16 orthogonal array were performed using different ranges of machining parameters namely spindle speed (600,900, 1200, 1600 rpm), feed rate (10, 15, 20, 25 mm/min) and drill diameter (6, 7, 8, 10 mm). Hole roundness error and surface-roughness are determined using ABC digital caliper and Zeta 20 profilometer, respectively. Optimum machining condition for drilling hybrid composite material (speed: 1600 rpm, feed rate: 25 mm/min and drill diameter: 6 mm) is determined, and the results are verified by conducting confirmation test which proves that the results are reliable.

    Citation: Nurhusien Hassen Mohammed, Desalegn Wogaso Wolla. Optimization of machining parameters in drilling hybrid sisal-cotton fiber reinforced polyester composites[J]. AIMS Materials Science, 2022, 9(1): 119-134. doi: 10.3934/matersci.2022008

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  • Machining natural fiber reinforced polymer composite materials is one of most challenging tasks due to the material's anisotropic property, non-homogeneous structure and abrasive nature of fibers. Commonly, conventional machining of composites leads to delamination, inter-laminar cracks, fiber pull out, poor surface finish and wear of cutting tool. However, these challenges can be significantly reduced by using proper machining conditions. Thus, this research aims at optimizing machining parameters in drilling hybrid sisal-cotton fibers reinforced polyester composite for better machining performance characteristics namely better hole roundness accuracy and surface finish using Taguchi method. The effect of machining parameters including spindle speed, feed rate and drill diameter on drill hole accuracy (roundness error) and surface-roughness of the hybrid composite are evaluated. Series of experiments based on Taguchi's L16 orthogonal array were performed using different ranges of machining parameters namely spindle speed (600,900, 1200, 1600 rpm), feed rate (10, 15, 20, 25 mm/min) and drill diameter (6, 7, 8, 10 mm). Hole roundness error and surface-roughness are determined using ABC digital caliper and Zeta 20 profilometer, respectively. Optimum machining condition for drilling hybrid composite material (speed: 1600 rpm, feed rate: 25 mm/min and drill diameter: 6 mm) is determined, and the results are verified by conducting confirmation test which proves that the results are reliable.



    In this paper we consider a spectral problem for the Laplace operator in an unbounded strip Π(,)×(0,H)R2 periodically perforated by a family of holes, which are also periodically distributed along a line, the so-called "perforation string". The perforated domain Πε is obtained by removing the double periodic family of holes ¯ωε from the strip Π, cf. Figure 1, a), (4)-(6). The diameter of the holes and the distance between them in the string is O(ε), while the distance between two perforation strings is 1. ε1 is a small positive parameter. A Dirichlet condition is prescribed on the whole boundary Πε. We study the band-gap structure of the essential spectrum of the problem as ε0.

    Figure 1. 

    a) The perforated strip Πε is obtained by removing the double periodic family of holes ¯ωε from the strip Π(,)×(0,H). The periodicities 1 and εH come from the width of he periodicity cell ϖε and the distance between two consecutive holes in the perforation string. b) The periodicity cell ϖε is obtained by removing a periodic family of holes of diameter O(ε) from ϖ0(1/2,1/2)×(0,H). It contains one perforation string

    .

    We provide asymptotic formulas for the endpoints of the spectral bands and show that these bands collapse asymptotically at the points of the spectrum of the Dirichlet problem in a rectangle obtained by gluing the lateral sides of the periodicity cell. These formulas show that the spectrum has spectral bands of length O(ε) that alternate with gaps of width O(1). In fact, there is a large number of spectral gaps and their number grows indefinitely when ε+0.

    It should be emphasized that waveguides with periodically perturbed boundaries have been the subject of research in the last decade: let us mention e.g. [34], [21], [22], [2] and [3] and the references therein. However the type of singular perturbation that we study in our paper has never been addressed. We consider a waveguide perforated by a periodic perforation string, which implies using a combination of homogenization methods and spectral perturbation theory.

    As usual in waveguide theory, we first apply the Gelfand transform (cf. [6], [30], [33], [26], [11] and (11)) to convert the original problem, cf. (7), into a family of spectral problems depending on the Floquet-parameter η[π,π] posed in the periodicity cell ϖε (cf. (13)-(16) and Fig. 1, b). Each one of these problems has a discrete spectrum, cf. (18), which describe the spectrum σε as the union of the spectral bands, cf. (20) and (9). One of the main distinguishing features of this paper is that each problem constitutes itself a homogenization problem with one perforation string. As a consequence, in the stretched coordinates, cf. (30), there appears a boundary value problem in an unbounded strip Ξ which contains the unit hole ω (cf. (2), (31)-(33) and Fig. 2).

    Figure 2. 

    The strip Ξ with the hole ω. Ξ is involved with the unit cell for the homogenization problem (13)-(16)

    .

    The above mentioned homogenization spectral problems have different boundary conditions from those considered in the literature (cf. [5], [14] and [16] for an extensive bibliography). Obtaining convergence for their spectra, correcting terms and precise bounds for discrepancies (cf. (10)), as ε0, prove essential for our analysis. We use matched asymptotic expansions methods, homogenization theory and basic techniques from the spectral perturbation theory.

    Let

    Π={x=(x1,x2):x1R,x2(0,H)} (1)

    be a strip of width H>0. Let ω be a domain in the plane R2 which is bounded by a simple closed contour ω which, for simplicity, we assume to be of class C, and that has the compact closure

    ¯ω=ωωϖ0, (2)

    where ϖ0 is a rectangle, the "limit periodicity" cell in Π,

    ϖ0=(1/2,1/2)×(0,H)Π. (3)

    We also introduce the strip Πε (see Figure 1, a) perforated by the holes

    ωε(j,k)={x:ε1(x1j,x2εkH)ω}with jZ,k{0,,N1}, (4)

    where ε=1/N is a small positive parameter, and NN is a big natural number that we will send to . The period of the perforation along the x1-axis in the domain

    Πε=ΠjZN1k=0¯ωε(j,k) (5)

    is made equal to 1 by rescaling, and similarly, the period is made equal to εH in the x2-direction. The periodicity cell in Πε takes the form

    ϖε=ϖ0N1k=0¯ωε(0,k),

    (see b) in Figure 1). For brevity, we shall denote by ωε the union of all the holes in (4), namely,

    ωε=jZN1k=0ωε(j,k), (6)

    while ω is referred to as the "unit hole", cf. (2).

    In the domain (5) we consider the Dirichlet spectral problem

    {Δuε(x)=λεuε(x),xΠε,uε(x)=0,xΠε. (7)

    The variational formulation of problem (7) refers to the integral identity

    (uε,v)Πε=λε(uε,v)ΠεvH10(Πε), (8)

    where (,)Πε is the scalar product in the space L2(Πε), and H10(Πε) denotes the completion, in the topology of H1(Πε), of the space of the infinitely differentiable functions which vanish on Πε and have a compact support in ¯Πε. Since the bi-linear form on the left of (8) is positive, symmetric and closed in H10(Πε), the problem (8) is associated with a positive self-adjoint unbounded operator Aε in L2(Πε) with domain H10(Πε)H2(Πε) (see Ch. 10 in [1]).

    Problem (7) gets a positive cutoff value λε and, therefore, its spectrum σε[λε,) (cf. (20) and Remark 5). It is known, see e.g. [30], [33], [11] and [26], that σε has the band-gap structure

    σε=nNBεn, (9)

    where Bεn are closed connected bounded segments in the real positive axis. The segments Bεn and Bεn+1 may intersect but also they can be disjoint so that a spectral gap becomes open between them. Recall that a spectral gap is a non empty interval which is free of the spectrum but has both endpoints in the spectrum.

    In Section 2 we address the setting of the Floquet parametric family of problems (13)-(16), obtained by applying the Gelfand transform (11) to the original problem (7). They are homogenization spectral problems in a perforated domain, the periodicity cell ϖε, with quasi-periodicity conditions (15)-(16) on the lateral sides of ϖε. Obviously, each problem of the parametric family (13)-(16) depends on the Floquet-parameter η, cf. (11), (19) and (20). For a fixed η[π,π], the problem has the discrete spectrum Λεi(η),i=1,2,, cf. (18). Section 2.2 contains a first approach to the eigenpairs (i.e., eigenvalues and eigenfunctions) of this problem via the homogenized problem, cf. (27). To get this homogenized problem, we use the energy method combined with techniques from the spectral perturbation theory. We show that its eigenvalues Λ0i,i=1,2, do not depend on η, since they constitute the spectrum of the Dirichlet problem in υ=(0,1)×(0,H), cf. (24). In particular, Theorem 2.1 shows that

    Λεi(η)Λ0i as ε0,η[π,π],i=1,2,.

    However, this result does not give information on the spectral gaps.

    Using the method of matched asymptotic expansions for the eigenfunctions of the homogenization problems (cf. Section 4) we are led to the unit cell boundary value problem (31)-(33), the so-called local problem, that is, a problem to describe the boundary layer phenomenon. Section 3 is devoted to the study of this stationary problem for the Laplace operator, which is independent of η and it is posed in an unbounded strip Ξ which contains the unit hole ω. Its two solutions, with a polynomial growth at the infinity, play an important role when determining correctors for the eigenvalues Λεi(η), i=1,2,. Further specifying, the definition and the properties of the so-called polarization matrix p(Ξ), which depend on the "Dirichlet hole" ω, cf. (38) and Section 3.1, are related with the first term of the Fourier expansion of certain solutions of the unit cell problem (cf. (39) and (42)). The correctors εΛ1i(η) depend on the polarization matrix and the eigenfunctions of the homogenized problem, and we prove that for sufficiently small ε,

    |Λεi(η)Λ0iεΛ1i(η)|ciε3/2, (10)

    with some ci>0 independent of η. These bounds are obtained in Section 5, see Theorems 5.1 and 5.2 depending on the multiplicity of the eigenvalues of (24). Λ1i(η) is a well determined function of η (see formulas (61), (62), (68), (69), (71) and Remarks 3 and 4); it is identified by means of matched asymptotic expansions in Section 4.

    As a consequence, we deduce that the bands Bεi={Λεi(η),η[π,π]} are contained in intervals

    [Λ0i+εBiciε3/2,Λ0i+εBi++ciε3/2],

    of length O(ε), where Bi,Bi+ are also well determined values for each eigenvalue Λ0i of (24) (cf. Corollaries 5.1 and 5.2 depending on the multiplicity). All of this together gives that for each i such that Λ0i<Λ0i+1, cf. (23), the spectrum σε opens a gap of width O(1) between the corresponding spectral bands Bεi and Bεi+1.

    Dealing with the precise length of the band, we note that the results rely on the fact that the elements of the antidiagonal of the polarization matrix do not vanish (cf. (70)-(75)), but this is a generic property for many geometries of the unit hole ω (see, e.g., (47) and (51)). Also note, that for simplicity, we have considered that ω has a smooth boundary but most of the results hold in the case where ω has a Lipschitz boundary or even when ω is a vertical crack, cf. Section 3.1.

    Summarizing, Section 2 addresses some asymptotics for the spectrum of the Floquet-parameter family of spectral problems; Section 3 considers the unit cell problem; Section 4 deals with the asymptotic expansions; in Section 5.1, we formulate the main asymptotic results of the paper, while the proofs are performed in Section 5.2.

    In this section, we deal with the setting of the Floquet-parameter dependent spectral problems and the limit behavior of their spectra, cf. Sections 2.1 and 2.2, respectively.

    The Floquet-Bloch-Gelfand transform (FBG-transform, in short)

    uε(x)Uε(x;η)=12πnZeinηuε(x1+n,x2), (11)

    see [6] and, e.g., [30], [33], [11], [26] and [4], converts problem (7) into a η-parametric family of spectral problems in the periodicity cell

    ϖε={xΠε:|x1|<1/2} (12)

    see Figure 1, b. Note that xΠε on the left of (11), while xϖε on the right. For each η[π,π], the spectral problem of the family is defined by the equations

    ΔUε(x;η)=Λε(η)Uε(x;η),xϖε, (13)
    Uε(x;η)=0,xΓε, (14)
    Uε(1/2,x2;η)=eiηUε(1/2,x2;η),x2(0,H), (15)
    Uεx1(12,x2;η)=eiηUεx1(12,x2;η),x2(0,H), (16)

    where Γε=ϖεΠε, η is the dual variable, i.e., the Floquet-parameter, while Λε(η) and Uε(;η) denote the spectral parameter and an eigenfunction, respectively. If no confusion arises, they can be denoted by Λε and Uε, respectively. Conditions (15)-(16) are the quasi-periodicity conditions on the lateral sides {±12}×(0,H) of ϖε.

    The variational formulation of the spectral problem (13)-(16) reads:

    (Uε,V)ϖε=Λε(Uε,V)ϖεVH1,ηper(ϖε;Γε), (17)

    where H1,ηper(ϖε;Γε) is a subspace of H1(ϖε) of functions which satisfy the quasi-periodicity condition (15) and vanish on Γε. In view of the compact embedding H1(ϖε)L2(ϖε), the positive, self-adjoint operator Aε(η) associated with the problem (17) has the discrete spectrum constituting the monotone unbounded sequence of eigenvalues

    0<Λε1(η)Λε2(η)Λεm(η), (18)

    which are repeated according to their multiplicities (see Ch. 10 in [1] and Ch. 13 in [30]). The eigenfunctions are assumed to form an orthonormal basis in L2(ϖε).

    The function

    η[π,π]Λεm(η) (19)

    is continuous and 2π-periodic (see, e.g., Ch. 7 of [9]). Consequently, the sets

    Bεm={Λεm(η):η[π,π]} (20)

    are closed, connected and bounded intervals of the real positive axis ¯R+. Results (9) and (20) for the spectrum of the operator Aε(η) and the boundary value problem (7) are well-known in the framework of the FBG-theory (see the above references). As a consequence of our results, we show that in our problem, depending on the geometry of the unit hole, and for certain lower frequency range of the spectrum, the spectral band (20) does not reduce to a point (cf. (72), (47), (70) and (74)).

    A first approach to the asymptotics for eigenpairs of (13)-(16) is given by the following convergence result, that we show adapting standard techniques in homogenization and spectral perturbation theory: see, e.g., Ch. 3 in [27] for a general framework and [14] for its application to spectral problems in perforated domains with different boundary conditions. Let us recall ϖ0 which coincides with ϖε at ε=0 (cf. (12), and (3)) and contains the perforation string

    ωε(0,0),,ωε(0,N1)ϖ0. (21)

    Theorem 2.1. Let the spectral problem (13)-(16) and the sequence of eigenvalues (18). Then, for any η[π,π], we have the convergence

    Λεm(η)Λ0m,as ε0, (22)

    where

    0<Λ01<Λ02Λ0m,as m, (23)

    are the eigenvalues, repeated according to their multiplicities, of the Dirichlet problem

    ΔU0(x)=Λ0U0(x),xυ,υ(0,1)×(0,H)U0(x)=0,xυ. (24)

    Proof. First, for each fixed m, we show that there are two constants C,Cm such that

    0<CΛεm(η)Cmη[π,π]. (25)

    To obtain the lower bound in (25), it suffices to consider (17) for the eigenpair (Λε,Uε) with ΛεΛε1(η) and apply the the Poincaré inequality in H1(ϖ0) once that Uε is extended by zero in ¯ωε. To get Cm in (25) we use the minimax principle,

    Λεm(η)=minEεmH1,ηper(ϖε;Γε)maxVEεm,V0(V,V)ϖε(V,V)ϖε,

    where the minimum is computed over the set of subspaces Eεm of H1,ηper(ϖε;Γε) with dimension m. Indeed, let us take a particular Eεm that we construct as follows. Consider the eigenfunctions corresponding to the m first eigenvalues of the mixed eigenvalue problem in the rectangle (1/4,1/2)×(0,H), with Neumann condition on the part of the boundary {1/2}×(0,H), and Dirichlet condition on the rest of the boundary. Extend these eigenfunctions by zero for x[0,1/4]×(0,H), and by symmetry for x[1/2,0]×(0,H). Finally, multiplying these eigenfunctions by eiηx1 gives Eεm and the rigth hand side of (25).

    Hence, for each η and m, we can extract a subsequence, still denoted by ε such that

    Λεm(η)Λ0m(η),Uεm(;η)U0m(;η) in H1(ϖ0)weak, as ε0, (26)

    for a certain positive Λ0m(η) and a certain function U0m(;η)H1,ηper(ϖ0), both of which, in principle, can depend on η. Obviously, U0m(;η) vanish on the lower and upper bases of ϖ0. Also, we use the Poincaré inequality in ϖ0ω, cf. (3),

    U;L2(ϖ0¯ω)CU;L2(ϖ0¯ω)UH1(ϖ0¯ω),U=0 on ω,

    and we deduce

    ε1Uεm(;η);L2({|x1|ε/2}ϖ0)2CεUεm(;η);L2({|x1|ε/2}ϖ0)2.

    Now, taking limits as ε0, we get U0m(,η)=0 on {0}×(0,H) (cf., e.g., [16] and (25)). Hence, we identify (Λ0m(η), U0m(;η)) with an eigenpair of the following problem:

    ΔU0m(x;η)=Λ0m(η)U0m(x;η),x1{(1/2,0)(0,1/2)},x2(0,H),U0m(x;η)=0 for x2{0,H},x1(1/2,1/2) and x1=0,x2(0,H),U0m(1/2,x2;η)=eiηU0m(1/2,x2;η),x2(0,H),U0mx1(1/2,x2;η)=eiηU0mx1(1/2,x2;η),x2(0,H), (27)

    where the differential equation has been obtained by taking limits in the variational formulation (17) for VC0((1/2,0)×(0,H)) and for VC0((0,1/2)×(0,H)).

    Now, from the orthonormality of Uεm(;η) in L2(ϖε), we get the orthonormality of U0m(,η) in L2(ϖ0). Also, an argument of diagonalization (cf., e.g., Ch. 3 in [27]) shows the convergence of the whole sequence of eigenvalues (18) towards those of (27) with conservation of the multiplicity, and that the set {U0m(;η)}m=1 forms a basis of L2(ϖ0).

    In addition, extending by η-quasiperiodicity the eigenfunctions U0m(;η),

    u0m(x;η)={U0m(x;η),x1(0,1/2),eiηU0m(x11,x2;η),x1(1/2,1), (28)

    we obtain a smooth function in the rectangle υ, and moreover that the pair (Λ0m(η), U0m(,η)) satisfies (24). In addition, the orthogonality of {U0m(;η)}m=1 in L2(ϖ0) implies that the extended functions {u0m(;η)}m=1 in (28) form an orthogonal basis in L2(υ), cf. also (55), and we have proved that Λ0m(η) coincides with Λ0m in the sequence (23) for any η[π,π]. Consequently, the result of the theorem holds.

    Remark 1. Note that the eigenpairs of (24) can be computed explicitly

    Λ0np=π2(n2+p2H2),U0np(x)=2Hsin(nπx1)sin(pπx2/H),p,nN. (29)

    The eigenvalues Λ0np are numerated with two indexes and must be reordered in the sequence (23); the corresponding eigenfunctions U0np are normalized in L2(υ). Also, we note that if H2 is an irrational number all the eigenvalues are simple.

    In this section, we study the properties of certain solutions of the boundary value problem in the unbounded strip Ξ, cf. (31)-(33) and Figure 2. This problem, the so-called unit cell problem, is involved with the homogenization problem (13)-(16) and the periodical distribution of the openings in the periodicity cell ϖε, but it remains independent of the Floquet-parameter.

    In order to obtain a corrector for the approach to the eigenpairs of (13)-(16) given by Theorem 2.1, we introduce the stretched coordinates

    ξ=(ξ1,ξ2)=ε1(x1,x2εkH). (30)

    which transforms each opening of the string ωε(0,k) into the unit opening ω. Then, we proceed as usual in two-scale homogenization when boundary layers arise (cf., e.g. [28], [18], [32] and [24]): assuming a periodic dependence of the eigenfunctions on the ξ2-variable, cf. (34), we make the change (30) in (13)-(16), and take into account (22), to arrive at the unit cell problem. This problem consists of the Laplace equation

    ΔξW(ξ)=0,ξΞ, (31)

    with the periodicity conditions

    W(ξ1,H)=W(ξ1,0),Wξ2(ξ1,H)=Wξ2(ξ1,0),ξ1R, (32)

    and the Dirichlet condition on the boundary of the hole ¯ω

    W(ξ)=0,ξω. (33)

    Regarding (31)-(33), it should be noted that, for any ΛεC, we have

    Δx+Λε=ε2(Δξ+ε2Λε),

    and ε2ΛεCε2 while the main part Δξ is involved in (31). Also, the boundary condition (33) is directly inherited from (14), while the periodicity conditions (32) have no relation to the original quasi-periodicity conditions (15)-(16), but we need them to support the standard asymptotic ansatz

    w(x2)W(ε1x), (34)

    for the boundary layer. Here, w is a sufficiently smooth function in x2[0,H] and W is H-periodic in ξ2=ε1x2.

    It is worth recalling that, according to the general theory of elliptic problems in domains with cylindrical outlets to infinity, cf., e.g., Ch. 5 in [26], problem (31)-(33) has just two solutions with a linear polynomial growth as ξ1±. Here, we search for these two solutions W±(ξ) by setting ±1 for the constants accompanying ξ1 (cf. Proposition 1). In order to do it, let us consider a fixed positive R such that

    ¯ω(R,R)×(0,H) (35)

    and define the cut-off functions χ±C(R) as follows

    χ±(y)={1, for ±y>2R,0, for ±y<R, (36)

    where the subindex ± represent the support in ±ξ1[0,).

    Proposition 3.1. There are two normalized solutions of (31)-(33) in the form

    W±(ξ)=±χ±(ξ1)ξ1+τ=±χτ(ξ1)pτ±+˜W±(ξ),ξΞ, (37)

    where the remainder ˜W±(ξ) gets the exponential decay rate O(e|ξ1|2π/H), and the coefficients pτ±pτ±(Ξ), with τ=±, which are independent of R and compose a 2×2-polarization matrix

    p(Ξ)=(p++(Ξ)p+(Ξ)p+(Ξ)p(Ξ)). (38)

    Proof. The existence of two linearly independent normalized solutions W± of (31)-(33) with a linear polynomial behavior ±ξ1+p±±, as ±ξ1, is a consequence of the Kondratiev theory [10] (cf. Ch. 5 in [26] and Sect. 3 [20]). Each solution has a linear growth in one direction and stabilizes towards a constant p± in the other direction. In addition, it lives in an exponential weighted Sobolev space which guarantees that, substracting the linear part, the remaining functions have a gradient in (L2(Ξ))2.

    Let us consider the functions

    ˆW±(ξ)=W±(ξ)χ±(ξ1)ξ1, (39)

    which, obviously, satisfy (32), (33) and

    ΔξˆW±(ξ)=F±(ξ),ξΞ, (40)

    with F±(ξ)=F±(ξ1)=±Δ(χ±(ξ1)ξ1)=±(2ξ1χ±ξ1+2ξ1χ±). By construction, F± has a compact support in ±ξ1[R,2R].

    Let Ccper(¯Ξ) be the space of the infinitely differentiable H-periodic functions, vanishing on ω, with compact support in ¯Ξ. Let us denote by H the completion of Ccper(¯Ξ) in the norm

    W,H=yW;L2(Ξ).

    The variational formulation of (40), (32) and (33) reads: to find ˆW±H satisfying the integral identity

    (yˆW±,yV)Ξ=(F±,V)ΞVH. (41)

    Since supp(F±) is compact, we can apply the Poincaré inequality to the elements of {VH1([2R,2R])×(0,H)):V|ω=0}, to derive that the right hand side of (41) defines a linear continuous functional on H. In addition, the left-hand side of the integral identity (41) implies a norm in the Hilbert space H, and consequently, the Riesz representation theorem assures that the problem (41) has a unique solution ˆWH satisfying (41).

    In addition, since for each τ, τ=±, function ˆWτ(ξ) in (39) is harmonic for |ξ1|>2R with gradient in L2((,2R)×(0,H))L2((2R,+)×(0,H)), the Fourier method (cf., e.g. [13] and [26]) ensures that

    ˆWτ(ξ)=cτ±+O(e(±ξ1)2π/H) as ±ξ1+,

    where the constants cτ± are defined by

    cτ±=limT1HH0ˆWτ(±T,ξ2)dξ2=limT1HH0(Wτ(±T,ξ2)τδτ,±T)dξ2. (42)

    Obviously, c+± (c± respectively) are independent of R and they provide all the constants appearing in (37); namely, cτ±=pτ±(Ξ). Hence, the result of the proposition holds.

    In this section, we detect certain properties of the matrix p(Ξ). This matrix represents an integral characteristics of the "Dirichlet hole" ¯ω in the strip Π. Its definition is quite analogous to the classical polarization tensor in the exterior Dirichlet problem, see Appendix G in [29]. Let us refer to [23] for further properties of matrix p(Ξ) as well as for examples on its dependence on the shape and dimensions of the hole.

    Proposition 3.2. The matrix p(Ξ)+RI is symmetric and positive, where I stands for the 2×2 unit matrix and R given in (35).

    Proof. We represent (37) in the form

    W±(ξ)=W±0(ξ)+{±ξ1R,±ξ1>R,0,±ξ1<R. (43)

    The function W±0 still satisfies the periodicity condition of (32) and the homogeneous Dirichlet condition (33) but remains harmonic in ΞΥ±(R), Υ±(R)={ξΞ:±ξ1=R}, and its derivative has a jump on the segment Υ±(R), namely

    [W±0]±(ξ2)=0,[W±0|ξ1|]±(ξ2)=1,ξ2(0,H),

    where [ϕ]±(ξ2)=ϕ(±R±0,ξ2)ϕ(±R0,ξ2).

    In what follows, we write the equations for τ=±. Since ΔW±0=0, we multiply it with Wτ0 and apply the Green formula in (ΞΥ±(R)){|ξ1|<T}. Finally, we send T to + and get

    H0Wτ0(±R,ξ2)dξ2=H0Wτ0(±R,ξ2)[W±0|ξ1|]±(ξ2)dξ2=(ξWτ0,ξW±0)Ξ. (44)

    On the other hand, on account of (43) and the definition of Wτ, we have

    Wτ0(±R,ξ2)=Wτ(±R,ξ2) and [Wτ|ξ1|]±(ξ2)=0.

    Consequently, we can write

    H0Wτ0(±R,ξ2)dξ2=H0Wτ(±R,ξ2)[W±0|ξ1|]±(ξ2)dξ2=H0(Wτ(±R,ξ2)[W±0|ξ1|]±(ξ2)W±0(±R,ξ2)[Wτ|ξ1|]±(ξ2))dξ2,

    and using again the Green formula for Wτ and W±0, in a similar way to (44) we get

    H0Wτ0(±R,ξ2)dξ2=+limTH0(Wτ(τT,ξ2)W±0|ξ1|(τT,ξ2)W±0(τT,ξ2)Wτ|ξ1|(τT,ξ2))dξ2=H(pτ±(Ξ)+δτ,±R). (45)

    Here, we have used the following facts: /|ξ1| is the outward normal derivative at the end of the truncated domain {ξΞ:|ξ1|<R}, the function Wτ0 is smooth near Υ±(R), the derivative W±0/|ξ1| decays exponentially and, according to (37) and (43), the function W±0 admits the representation when ±ξ1>2R (cf. (37))

    W±0(ξ)=χ±(ξ1)(p±±+R)+χ(ξ1)p±+˜W±(ξ).

    Considering (44) and (45) we have shown the equality for the Gram matrix

    (ξWτ0,ξW±0)Ξ=H(pτ±(Ξ)+δτ,±R),

    which gives the symmetry and the positiveness of the matrix p(Ξ)+RI.

    Let us note that our results above apply for Lipschitz domains or even cracks as it was pointed out in Section 2.1. Now, we get the following results in Propositions 3.3 and 3.4 depending on whether ω is an open domain in the plane with a positive measure mes2(ω), or it is a crack with mes2(ω)=0.

    Proposition 3.3. Let ω be such that mes2(ω)>0. Then, the coefficients of the polarization matrix p(Ξ) satisfy

    H(2p+p++p)>mes2(ω).

    Proof. We consider the linear combination

    W0(ξ)=W+(ξ)W(ξ)ξ1=χ+(ξ1)(p++p+)χ(ξ1)(pp+)+˜W0(ξ).

    It satisfies

    ΔξW0(ξ)=0,ξΞ,W0(ξ)=ξ1,ξω,

    with the periodicity conditions in the strip, and ~W0(ξ)=˜W+(ξ)˜W(ξ) gets the exponential decay rate O(e|ξ1|2π/H). Considering the equations ΔW0=0 and Δ(W0+ξ1)=0 in Ξ{|ξ1|<T}, and Δξ1=0 in ω, we apply the Green formula taking into account the boundary condition for W0. Then, taking limits as T, we have

    0<W0;L2(Ξ)2+mes2(ω)=ωξ1ν(ξ1)dν+ωW0ν(W0(ξ))dν=ωξ1ν(ξ1+W0(ξ))dν=ω(νξ1(ξ1+W0(ξ))ξ1ν(ξ1+W0(ξ)))dν=limT±±H0W0(±T,ξ2)dξ2=H(p+++pp+p+).

    Remark 2. observe that for a hole ω, which is symmetric with respect to the x1-axis, the matrix p(Ξ) becomes symmetric with respect to the anti-diagonal, namely,

    p++=p. (46)

    Indeed, this is due to the fact that each one of the two normalized solutions in (37) are related with each other by symmetry. Also, we note that, on account of Proposition 3.2, the symmetry p+=p+ holds for any shape of the hole ω.

    Proposition 3.4. Let ω be the crack ω={ξR2:ξ1=0,ξ2(h,Hh)}, where h<H/2. Then,

    p+=p+>0. (47)

    In addition, p=p++=p+=p+.

    Proof. First, let us note that due to the symmetry W+(ξ1,ξ2)=W(ξ1,ξ2), and the construction (43) when R=0 reads

    W(ξ1,ξ2)={ξ1+W(ξ1,ξ2),ξ1<0,W(ξ1,ξ2),ξ1>0. (48)

    where W(ξ1,ξ2) is the function defined in Π+={ξ:ξ1>0,ξ2(0,H)} satisfying the periodicity condition (32) and equations

    ΔξW(ξ)=0, for ξΠ+,W(0,ξ2)=0, for ξ2(h,Hh),ξ1W(0,ξ2)=1/2, for ξ2(0,h)(Hh,H). (49)

    Indeed, denoting by ˜W the extension of W to Π={ξ:ξ1<0,ξ2(0,H)}, in order to verify the representation (48), it suffices to verify that the jump of ˜W and its the derivative of through Υ(0)={ξΞ:ξ1=0} is given by

    [˜W](0,ξ2)=0,[˜Wξ1](0,ξ2)=1,

    and hence, the function on the right hand side of (48) is a harmonic function in Ξ.

    Now, considering (49), integrating by parts on (0,T)×(0,H), and taking limits as T+ provide

    Υ(0)W(0,ξ2)dξ2=limTH0W(T,ξ2)dξ2=Hp+(Ξ).

    Similarly, from (49), we get

    0=Π+W(ξ)ΔξW(ξ)dξ=Π+|ξW(ξ)|2dξ12Υ(0)W(0,ξ2)dξ2.

    Therefore, we deduce

    H2p+(Ξ)=Π+|ξW(ξ)|2dξ>0 (50)

    and from the symmetry of p(Ξ) (cf. Proposition 3.2), we obtain (47).

    Also, from the definition (48), we have p(Ξ)=p+(Ξ), and (cf. (46)) all the elements of the polarization matrix p(Ξ) coincide. Thus, the proposition is proved.

    From Proposition 3.4, note that when ω is a vertical crack, the inequality in Proposition 3.3 must be replaced by H(2p+p++p)=mes2(ω)=0. Also, we observe that in order to get property (47) for a domain ω with a smooth boundary, we may apply asymptotic results on singular perturbation boundaries (cf. [7], Ch. 3 in [8] and Ch. 5 in [17]) which guarantee that for thin ellipses

    ¯ω={ξ:δ2ξ21+(ξ2H/2)2τ2},τ=H/2h, (51)

    (47) holds true, for a small δ>0.

    In this section we construct asymptotic expansions for the eigenpairs (Λεm(η), Uεm(;η)) of problem (13)-(16) on the periodicity cell ϖε. The parameters mN and η[π,π] are fixed in this analysis. In Sections 4.1-4.2 we consider the case in which the eigenvalue Λ0m of (24) is simple. Note that for many values of H, all the eigenvalues are simple (cf. Remark 1). Section 4.3 contains the asymptotic ansatz for the eigenpairs case where Λ0m is an eigenvalue of (24) of multiplicity κm2.

    Let Λ0m be a simple eigenvalue in sequence (23) and let U0m be the corresponding eigenfunction of problem (24) normalized in L2(υ). Then, on account of the Theorem 2.1, for the eigenvalue Λεm of problem (13)-(16) we consider the asymptotic ans¨atze

    Λεm=Λ0m+εΛ1m(η)+. (52)

    To construct asymptotics of the corresponding eigenfunctions Uεm(x;η), we employ the method of matched asymptotic expansions, see, e.g., the monographs [35] and [8], and the papers [32], [18] and [24] where this method has been applied to homogenization problems. Namely, we take

    Uεm(x;η)=U0m(x;η)+εU1m(x;η)+ (53)

    as the outer expansion, and

    Uεm(x;η)=ε±wm±(x2;η)W±(xε)+ (54)

    as the inner expansion near the perforation string, cf. (4) and (21).

    Above, U0m(x;η) is built from the eigenfunction U0m of (24) by formula

    U0m(x;η)={U0m(x),x1(0,1/2),eiηU0m(x1+1,x2),x1(1/2,0), (55)

    W± are the solutions (37) to problem (31)-(33), while the functions U1m, wm± and the number Λ1m(η) are to be determined applying matching principles, cf. Section 4.2. Note that near the perforation string, cf. (4), (21), the Dirichlet condition satisfied by U0m(x;η) implies that the term accompanying ε0 in the inner expansion vanishes (see, e.g., [24]); this is why the first order function in (54) is ε. Also, above and in what follows, the ellipses stand for higher-order terms, inessential in our formal analysis.

    First, let us notice that U0mC(¯υ), and the Taylor formula applied in the outer expansion (53) yields

    Uεm(x;η)=0+x1U0mx1(0,x2)+εU1m(+0,x2;η)+,x1>0,Uεm(x;η)=0+x1eiηU0mx1(1,x2)+εU1m(0,x2;η)+,x1<0, (56)

    where, for second formula (56), we have used (55).

    The inner expansion (54) is processed by means of decompositions (37). We have

    Uεm(x;η)=εwm+(x2;η)(ξ1+p++)+εwm(x2;η)p++,ξ1>0,Uεm(x;η)=εwm(x2;η)(ξ1+p)+εwm+(x2;η)p++,ξ1<0. (57)

    Recalling relationship between x1 and ξ1, we compare coefficients of ε and x1=εξ1 on the right-hand sides of (56) and (57). As a result, we identify wm± by

    wm+(x2;η)=U0mx1(0,x2),wm(x2;η)=eiηU0mx1(1,x2), (58)

    and also obtain the equalities

    U1m(+0,x2;η)=τ=±wmτ(x2;η)pτ+,U1m(0,x2;η)=τ=±wmτ(x2;η)pτ. (59)

    Formulas (58) define coefficients of the linear combination (54) while formulas (59) are the boundary conditions for the correction term in (53). Moreover, inserting ans¨atze (52) and (53) into (13)-(14), we derive that

    {ΔxU1m(x;η)Λ0mU1m(x;η)=Λ1m(η)U0m(x;η),xϖ0,x10,U1m(x1,H;η)=U1m(x1,0;η)=0,x1(1/2,0)(0,1/2), (60)

    and the quasi-periodic conditions with η (cf. (15)-(16)).

    Since U0m(x;η) is defined by (55), (Λ0m,U0m(x)) is an eigenpair of (24), and U0m;L2(υ)=U0m(;η);L2(ϖ0)=1, we multiply by U0m(x;η) in the differential equation of (60), integrate by parts and obtain

    ϖ0Λ1m(η)U0m(x;η)¯U0m(x;η)dx=H0U1m(0,x2;η)¯U0mx1(0,x2;η)dx2H0U1m(+0,x2;η)¯U0mx1(+0,x2;η)dx2.

    Thus, by (55) and (59), the only compatibility condition in (60) (recall that Λ0m is a simple eigenvalue) converts into

    Λ1m(η)=H0¯Bm(x2;η)p(Ξ)Bm(x2;η)dx2 (61)

    where

    Bm(x2;η)=(U0mx1(0,x2),eiηU0mx1(1,x2))TC2, (62)

    and it determines uniquely the second term of the ansatz (52). Here and in what follows, the top index T indicates the transpose vector.

    Also, from (53), (54) and (57) the composite expansion approaching Uεm(x;η) in the whole domain ϖ0 reads

    Uεm(x;η)U0m(x;η)+εU1m(x;η)+ετ=±wmτ(x2;η)Wτ(xε)(εwm±(x2;η)(ε1|x1|+p±±)+εwm(x2;η)p±),±x10. (63)

    We address the case where Λ0m is an eigenvalue of (24) with multiplicity κm2. Let us consider Λ0m==Λ0m+κm1 in the sequence (23) and the corresponding eigenfunctions U0m,, U0m+κm1 which are orthonormal in L2(υ). On account of Theorem 2.1 there are κm eigenvalues of problem (13)-(16), which we denote by Λεm+l(η), l=0,,κm1, satisfying

    Λεm+l(η)Λ0m+l as ε0, for l=0,,κm1. (64)

    Let Uεm+l(;η), l=0,,κm1, be the corresponding eigenfunctions among the set of the eigenfunctions which form an orthonormal basis in L2(ϖε), cf. (17).

    Following Section 4.1, for each l=0,,κm1, we take the ansatz for Λεm+l(η)

    Λεm+l=Λ0m+εΛ1m+l(η)+, (65)

    the outer expansion for Uεm+l(;η)

    Uεm+l(x;η)=U0m+l(x;η)+εU1m+l(x;η)+, (66)

    and the inner expansion

    Uεm(x;η)=ε±wm+l±(x2;η)W±(xε)+, (67)

    where the terms Λ1m+l(η), U1m+l(x;η) and wm+l±(x2;η) have to be determined by the matching procedure, cf. Section 4.2, while U0m+l(x;η) is constructed from U0m+l(x) replacing U0m by U0m+l in formula (55), and W± are the solutions (37) to problem (31)-(33).

    By repeating the reasoning in Section 4.2, we obtain formulas for the above mentioned terms in (65), (66) and (67) by replacing index m by m+l in (56)-(62), while we realize that the compatibility condition for each Λ1m+l(η) is satisfied. Indeed, multiplying by U0m+l(x;η), l=0,,κm1 in the partial differential equation satisfied by U1m+l(x;η) (cf. (60))

    ΔxU1m+l(x;η)Λ0mU1m+l(x;η)=Λ1m+l(η)U0m+l(x;η),xϖ0,x10,

    and integrating by parts, we obtain

    ϖ0Λ1m+l(η)U0m+l(x;η)¯U0m+l(x;η)dx=H0(U0m+lx1(0,x2),eiηU0m+lx1(1,x2))p(Ξ)×(U0m+lx1(0,x2),eiηU0m+lx1(1,x2))Tdx2.

    Since the eigenfunctions U0m+l and U0m+l have been computed (cf. the explicit formulas (29) in Remark 1), we conclude now that

    H0U0m+lx1(x1,x2)U0m+lx1(x1,x2)dx2=0, with x1{0,1},ll,

    and, hence, for each l=0,,κm1, the κm compatibility conditions to be satisfied by the pairs (Λ1m+l(η), U1m+l(x;η)), cf. (60), provide Λ1m+l(η) given by

    Λ1m+l(η)=H0¯Bm+l(x2;η)p(Ξ)Bm+l(x2;η)dx2, (68)

    where Bm+l(x2;η) is defined by

    Bm+l(x2;η)=(U0m+lx1(0,x2),eiηU0m+lx1(1,x2))T. (69)

    Therefore we have determined completely all the terms in the asymptotic ans¨atze (65), (66) and (67) for l=0,,κm1.

    In this section, we justify the results obtained by means of matched asymptotic expasions in Section 4. Since the case in which all the eigenvalues of the Dirichlet problem (24) are simple can be a generic property, we first consider this case, cf. Theorem 5.1 and Corollary 5.1, and then the case in which these eigenvalues have a multiplicity greater than 1, cf. Theorem 5.2 and Corollary 5.2. We state the results in Section 5.1 while we perform the proofs in Section 5.2.

    Theorem 5.1. Let mN, let Λ0m be a simple eigenvalue of the Dirichlet problem (24), and let Λ1m(η) be defined in (61) and (62). There exist positive εm and cm independent of η such that, for any ε(0,εm], the eigenvalue Λεm(η) of problem (13)-(16) meets the estimate

    |Λεm(η)Λ0mεΛ1m(η)|cmε3/2 (70)

    and there are no other different eigenvalues in the sequence (18) satisfying (70).

    Theorem 5.1 shows that εΛ1m(η) provides a correction term for Λεm(η) improving the approach to λ0m shown in Theorem 2.1. In particular, it justifies the asymptotic ansatz (52) and formula (61). This corrector depends on the polarization matrix p(Ξ), which is given by the coefficients pτ±pτ±(Ξ), with τ=±, in the decomposition (37), and on the eigenfunction U0m of problem (24), which corresponds to Λ0m and is normalized in L2(υ) (cf. (61) and (62)).

    In order to detect the gaps between consecutive spectral bands (20) it is worthy writing formulas

    Λ1m(η)=B0(m)+B1(m)cos(η), with B0(m)=H0(p++|U0mx1(0,x2)|2+p|U0mx1(1,x2)|2)dx2,B1(m)=2p+H0U0mx1(0,x2)U0mx1(1,x2)dx2, (71)

    which are obtained from (61) and (62). Formula (29) demonstrates that

    B0(m)=(p+++p)H0|U0mx1(0,x2)|2dx2,

    and that the integral in B1(m) does not vanish. We note that B1(m)=0 only in the case when p+=0; if so, p(Ξ) is diagonal and the solutions of (37), W±, decay exponentially when ξ1, respectively. However, we have given examples of cases where p+0 (cf. (47) and (51)).

    Remark 3. Let us consider that the eigenvalue Λ0m coincides with Λ0nq in formula (29) for certain natural n and q. Then, we obtain

    B0(m)=2(p+++p)n2π2,B1(m)=(1)n4p+n2π2,

    and, consequently,

    Λ1m(η)=2(p+++p)n2π2+(1)n4p+n2π2cos(η). (72)

    Corollary 5.1. Under the hypothesis of Theorem 5.1, the endpoints Bε±(m) of the spectral band (20) satisfy the relation

    |Bε±(m)Λ0mε(B0(m)±|B1(m)|)|cmε3/2. (73)

    Hence, the length of the band Bεm is 2ε|B1(m)|+O(ε3/2).

    Note that for the holes such that the polarization matrix (38) satisfies p+0, asymptotically, the bands Bεm have the precise length 2ε|B1(m)|+O(ε3/2) and they cannot reduce to a point, namely to the point Λ0m+εB0(m) (cf. (71) and Remark 3). Also note that if p+=0, Theorem 5.1 still provides a correction term for Λεm(η) which however does not depend on η (cf. (70), (71) and Remark 3), the width of the band being O(ε3/2). Although the length of the band is shorter than in the cases where p+0, bounds in Corollary 5.1 may not be optimal (cf. Remark 3) and further information on the corrector depending on η can be obtained by constructing higher-order terms in the asymptotic ansatz (53).

    Theorem 5.2. Let mN, let Λ0m be an eigenvalue of the Dirichlet problem (24) with multiplity κm>1. Let Λ1m+l(η) defined in (68) and (69) for l=0,,κm1. There exist positive εm and cm independent of η such that, for any ε(0,εm], and for each l=0,,κm1, at least one eigenvalue Λεm+l0(η) of problem (13)-(16) satisfying (64) meets the estimate

    |Λεm+l0(η)Λ0mεΛ1m+l(η)|cmε3/2. (74)

    In addition, when l{0,1,,κm1}, the total multiplicity of the eigenvalues in (18) satisfying (74) is κm.

    Corollary 5.2. Under the hypothesis in Theorem 5.2, the spectral bands Bεm+l associated with Λεm+l(η), for l=0,,κm1, cf. (20), are contained in the interval

    [Λ0m+εmin0lκm1η[π,π]Λ1m+l(η)cmε3/2,Λ0m+εmax0lκm1η[π,π]Λ1m+l(η)+cmε3/2]. (75)

    Hence, the length of the the bands Bεm+l are O(ε) but they may not be disjoint.

    Remark 4. Under the hypothesis of Theorem 5.2, it may happen that, for l=0,,κm1, only the eigenvalue Λεm+l0(η) in the sequence (18) satisfies (74). This depends on the polarization matrix p(Ξ). As a matter of fact, it can be shown by contradiction under the assumption that for two different l the functions Λ1m+l(η) do not intersect at any point η, cf. (71) and (72). For instance, this follows for ω with p+(Ξ)=0.

    Remark 5. Notice that the positive cutoff value λε such that the spectrum σε[λε,) is bounded from above by a positive constant, cf. (9), (20) and (25). In addition, from Theorem 5.2 (cf. Remark 1), we have proved that λεπ2(1+H2) as ε0.

    In this section we prove the results of Theorems 5.1 and 5.2 and of their respective corollaries.

    Proof of Theorem 5.1. Let us fix η in [π,π]. Let us endow the space H1,ηper(ϖε;Γε), with the scalar product Uε,Vε=(Uε,Vε)ϖε+(Uε,Vε)ϖε, and the positive, symmetric and compact operator Tε(η),

    Tε(η)Uε,Vε=(Uε,Vε)ϖεUε,VεH1,ηper(ϖε;Γε). (76)

    The integral identity (17) for problem (13)-(16) can be rewritten as the abstract equation

    Tε(η)Uε(;η)=τε(η)Uε(;η)in H1,ηper(ϖε;Γε),

    with the new spectral parameter

    τε(η)=(1+Λε(η))1. (77)

    Since Tε(η) is compact (cf. e.g, Section I.4 in [31] and Ⅲ.9 in [1]), its spectrum consists of the point τ=0, the essential spectrum, and of the discrete spectrum {τεm(η)}mN which, in view of (18) and (77), constitutes the infinitesimal sequence of positive eigenvalues

    {τεm(η)=(1+Λεm(η))1}mN.

    For the point

    tεm(η)=(1+Λ0m+εΛ1m(η))1, (78)

    cf. (52) and (61), we construct a function UεmH1,ηper(ϖε;Γε) such that

    Uεm;H1,ηper(ϖε;Γε)cm, (79)
    Tε(η)Uεmtεm(η)Uεm;H1,ηper(ϖε;Γε)Cmε3/2, (80)

    where cm and Cm are some positive constants independent of ε(0,εm], with εm>0. These inequalities imply the estimate for the norm of the resolvent operator (Tε(η)tεm(η))1

    (Tε(η)tεm(η))1;H1,ηper(ϖε;Γε)H1,ηper(ϖε;Γε)c1mε3/2,

    with cm=c1mCm>0. According to the well-known formula for self-adjoint operators

    dist(tεm(η),σ(Tε(η))=(Tε(η)tεm(η))1;H1,ηper(ϖε;Γε)H1,ηper(ϖε;Γε)1

    supported by the spectral decomposition of the resolvent (cf., e.g., Section V.5 in [9] and Ch. 6 in [1]), we deduce that the closed segment

    [t^{ \varepsilon}_{m}(\eta)- \mathbf{c}_m \varepsilon^{3/2}, t^{ \varepsilon}_{m}(\eta) + \mathbf{c}_m \varepsilon^{3/2}]

    contains at least one eigenvalue \tau^{ \varepsilon}_{p}(\eta) of the operator T^ \varepsilon(\eta) . Since the eigenvalues of T^ \varepsilon(\eta) satisfy (77) and we get the definition (78), we derive that

    \begin{equation} \left|(1+\Lambda^{ \varepsilon}_{p}(\eta))^{-1}- (1+ \Lambda^{0}_{m}+ \varepsilon\Lambda^{1}_{m}(\eta))^{-1}\right| \leq \mathbf{c}_m \varepsilon^{3/2}. \end{equation} (81)

    Then, simple algebraic calculations (cf. (81) and (25)) show that, for a \varepsilon_m>0 , the estimate

    \begin{equation} \left| \Lambda^{ \varepsilon}_{p}(\eta)- \Lambda^{0}_{m}- \varepsilon\Lambda^{1 }_{m}(\eta)\right| \leq \mathcal{C}_m \varepsilon^{3/2} \end{equation} (82)

    is satisfied with a constant \mathcal{C}_m independent of \varepsilon\in(0, \varepsilon_m] . Due to the convergence with conservation of the multiplicity (22), p = m in (82) and this estimate becomes (70).

    To conclude with the proof of Theorem 5.1, there remains to present a function \mathcal{U}^{ \varepsilon}_{m}\in H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon) enjoying restrictions (79) and (80). In what follows, we construct \mathcal{U}^{ \varepsilon}_{ m} using (63) suitably modified with the help of cut-off functions with "overlapping supports", cf. [19], Ch. 2 in [17] and others. We define

    \begin{equation} \mathcal{V}^{ \varepsilon m}_{out}(x;\eta) = U^{0}_{m}(x;\eta)+ \varepsilon U^{1}_{m}(x;\eta), \end{equation} (83)

    where U^0_m satisfies (55) and U^1_m is the solution of (60) satisfying the boundary conditions (15)-(16) and (59). Similarly, we define

    \begin{equation} \mathcal{V}^{ \varepsilon m}_{in}(x;\eta) = \varepsilon \sum\limits_{\pm}w^m_\pm(x_2;\eta)W^\pm( \varepsilon^{-1}x), \end{equation} (84)

    and

    \begin{equation} \mathcal{V}^{ \varepsilon m}_{mat}(x;\eta) = \varepsilon w^m_\pm(x_2;\eta)( \varepsilon^{-1}|x_1|+p_{\pm\pm})+ \varepsilon w^m_\mp (x_2;\eta) p_{\mp\pm}, \quad \pm x_1 > 0, \end{equation} (85)

    with w^m_\pm defined in (58), and W^\pm and matrix p(\Xi) in Proposition 1 (cf. (38)). Finally, we set

    \begin{equation} \mathcal{U}^{ \varepsilon}_{m}(x;\eta) = X^ \varepsilon(x_1) \mathcal{V}^{ \varepsilon m}_{out}(x;\eta) + \mathcal{X}(x_1) \mathcal{V}^{ \varepsilon m}_{in}(x;\eta) -X^ \varepsilon(x_1) \mathcal{X}(x_1) \mathcal{V}^{ \varepsilon m}_{mat}(x;\eta), \end{equation} (86)

    where X^ \varepsilon and \mathcal{X} are two cut-off functions, both smoothly dependent on the x_1 variable, such that

    \begin{equation} X^ \varepsilon(x_1) = \left\{ \begin{array}{ll} 1, &\hbox{ for } |x_1| > 2R \varepsilon, \\ 0, &\hbox{ for } |x_1| < R \varepsilon, \end{array} \right. \mbox{ and } \quad \mathcal{X}(x_1) = \left\{ \begin{array}{ll} 1, &\hbox{ for } |x_1| < 1/6, \\ 0, &\hbox{ for } |x_1| > 1/3. \end{array} \right. \end{equation} (87)

    Note that (85) takes into account components in both expressions (83) and (84), but the last subtrahend in \mathcal{U}^{ \varepsilon}_{m} compensates for this duplication. In further estimations, term (85) will be joined to either \mathcal{V}^{ \varepsilon m}_{out} or \mathcal{V}^{ \varepsilon m}_{in} in order to obtain suitable bounds.

    First, let us show that \mathcal{U}^{ \varepsilon}_{m} \in H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon) . Indeed, the function defined in (86) enjoys the conditions (15)-(16) and (14). This is due to the fact that \mathcal{U}^{ \varepsilon}_{m} = \mathcal{V}^{ \varepsilon m}_{out} near the sides \{x_1 = \pm1/2, \, x_2\in (0, H)\} and the quasi-periodicity conditions (15)-(16) are verified by both terms in (83). Also, \mathcal{U}^{ \varepsilon}_{m} = \mathcal{V}^{ \varepsilon m}_{in} near the perforation string (21) so that the Dirichlet conditions are fulfilled on the boundary of the perforation string \Gamma^ \varepsilon \cap \varpi^0 because W^\pm satisfy (33). Finally, formulas (58) and (29) assure that w^m_\pm(H;\eta) = w^m_\pm(0;\eta) = 0 and hence the Dirichlet condition is met on \Gamma^ \varepsilon\cap\partial\varpi^0 as well.

    First of all, we recall (83) and (87) to derive

    \begin{multline*} \| \mathcal{U}^{ \varepsilon}_{m} ; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)\| \\\geq \left\| \mathcal{U}^{ \varepsilon}_{m} ; L^2((1/3, 1/2)\times (0, H))\right\| = \| \mathcal{V}^{ \varepsilon m}_{out} ; L^2((1/3, 1/2)\times (0, H))\| \\ \geq \left\|U^{0}_{m} ; L^2((1/3, 1/2)\times (0, H))\right\| - \varepsilon\|U^{1}_{m} ; L^2((1/3, 1/2)\times (0, H))\| \geq c > 0, \end{multline*}

    for a small \varepsilon>0 . Thus, (79) is fulfilled.

    Using (76) and (78), we have

    \begin{multline} \|T^ \varepsilon(\eta) \mathcal{U}^{ \varepsilon}_{m} -t^{ \varepsilon}_{m}(\eta) \mathcal{U}^{ \varepsilon}_{m}; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)\| = \sup\left|\langle T^ \varepsilon(\eta) \mathcal{U}^{ \varepsilon}_{m} -t^{ \varepsilon}_{m}(\eta) \mathcal{U}^{ \varepsilon}_{m}, \mathcal{W}^{ \varepsilon}\rangle\right|\\ = (1+ \Lambda^{0}_{m}+ \varepsilon\Lambda^{1}_{m}(\eta))^{-1}\sup \left|(\nabla \mathcal{U}^{ \varepsilon}_{m}, \nabla \mathcal{W}^{ \varepsilon})_{\varpi^ \varepsilon} -(\Lambda^{0}_{m}+ \varepsilon\Lambda^{1}_{m}(\eta))( \mathcal{U}^{ \varepsilon}_{m}, \mathcal{W}^{ \varepsilon})_{\varpi^ \varepsilon}\right|, \end{multline} (88)

    where the supremum is computed over all \mathcal{W}^{ \varepsilon}\in H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon) such that

    \| \mathcal{W}^{ \varepsilon}; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)\|\leq 1.

    Taking into account the Dirichlet conditions on \partial\omega^ \varepsilon we use the Poincaré and Hardy inequalities, namely, for a fixed T such that \omega\subset \Pi_T\equiv\Pi\cap \{y_1<T \} ,

    \int_{ \Pi_T\setminus\overline{\omega} } \vert U \vert^2\, dy \leq C_{T}\int_{ \Pi_T\setminus\overline{\omega} } \vert \nabla_y U \vert^2 dy \quad \forall U \in H^1( \Pi_T\setminus\overline{\omega} ) , \, U = 0 \mbox{ on } \partial \omega,

    where C_{T} is a constant independent of U , and

    \int_0^\infty \frac{1}{t^2} z(t)^2\, dt\leq 4 \int_0^\infty \left\vert \frac{dz}{dt}(t)\right\vert^2\, dt \quad \forall z\in C^1[0, \infty), \, z(0) = 0.

    Then, we have

    \begin{equation} \|( \varepsilon+|x_1|)^{-1} \mathcal{W}^{ \varepsilon}; L^2(\varpi^ \varepsilon)\|\leq c \|\nabla \mathcal{W}^{ \varepsilon}; L^2(\varpi^ \varepsilon)\|\leq c. \end{equation} (89)

    Clearly, from (71), (1+ \Lambda^{0}_{m}+ \varepsilon\Lambda^{1}_{m}(\eta))^{-1}\leq 1 for a small \varepsilon>0 independent of \eta , and there remains to estimate the last supremum in (88). We integrate by parts, take the Dirichlet and quasi-periodic conditions into account and observe that

    \begin{multline*} \left|(\nabla \mathcal{U}^{ \varepsilon}_{m}, \nabla \mathcal{W}^{ \varepsilon})_{\varpi^ \varepsilon} -(\Lambda^{0}_{m}+ \varepsilon\Lambda^{1}_{m}(\eta))( \mathcal{U}^{ \varepsilon}_{m}, \mathcal{W}^{ \varepsilon})_{\varpi^ \varepsilon}\right|\\ = \left|(\Delta \mathcal{U}^{ \varepsilon}_{m}+(\Lambda^{0}_{m}+ \varepsilon\Lambda^{1}_{m}(\eta)) \mathcal{U}^{ \varepsilon}_{m}, \mathcal{W}^{ \varepsilon})_{\varpi^ \varepsilon}\right|. \end{multline*}

    On the basis of (83)-(86) we write

    \begin{multline} \Delta \mathcal{U}^{ \varepsilon}_{m} + (\Lambda^{0}_{m}+ \varepsilon\Lambda^{1}_{m}(\eta)) \mathcal{U}^{ \varepsilon}_{m}\\ = X^ \varepsilon \left(\Delta U^{0}_{m}+\Lambda^{0}_{m} U^{0}_{m} + \varepsilon(\Delta U^{1}_{m}+\Lambda^{0}_{m} U^{1}_{m} +\Lambda^{1}_{m} U^{0}_{m})+ \varepsilon^2 \Lambda^{1}_{m} U^{1}_{m}\right)\\ \quad \quad\quad + [\Delta, X^ \varepsilon]( \mathcal{V}^{ \varepsilon m}_{out}- \mathcal{V}^{ \varepsilon m}_{mat})+ \mathcal{X}(\Delta \mathcal{V}^{ \varepsilon m}_{in}- X^ \varepsilon\Delta \mathcal{V}^{ \varepsilon m}_{mat})+[\Delta, \mathcal{X}]( \mathcal{V}^{ \varepsilon m}_{in}- \mathcal{V}^{ \varepsilon m}_{mat})\\ + (\Lambda^{0}_{m} + \varepsilon\Lambda^{1}_{m}) \mathcal{X}( \mathcal{V}^{ \varepsilon m}_{in}-X^ \varepsilon \mathcal{V}^{ \varepsilon m}_{mat}) = : S_1^ \varepsilon + S^ \varepsilon_2 + S_3^ \varepsilon + S^ \varepsilon_4 + S_5^ \varepsilon. \quad \quad\, \, \end{multline} (90)

    Here, [\Delta, \chi] = 2\nabla\chi \cdot\nabla+ \Delta \chi is the commutator of the Laplace operator with a function \chi , and the equality [\Delta, X^ \varepsilon \mathcal{X}] = [\Delta, \mathcal{X}]+[\Delta, X^ \varepsilon], which is valid due to the position of supports of functions in (87), is used when distributing terms originated by the last subtrahend in (86). Let us estimate the scalar products (S^ \varepsilon_k , \mathcal{W}^ \varepsilon)_{\varpi^ \varepsilon} for S_k^ \varepsilon in (90).

    Considering S^ \varepsilon_1 , because of (27), (60), (87) and (71), we have that in fact S^ \varepsilon_1 = \varepsilon^2 X^ \varepsilon\Lambda^{1}_{m} U^{1}_{m} , hence

    \left|(S^ \varepsilon_1, \mathcal{W}^ \varepsilon)_{\varpi^ \varepsilon}\right|\leq \varepsilon^2 \Lambda^{1}_{m}(\eta) \|U^{1}_{m} ; L^2(\varpi^ \varepsilon)\|\| \mathcal{W}^{ \varepsilon}; L^2(\varpi^ \varepsilon)\|\leq C_m \varepsilon^2.

    As regards S^ \varepsilon_2 , we take into account that the supports of the functions \partial_{x_1} X^ \varepsilon and \Delta X^ \varepsilon belong to the adherence of the thin domain \varpi^ \varepsilon_{ \varepsilon R} = \{x\in\overline{\varpi^ \varepsilon}:\: |x_1|\in ( \varepsilon R, 2 \varepsilon R)\} , cf. (87). Thus, the error in the Taylor formula up to the second term, and relations (58), (59) and (85) provide

    \begin{eqnarray*} \left| \mathcal{V}^{ \varepsilon m}_{out}(x;\eta)- \mathcal{V}^{ \varepsilon m}_{mat}(x;\eta)\right|&\leq & c(|x_1|^2+ \varepsilon |x_1|), \\ \left|\frac{\partial \mathcal{V}^{ \varepsilon m}_{out }}{\partial x_1}(x;\eta)-\frac{\partial \mathcal{V}^{ \varepsilon m}_{mat }}{\partial x_1}(x;\eta)\right|&\leq &c(|x_1|+ \varepsilon), \quad\pm x_1\in [ \varepsilon R, 2 \varepsilon R]. \end{eqnarray*}

    Above, we have also used the smoothness of the function U_m^1 which holds on account that V = U_m^1e^{-{\rm i} \eta y_1} is a periodic function in the y_1 variable, solution of an elliptic problem with constant coefficients (cf. (60), (15)-(16) and (91)), and therefore it is smooth. Then, we make use of the weighted inequality (89) and write

    \begin{eqnarray*} \left|(S^ \varepsilon_2, \mathcal{W}^ \varepsilon)_{\varpi^ \varepsilon}\right| \leq & \|S^ \varepsilon_2 ; L^2(\varpi^ \varepsilon_{ \varepsilon R})\|\| \mathcal{W}^{ \varepsilon}; L^2(\varpi^ \varepsilon_{ \varepsilon R})\|\leq c \varepsilon \|( \varepsilon+|x_1|)^{-1} \mathcal{W}^{ \varepsilon}; L^2(\varpi^ \varepsilon)\|\\ & \times \left(\int\limits^H_0\int\limits^{2 \varepsilon R}_{ \varepsilon R}\left(\frac{1}{ \varepsilon^2}\left|\frac{ \mathcal{V}^{ \varepsilon m}_{out }}{\partial x_1}-\frac{ \mathcal{V}^{ \varepsilon m}_{mat }}{\partial x_1}\right|^2+\frac{1}{ \varepsilon^4}\left| \mathcal{V}^{ \varepsilon m}_{out}- \mathcal{V}^{ \varepsilon m}_{mat}\right|^2\right)d|x_1|dx_2\right)^\frac 12 \\ \leq & c\left(\frac{1}{ \varepsilon^2} \varepsilon^2+\frac{1}{ \varepsilon^4} \varepsilon^4\right)^\frac 12\left(mes_2 \varpi^ \varepsilon_{ \varepsilon R} \right)^\frac 12 \varepsilon \|( \varepsilon+|x_1|)^{-1} \mathcal{W}^{ \varepsilon}; L^2(\varpi^ \varepsilon)\|\leq c \varepsilon^\frac{3}{2}. \end{eqnarray*}

    Dealing with S^ \varepsilon_3 , we match the definitions of the cut-off functions \chi_\pm and X^ \varepsilon such that X^ \varepsilon(x_1) = \chi_\pm(x_1/ \varepsilon) for \pm x_1>0 (cf. (36)). Recalling formulas (37), (84) and (85), we write

    \begin{multline*} \Delta \mathcal{V}^{ \varepsilon m}_{in}(x;\eta)- X^ \varepsilon(x_1)\Delta \mathcal{V}^{ \varepsilon m}_{mat}(x;\eta)\\ = 2\sum\limits_{\pm} \frac{\partial w^m_\pm}{\partial x_2}(x_2;\eta)\frac{\partial W^\pm}{\partial \xi_2}(y) + \varepsilon \sum\limits_{\pm} \frac{\partial^2 w^m_\pm}{\partial x_2^2}(x_2;\eta)\widetilde{W}^\pm(y), \end{multline*}

    when \pm x_1>0 , respectively. Note that W^\pm are harmonics and both, \partial W^\pm/\partial \xi_2 and \widetilde{W}^\pm decay exponentially as |\xi_1|\to\infty , see Proposition 3.1. Thus,

    \begin{multline*} \left|(S^ \varepsilon_3, \mathcal{W}^ \varepsilon)_{\varpi^ \varepsilon}\right| \leq \\ c\left( \left\|( \varepsilon+|x_1|)\frac{\partial W^\pm}{\partial \xi_2}; L^2(\varpi^ \varepsilon)\right\| + \varepsilon \left\|( \varepsilon+|x_1|)\widetilde{W}^\pm; L^2(\varpi^ \varepsilon)\right\| \right) \left\|\frac{1}{ \varepsilon+|x_1|} \mathcal{W}^{ \varepsilon}; L^2(\varpi^ \varepsilon)\right\| \\ \leq c\left( \int\limits_0^{1/2} ( \varepsilon + t)^2e^{-2\delta t/ \varepsilon} dt\right)^\frac 12 \|\nabla \mathcal{W}^{ \varepsilon}; L^2(\varpi^ \varepsilon)\|\leq c \varepsilon^\frac{3}{2}. \end{multline*}

    Above, obviously, we take the positive constant \delta to be 2\pi /H , cf. Proposition 3.1, and we note that the last integral has been computed to obtain the bound. With the same argument on the exponential decay of \mathcal{V}^{ \varepsilon m}_{in}- X^ \varepsilon \mathcal{V}^{ \varepsilon m}_{mat} , one derives that

    \left|(S^ \varepsilon_5, \mathcal{W}^ \varepsilon)_{\varpi^ \varepsilon}\right|\leq c \varepsilon^\frac{3}{2}.

    Moreover, the supports of the coefficients \partial_{ x_1} \mathcal{X} and \Delta \mathcal{X} in the commutator [\Delta, \mathcal{X}] are contained in the set \overline{\varpi^ \varepsilon } \cap \{1/6<|x_1|<1/3\} while the above-mentioned decay brings the estimate

    \left|(S^ \varepsilon_4, \mathcal{W}^ \varepsilon)_{\varpi^ \varepsilon}\right|\leq c{e^{-2\delta/(3 \varepsilon) }}.

    Revisiting the obtained estimates we find the worst bound c \varepsilon^{3/2} , and this shows (80).

    The fact that the constants \varepsilon_m and c_m of the statement of the theorem are independent of \eta follows from the independence of \eta of the above inequalities throughout the proof. Indeed, we use formulas (55) and (71) for the boundedness of U_m^0 and \Lambda_m^1(\eta) , while we note that the fact that \|U^{1}_{m} ; H^1(\varpi^ \varepsilon)\| is bounded by a constant independent of \eta follows from the definition of the solution of (60) with the quasi-periodic boundary conditions (15)-(16). Further specifying, the change V = U_m^1e^{-{\rm i} \eta y_1} converts the Laplacian into the differential operator

    \begin{equation} -\big(\frac{\partial}{\partial y_1} + {\rm i} \eta \big) \big(\frac{\partial}{\partial y_1} + {\rm i}\eta \big)- \frac{\partial^2}{\partial y_2^2}, \end{equation} (91)

    and therefore, performing this change in (60), gives the solution V\in H^{1 }_{per} (\varpi^0) . Then, as a consequence of the variational formulation of the problem for V in the set of spaces L^2 (\varpi_0)\subset H^{1 }_{per} (\varpi^0) , the bound of \|U^{1}_{m} ; H^1(\varpi^ \varepsilon)\| independently of \eta\in [-\pi, \pi] holds true. Hence, the proof of Theorem 5.1 is completed.

    Proof of Corollary 5.1. Due to the continuity of the function (19), the maximum and minimum of \Lambda_m^ \varepsilon(\eta) for \eta\in [-\pi, \pi] are achieved at two points \eta^\pm_{ \varepsilon, m}\in [-\pi, \pi] . Thus, the endpoints B^{ \varepsilon\pm}(m) of the spectral band (20) are given by B^{ \varepsilon\pm}(m) = \Lambda_m^ \varepsilon(\eta^\pm_{ \varepsilon, m}) .

    In order to show (73) for the maximum B^{ \varepsilon+}(m) , we consider \eta = \eta^+ to be \pi or -\pi in such a way that \Lambda^1(\eta^+) = B_0(m) +|B_1(m)| . Since (70) is satisfied for \eta = \eta^\pm_{ \varepsilon, m} and for \eta = \pm \pi , we write

    \Lambda^{0}_{m}+ \varepsilon B_0(m) + \varepsilon |B_1(m)|-c_m \varepsilon^{3/2} \leq \Lambda_m^{ \varepsilon}(\eta^+) \leq \Lambda^{0}_{m}+ \varepsilon B_0(m) + \varepsilon |B_1(m)| + c_m \varepsilon^{3/2}

    and

    \Lambda^{0}_{m} + \varepsilon \Lambda^1_m(\eta^+_{ \varepsilon, m}) -c_m \varepsilon^{3/2} \leq\Lambda_m^{ \varepsilon}(\eta^+_{ \varepsilon, m}) \leq \Lambda^{0}_{m}+ \varepsilon \Lambda^1_m(\eta^+_{ \varepsilon, m}) + c_m \varepsilon^{3/2}.

    Consequently, from (71), we derive

    \begin{multline*} \Lambda^{0}_{m}+ \varepsilon B_0(m) + \varepsilon |B_1(m)|-c_m \varepsilon^{3/2} \leq \Lambda_m^{ \varepsilon}(\eta^+ ) \leq \Lambda_m^{ \varepsilon}(\eta^+_{ \varepsilon, m}) \\ \leq \Lambda^{0}_{m}+ \varepsilon B_0(m) + \varepsilon |B_1(m)| + c_m \varepsilon^{3/2}, \end{multline*}

    which gives (73) for B^{ \varepsilon+}(m) .

    We proceed in a similar way for the minimum B^{ \varepsilon-}(m) and \eta = \eta^- such that \Lambda^1(\eta^-) = B_0(m) -|B_1(m)| and we obtain (73). Obviously, this implies that B^{ \varepsilon\pm}(m) belong to the interval

    \left [\Lambda^{0}_{m}+ \varepsilon B_0(m) - \varepsilon |B_1(m)|-c_m \varepsilon^{3/2} \, , \, \Lambda^{0}_{m}+ \varepsilon B_0(m) + \varepsilon |B_1(m)| + c_m \varepsilon^{3/2}\right]

    Therefore, the whole band B_m^ \varepsilon is contained in the interval above whose length is 2 \varepsilon |B_1(m)| + O( \varepsilon^{3/2}) and the corollary is proved.

    Proof of Theorem 5.2. This proof holds exactly the same scheme of Theorem 5.1. Indeed, for each l = 0, \cdots, \kappa_m-1 , we follow the reasoning in (76)-(82) and we deduce (cf. (81) and (25)) that for each l , and for a \varepsilon_{m, l}>0 , the estimate

    \begin{equation} \left| \Lambda^{ \varepsilon}_{p}(\eta)- \Lambda^{0}_{m}- \varepsilon\Lambda^{1 }_{m+l}(\eta)\right| \leq \mathcal{C}_{m, l} \varepsilon^{3/2} \end{equation} (92)

    is satisfied for a certain natural p{ \equiv p(l)} and \mathcal{C}_{m, l} independent of \varepsilon\in(0, \varepsilon_{m, l}] . Due to the convergence with conservation of the multiplicity (64), the only possible eigenvalues \Lambda^{ \varepsilon}_{p}(\eta) of problem (13)-(16) satisfying (92) are the set \{ \Lambda^{ \varepsilon}_{m+l}(\eta) \}_{l = 0, \cdots, \kappa_m-1} . Then, it suffices to show that there are \kappa_m linearly independent eigenfunctions associated with the eigenvalues \{ \Lambda^{ \varepsilon}_{p(l)}(\eta) \}_{l = 0, \cdots, \kappa_m-1} in (92), to deduce the result of the theorem.

    We use a classical argument of contradiction (cf. [15] and [25]). We consider the set of functions \{ \mathcal{U}^{ \varepsilon}_{m+l}(x;\eta)\}_{l = 0, \cdots, \kappa_m-1} , constructed in (86), and we verify that they satisfy the almost orthogonality conditions

    \begin{equation} \| \mathcal{U}^{ \varepsilon}_{m+l} ; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)\|\geq \widetilde c_m \quad \mbox{ and }\quad \left \vert \langle \mathcal{U}^{ \varepsilon}_{m+l} \, , \, \mathcal{U}^{ \varepsilon}_{m+l'} \rangle \right \vert \leq \widetilde C_m \varepsilon^{1/2}, \mbox{ with } l\not = l'\, , \end{equation} (93)

    for certain constants \widetilde c_m and \widetilde C_m . Indeed, the first inequality above is a consequence of (79), for l = 0, \cdots, \kappa_m-1 , while the second one follows from the orthogonality of the set of eigenfunctions \{U^0_{m+l}(x, \eta)\}_{l = 0, \cdots, \kappa_m-1} and the definitions (83)-(87).

    Then, we define \widetilde{\mathcal{U }}^ \varepsilon_{m+l} = \mathcal{U}^ \varepsilon_{m+l} \, \| \mathcal{U}^{ \varepsilon}_{m+l} ; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)\|^{-1} and consider \mathcal{W}^ \varepsilon_{m+ l } the projection of T^ \varepsilon(\eta)\widetilde{\mathcal{U }}^{ \varepsilon}_{m+l} -t^{ \varepsilon}_{m}(\eta)\widetilde{\mathcal{U }}^{ \varepsilon}_{m+l} in the space of the eigenfunctions of T^ \varepsilon(\eta) associated with all the eigenvalues \{ \Lambda^{ \varepsilon}_{p(l)}(\eta)\}_{l = 0, \cdots, \kappa_m-1} in (92) for certain constants \mathcal{C}_{m, l} , and more precisely, satisfying

    \begin{equation} \left|(1+\Lambda^{ \varepsilon}_{p(l)}(\eta))^{-1}- (1+ \Lambda^{0}_{m}+ \varepsilon\Lambda^{1}_{m+l}(\eta))^{-1}\right| \leq \widetilde{\mathbf{c }}_m \varepsilon^{3/2}, \end{equation} (94)

    for a constant \widetilde{\mathbf{c }}_m that we shall set later in the proof, cf. (81) and definitions (76)-(78) for the operator T^ \varepsilon(\eta) and the "almost eigenvalue" t^{ \varepsilon}_{m}(\eta) . Then, we show

    \begin{equation} \left \| \, \widetilde{\mathcal{W }}^ \varepsilon_{m+l} -\widetilde{\mathcal{U }}^ \varepsilon_{m+l} ; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon) \right \|\leq \widetilde{\mathcal{C }}_m , \end{equation} (95)

    where \widetilde{\mathcal{W }}^ \varepsilon_{m+l} = { \mathcal{W}^ \varepsilon_{m+l }}\| \mathcal{W}^{ \varepsilon}_{m+l} ; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon)\|^{-1} , and \widetilde{\mathcal{C }}_m = 2\widetilde{\mathbf{c }}_m^{-1} \max_{0\leq l\leq \kappa_m-1} \mathcal{C}_{l, m} . This is due to the fact that

    \left \| \, \widetilde{\mathcal{U }}^ \varepsilon_{m+l} - \mathcal{W}^ \varepsilon_{m+l} ; H^{1, \eta}_{per} (\varpi^ \varepsilon;\Gamma^ \varepsilon) \right \|\leq \widetilde{\mathbf{c }}_m^{-1} \max\limits_{0\leq l\leq \kappa_m-1} \mathcal{C}_{l, m} ,

    and some straightforward computation (cf., eg., Lemma 1 in Ch. 3 of [27]). Now, from (93) and (95) and straightforward computations we obtain

    \begin{equation} \left \vert \langle\widetilde{\mathcal{W }}^{ \varepsilon}_{m+l} \, , \, \widetilde{\mathcal{W }}^{ \varepsilon}_{m+l'} \rangle \right \vert \leq 5\widetilde{\mathcal{C }}_m \, \mbox{ with } \, l\not = l'\, , \end{equation} (96)

    and this allows us to assert that set \{ \widetilde{\mathcal{W }}^ \varepsilon_{m+l}\}_{l = 0, \cdots, \kappa_m-1} defines \kappa_m linearly independent functions. Indeed, to prove it, we proceed by contradiction, by assuming that there are constants \alpha_l^ \varepsilon different from zero such that

    \sum\limits_{l = 0}^{\kappa_m-1} \alpha_l^ \varepsilon \widetilde{\mathcal{W }}^ \varepsilon_{m+l} = 0.

    Let us consider \alpha^{*, \varepsilon} = \max_{0\leq l\leq \kappa_m-1} \vert \alpha_l^ \varepsilon \vert and assume, without any restriction that \alpha^{*, \varepsilon} = \alpha_0^ \varepsilon . Then, we write

    \langle \widetilde{\mathcal{W }}^ \varepsilon_{m}, \widetilde{\mathcal{W }}^ \varepsilon_{m} \rangle \leq \sum\limits_{l = 1}^{\kappa_m-1} \left\vert \frac{\alpha_l^ \varepsilon}{\alpha_0^ \varepsilon} \right\vert \left\vert \langle \widetilde{\mathcal{W }}^ \varepsilon_{m+l}, \widetilde{\mathcal{W }}^ \varepsilon_{m} \rangle \right \vert \leq (\kappa_m-1) 5\widetilde{\mathcal{C }}_m .

    Now, setting (\kappa_m-1)5\widetilde{\mathcal{C }}_m <1 gives a contradiction, since the left hand side takes the value 1 , cf. (96). In this way, we also have fixed \widetilde{\mathbf{c}}_m in (94).

    Thus, \{ \widetilde{\mathcal{W }}^ \varepsilon_{m+l}\}_{l = 0, \cdots, \kappa_m-1} define \kappa_m linearly independent functions, which obviously implies that they are associated with \kappa_m eigenvalues; hence the set of eigenvalues \{ \Lambda^{ \varepsilon}_{p(l)}(\eta) \}_{l = 0, \cdots, \kappa_m-1} coincides with \{ \Lambda^{ \varepsilon}_{m+l}(\eta) \}_{l = 0, \cdots, \kappa_m-1} and this concludes the proof of the theorem.



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