# Spectral gaps in a double-periodic perforated
Neumann waveguide

Delfina Gómeza, Sergei A. Nazarovb, Rafael Orive-Illerac,d, Maria-Eugenia
Pérez-Martíneze
a Departamento de Matemáticas, Estadística y Computación, Universidad de
Cantabria,
Santander, Spain<EMAIL_ADDRESS>
b Institute for Problems in Mechanical Engineering of the Russian Academy of
Sciences,
Saint Petersburg, Russia<EMAIL_ADDRESS>
c Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Madrid, Spain,
d Departamento de Matemáticas, Universidad Autónoma de Madrid,
Madrid, Spain<EMAIL_ADDRESS>
e Departamento de Matemática Aplicada y Ciencias de la Computación,
Universidad de Cantabria,
Santander, Spain<EMAIL_ADDRESS>
Abstract We examine the band-gap structure of the spectrum of the Neumann
problem for the Laplace operator in a strip with periodic dense transversal
perforation by identical holes of a small diameter $\varepsilon>0$. The
periodicity cell itself contains a string of holes at a distance
$O(\varepsilon)$ between them. Under assumptions on the symmetry of the holes,
we derive and justify asymptotic formulas for the endpoints of the spectral
bands in the low-frequency range of the spectrum as $\varepsilon\to 0$. We
demonstrate that, for $\varepsilon$ small enough, some spectral gaps are open.
The position and size of the opened gaps depend on the strip width, the
perforation period, and certain integral characteristics of the holes. The
asymptotic behavior of the dispersion curves near the band edges is described
by means of a ‘fast Floquet variable’ and involves boundary layers in the
vicinity of the perforation string of holes. The dependence on the Floquet
parameter of the model problem in the periodicity cell requires a serious
modification of the standard justification scheme in homogenization of
spectral problems. Some open questions and possible generalizations are
listed.

Keywords: band-gap structure, spectral perturbations, homogenization,
perforated media, Neumann-Laplace operator, waveguide

MSC: 35B27, 35P05, 47A55, 35J25, 47A10

## 1 Introduction

In this section, we formulate the spectral problem under consideration, cf.
Section 1.1, and provide some background which relates it with a parametric
family of homogenization problems, the so-called model problem. In Section 1.3
we provide the structure of the paper while its framework in the literature is
in Section 1.2.

### 1.1 Formulation of the problem

Let

$\Pi=\\{x=(x_{1},x_{2})\in{\mathbb{R}}^{2}:\,x_{1}\in{\mathbb{R}},x_{2}\in(0,H)\\}$
(1.1)

be an open strip of width $H>0$ and let $\omega$ be a domain in the plane
${\mathbb{R}}^{2}$ which is bounded by a smooth simple closed curve
$\partial\omega$ and has the compact closure
$\overline{\omega}=\omega\cup\partial\omega$ inside $\Pi$. Let
$\varepsilon=N^{-1}$ where $N$ is a large natural number. We introduce the
strip $\Pi^{\varepsilon}$, see Figure 1 a), obtained from $\Pi$ perforated by
the family of holes

$\omega^{\varepsilon}(j,k)=\\{x:\,\varepsilon^{-1}(x_{1}-j,x_{2}-\varepsilon
kH)\in\omega\\},\quad j\in{\mathbb{Z}},\,\,k=0,1,\dots,N-1,$ (1.2)

distributed periodically along line segments parallel to the ordinate
$x_{2}$-axis. Each hole is homothetic to $\omega$ of ratio $\varepsilon$ and
translation of $\varepsilon\omega=\omega^{\varepsilon}(0,0).$ Namely,

$\Pi^{\varepsilon}=\Pi\setminus\overline{\Omega^{\varepsilon}}\quad\mbox{\rm
where}\quad\Omega^{\varepsilon}=\bigcup\limits_{j\in{\mathbb{Z}}}\bigcup\limits_{k=0}^{N-1}\omega^{\varepsilon}(j,k).$
(1.3)

The period of perforation along the abscissa $x_{1}$-axis in the domain
$\Pi^{\varepsilon}$ is made equal to one by rescaling, which also fixes the
dimensionless width $H>0$. The period along the $x_{2}$-axis is $\varepsilon
H$ with $\varepsilon\ll 1.$

Figure 1:  a) The perforated strip $\Pi^{\varepsilon}$. b) The periodicity
cell $\varpi^{\varepsilon}$.

We consider the spectral Neumann problem

$-\Delta_{x}u^{\varepsilon}(x)=\lambda^{\varepsilon}u^{\varepsilon}(x),\,\,x\in\Pi^{\varepsilon},$
(1.4) $\partial_{\nu}u^{\varepsilon}(x)=0,\,\,x\in\partial\Pi^{\varepsilon},$
(1.5)

where $\partial_{\nu}$ is the directional derivative along the outward normal
while $\partial_{\nu}=\pm\partial/\partial x_{2}$ at the lateral sides
$\Upsilon_{\pm}=\\{x:\,x_{1}\in{\mathbb{R}},x_{2}=(H\pm H)/2\\}$ of the strip
(1.1). The variational formulation of the problem (1.4), (1.5) reads: to find
a function $u^{\varepsilon}$ in the Sobolev space $H^{1}(\Pi^{\varepsilon})$,
$u^{\varepsilon}\not\equiv 0$, and a number
$\lambda^{\varepsilon}\in{\mathbb{C}}$ such that the integral identity

$\begin{array}[]{c}(\nabla_{x}u^{\varepsilon},\nabla_{x}v^{\varepsilon})_{\Pi^{\varepsilon}}=\lambda^{\varepsilon}(u^{\varepsilon},v^{\varepsilon})_{\Pi^{\varepsilon}}\quad\forall
v^{\varepsilon}\in H^{1}(\Pi^{\varepsilon})\end{array}$ (1.6)

is valid, cf. [19]. Here, $\nabla_{x}=\mbox{\rm grad}$,
$\Delta_{x}=\nabla_{x}\cdot\nabla_{x}$ is the Laplace operator and
$(\,,)_{\Pi^{\varepsilon}}$ stands for the natural scalar product in the
Lebesgue space $L^{2}(\Pi^{\varepsilon})$.

Since the bi-linear form on the left of (1.6) is positive, symmetric, and
closed in $H^{1}(\Pi^{\varepsilon})$, problem (1.6) is associated with a
positive self-adjoint operator $A^{\varepsilon}$ in the Hilbert space
$L^{2}(\Pi^{\varepsilon})$ with the domain

${\cal D}(A^{\varepsilon})=\\{u^{\varepsilon}\in
H^{2}(\Pi^{\varepsilon}):\,\eqref{(5)}\,\,\mbox{\rm is verified}\\}.$

Clearly, the spectrum $\sigma(A^{\varepsilon})$ belongs to the closed real
positive semi-axis
$[0,+\infty)=\overline{{\mathbb{R}}_{+}}\subset{\mathbb{C}}$. Moreover,
according to the Floquet–Bloch–Gelfand theory, see for instance [40, 42, 33,
18, 1], the spectrum gets the band-gap structure

$\begin{array}[]{c}\sigma(A^{\varepsilon})=\bigcup\limits_{p\in{\mathbb{N}}}\beta_{p}^{\varepsilon}\,,\end{array}$
(1.7)

where the bands $\beta_{p}^{\varepsilon}$ are connected and compact sets in
$\overline{{\mathbb{R}}_{+}}=[0,+\infty)$. The $\beta_{p}^{\varepsilon}$ are
related to the eigenvalues, cf. (2.9), of the model problem in the periodicity
cell

$\begin{array}[]{c}\varpi^{\varepsilon}=\\{x\in\Pi^{\varepsilon}:\,|x_{1}|<1/2\\},\end{array}$
(1.8)

see Figure 1 b), which itself constitutes a homogenization problem, cf.
(2.2)–(2.5). The spectral bands $\beta_{p}^{\varepsilon}$ and
$\beta_{p+1}^{\varepsilon}$ may intersect each other but can also be disjoint
so that the spectral gap $\gamma_{p}^{\varepsilon}$ becomes open between them.
Recall that an open spectral gap is recognized as a nontrivial open interval
in ${\mathbb{R}}_{+}$ which is free of the essential spectrum but has both
endpoints in it. If
$\beta^{\varepsilon}_{p}\cap\beta^{\varepsilon}_{p+1}\not=\varnothing$, then
we say that the gap $\gamma^{\varepsilon}_{p}$ is closed. In Figure 5 the open
spectral gaps correspond with the projections of the shaded bands on the
ordinate axis.

The main goal of our paper is to show that, under certain restrictions on the
width $H$ and the perforation shape, the problem (1.4), (1.5) can get at least
one open gap in its spectrum. Also, we aim to derive asymptotic formulas for
the position and geometric characteristics of several bands and gaps in the
low-frequency range of the spectrum. It should be mentioned that the
traditional homogenization procedure in the problem (1.4), (1.5) does not help
to detect open gaps. The crucial role is played by the boundary layer
phenomenon, cf. Section 3, while the width of the gaps is expressed in terms
of certain integral characteristics of the Neumann hole $\omega$ of unit size
in the strip $\Pi$ with the periodicity conditions at its lateral sides, cf.
(7.3), (7.7) and Remark 3.4. At the same time, we construct explicitly only
the main correction term in the asymptotics of eigenvalues of the model
problem in the periodicity cell and analyze different situations when this
term is not sufficient to conclude whether a concrete spectral gap is actually
open or not (see Section 8). Moreover, for a technical reason, cf. Section
4.5, and for simplification of asymptotic structures, we make the assumption

$\omega=\\{\xi=(\xi_{1},\xi_{2})\in{\mathbb{R}}^{2}:\,(\xi_{1},H-\xi_{2})\in\omega\\}.$
(1.9)

which means that the holes possess the mirror symmetry (see Figure 2). Also,
for simplicity, we assume that the boundary of $\omega$ is of class
$C^{\infty}.$

Figure 2: The strip $\Xi$ with two different possible geometries for the hole
$\omega$.

### 1.2 State of art

The continuous spectrum in a cylindrical waveguides of different physical
nature is always a ray $[\lambda_{\dagger},+\infty)$ so that, above the cutoff
value $\lambda_{\dagger}\geq 0$ wave processes surely occur. The spectrum of a
periodic waveguide gets far complicated band-gap structure (1.7) and the
spectral bands implying passing zones for waves can be separated from each
other by spectral gaps which do not permit propagation of waves with the
corresponding frequencies and, therefore, become stopping zones. This
phenomenon is used in different engineering devices, such as wave filters and
wave dampers.

Within the Floquet–Bloch–Gelfand theory, see e.g. [8, 40, 42, 41, 18, 33, 12],
mathematical studies of spectra with the band-gap structures need to find out
the eigenvalues of spectral elliptic boundary value problems which are posed
in the periodicity cell and involve an additional continuous parameter
$\eta\in[-\pi,\pi]$, the Floquet parameter or the Gelfand dual variable. It is
a very rare situation when such a problem admits explicit solutions while
computational methods become rather expensive to present the whole family of
dispersion curves, projections of which on the ordinate $\lambda$-axis involve
the spectral bands. As usual, variational and asymptotic methods help to prove
or disprove the existence of open spectral gaps in a certain range of the
spectrum and to estimate their geometrical characteristics.

There are numerous publications in which open spectral gaps are detected due
to high-contrast of coefficients in differential operators or shape
irregularities of the periodicity cells, see [15, 16, 45, 29, 5, 4, 3, 6] and
[26, 34, 39, 35, 7] among others. Such singular perturbations often provide
disintegration of the periodicity cells in the limit and, as a result, the
appearance of sufficiently wide gaps in the low- and/or middle-frequency
ranges of the spectrum. Both variational and asymptotic methods have been
employed in the cited papers to detect and describe those gaps.

Another way to open spectral gaps related to the splitting of band edges, is
used in our paper. In the case when two spectral bands of the limit problem,
cf. Section 2.2, intersect but just touch each other at a point, that is,
there is a common edge of the bands, small perturbations of the coefficients
or of the boundary may lead to a separation of these bands and the opening of
a narrow gap between them, cf. Sections 6.1, 6.4, 7.1 and 7.2. This effect is
well-known in the physical literature, but its mathematical study using
operators theory and spectral perturbation methods started in [27, 10, 11,
31]. In this paper a new type of the singular perturbation of the periodicity
cell is analyzed by means of the homogenization technique and several ways to
open spectral gaps are highlighted.

Finally, let us mention that, from a geometrical viewpoint, [32] is the
closest paper in the literature. It addresses the Dirichlet perforation in a
quantum waveguide. The asymptotic of the spectrum of the equation (1.4), with
the Dirichlet condition $u^{\varepsilon}=0$ on $\partial\Pi^{\varepsilon}$ is
considered, finding out the position and sizes of the spectral gaps and bands.
However, the results differ very much from those in this paper. Indeed,
roughly speaking, the Dirichlet spectrum consists of small, of order
$O(\varepsilon)$, spectral bands which are separated from each other by
spectral gaps of width $O(1)$. In contrast, the Neumann spectrum here
considered consists of long, of order $O(1)$, bands which are separated from
each other by short spectral gaps of order $O(\varepsilon)$, or even less. The
latter makes the asymptotic analysis much more complicated and delicate; in
particular, it becomes multiscale in several variables, not only in the
geometrical ones, but also in the Floquet parameter. As outlined above, the
justification procedure also becomes much more complicated. For a link between
the model problem in the waveguide with Neumann or Dirichlet conditions, let
us mention [14].

Figure 3: The dispersion curves of the limit problem in the cases $H<1/3$,
$H=1/\sqrt{8}$ and $H=1/2.$

### 1.3 Architecture of the paper

In Section 2 we formulate the model spectral problem in the periodicity cell,
b) in Figure 1, which is itself a parametric spectral homogenization problem.
We obtain the homogenized problem by the classical homogenization theory in
perforated media, see, e.g., [20], that is, a problem in the rectangular
periodicity cell without perforations. We list explicit solutions of the
homogenized problem and we study the dispersion curves which form the trusses
in Figures 3 and 4, while we classify the truss nodes, namely, the crossing
points of the dispersion curves. In Section 2.3, we show the convergence
result for the spectrum of the model problem towards that of the homogenized
one as a consequence of another stronger one, which also allows a perturbation
of the Floquet-parameter.

In Section 3 we discuss the boundary layer phenomenon arising in the vicinity
of the perforation. In particular, we examine several solutions of the Laplace
equation in the unbounded strip $\Pi$ with the only hole $\overline{\omega}$,
and we introduce the integral characteristics $m_{1}(\Xi)$, $m_{2}(\Xi)$ and
$m_{3}(\Xi)$ for the Neumann problem in the domain
$\Xi=\Pi\setminus{\overline{\omega}}$ (see Figure 2 and (3.1)) with the
periodicity conditions on the lateral sides, cf. the traditional harmonic
polarization and virtual mass tensors in the exterior domain
${\mathbb{R}}^{2}\setminus\overline{\omega}$ in [38].

In Section 4 we perform the preliminary formal asymptotic analysis for simple
eigenvalues using the method of matched asymptotic expansions, cf. [43, 41,
17, 21] for two scale asymptotic expansions. In Section 5 we derive error
estimates in the case of simple eigenvalues which will help us to detect open
gaps after a much more thorough analysis of multiple eigenvalues. The
perturbation of crossing dispersion curves require serious modifications of
the standard asymptotic procedures because we can no longer deal with a fixed
Floquet parameter but we must investigate the asymptotic behavior of the
eigenvalues in a neighborhood of each truss node, i.e., with the Floquet
parameter in a certain short interval. Recalling an idea from paper [27], in
Section 6 we introduce a fast Floquet parameter to describe this behavior and
detect, in different situations, open spectral gaps of width $O(\varepsilon)$,
cf. Figure 5 a)–b), which appear due to splitting of the nodes marked with
$\circ$ and $\square$ in Figure 4. This involves the characterization of the
projections of the shaded rectangles on the ordinate axis in Figure 5, which
represent the narrow gaps.

Figure 4: The dispersion curves in the limit problem in the cases a)
$H\in(1/\sqrt{8},1/2)$, b) $H\in(1/2,1)$ but $H\neq 1/\sqrt{3}$ and c)
$H\in(1,+\infty).$

It should be noted that our detailed calculation in Section 5 demonstrates
that the first correction term in the eigenvalue asymptotics is not able to
assure the gap opening and we need to discuss higher-order asymptotic terms.
In fact, Section 6 is devoted to deriving the formal asymptotic analysis and
its justification in the case where the eigenvalue under consideration is
multiple and therefore gives rise to a node of the dispersion curves in Figure
4 a)–b) for the homogenized problem; in particular, we consider the nodes
$(\eta_{\mbox{\small{$\circ$}}},\Lambda_{\mbox{\small{$\circ$}}})=(0,4\pi^{2})$
and
$(\eta_{\mbox{\tiny$\square$}},\Lambda_{\mbox{\tiny$\square$}})=(\pm\pi,\pi^{2})$.
Providing the error estimates for the whole range of the Floquet parameter
adds the most complication to the justification scheme (see Theorems 5.1, 6.1
and 6.3). The common procedure for deriving error estimates in the
homogenization theory does not support our conclusions of opening spectral
gaps (see Section 7) because the model problem in the periodicity cell
$\varpi^{\varepsilon}$ depends on the Floquet parameter $\eta\in[-\pi,\pi]$
and the eigenvalues (2.11) of the limit problem in $\varpi^{0}$ change their
multiplicity at the nodes. As usual, to provide appropriate error estimates,
we use a well-known result on almost eigenvalues and eigenfunctions from the
spectral perturbation theory (see [44] and Lemma 5.3). However, we need to
construct different approximations for eigenfunctions in the vicinity of the
nodes and at a certain distance from them. This is performed in Sections 6.1
and 6.4. As a result, we find proper small bounds for asymptotic remainders
that justify our formal computations of the band edges and gap width. It turns
out that these bounds are uniform in $\eta$ but in different regions.

As regards the spectral model problem, the somehow classical convergence of
the spectrum towards that of the homogenized problem is in Corollary 2.2. We
obtain this result as a consequence of a more general convergence result, cf.
Theorem 2.1, which allows a certain perturbation of the Floquet variable.
This result is new in the literature of model problems for waveguides, and
shows somehow a strong stability of the model problem on the parameter $\eta$.
It becomes essential to control the number of eigenvalues below certain
constants, cf. Propositions 2.3 and 2.4. Theorem 5.1 provides some estimates
which establish the closeness of eigenvalues depending on $\varepsilon$ and
the first three dispersion curves. As a consequence, Corollary 5.2 gives a
uniform bound for the convergence rate of the first eigenvalue at a certain
distance from the nodes. Theorems 6.1 and 6.3 involve a correcting term and
improve convergence rates in a small neighborhood of the above mentioned nodes
$(0,4\pi^{2})$ and $(\pm\pi,\pi^{2})$. Combining the results in Sections 5 and
6, in Section 7, we determine the existence of opening gaps and their width
depending on $H$, cf. Theorems 7.1 and 7.2. Finally, in Section 8, we provide
some hints on open problems for other nodes in Figure 4 and other geometrical
configurations, cf. Figures 13 and 14.

## 2 The model problem in the periodicity cell

In this section, we introduce the spectral model problem and its limit
problem, both of which depend on the Floquet parameter $\eta\in[-\pi,\pi]$,
see Sections 2.1 and 2.2 respectively. In Section 2.3, we show the spectral
convergence as $\varepsilon\to 0$, and its stability under a certain
perturbation of the parameter $\eta$. In particular, this proves useful for
controlling the eigenvalue number of the model problem below some bounds.

### 2.1 The FBG-transform and the quasi-periodicity conditions

The Floquet–Bloch–Gelfand transform (the FBG-transform in short), see [13, 40,
33, 42, 18],

$\begin{array}[]{c}\displaystyle
u^{\varepsilon}(x)\,\,\mapsto\,\,U^{\varepsilon}(x;\eta)=\frac{1}{\sqrt{2\pi}}\,\sum\limits_{p\in{\mathbb{Z}}}e^{-ip\eta}u^{\varepsilon}(x_{1}+p,x_{2})\end{array}$
(2.1)

converts the problem (1.4), (1.5) in the infinite waveguide
$\Pi^{\varepsilon}$ into a boundary value problem in the periodicity cell
$\varpi^{\varepsilon}$ defined by (1.8), cf. Figure 1 b).

This problem consists of the differential equation

$-\Delta_{x}U^{\varepsilon}(x;\eta)=\Lambda^{\varepsilon}(\eta)U^{\varepsilon}(x;\eta),\,\,x\in\varpi^{\varepsilon},$
(2.2)

the quasi-periodicity conditions on the lateral walls

$U^{\varepsilon}\Big{(}\frac{1}{2},x_{2};\eta\Big{)}=e^{i\eta}U^{\varepsilon}\Big{(}-\frac{1}{2},x_{2};\eta\Big{)},\,\,x_{2}\in(0,H),$
(2.3) $\frac{\partial U^{\varepsilon}}{\partial
x_{1}}\Big{(}\frac{1}{2},x_{2};\eta\Big{)}=e^{i\eta}\frac{\partial
U^{\varepsilon}}{\partial
x_{1}}\Big{(}-\frac{1}{2},x_{2};\eta\Big{)},\,\,x_{2}\in(0,H),$ (2.4)

and the Neumann condition on the remaining part of the boundary of the
periodicity cell (1.8)

$\partial_{\nu}U^{\varepsilon}(x{\color[rgb]{0,0,0};\eta})=0,\,\,x\in\\{x\in\partial\varpi^{\varepsilon}:\,|x_{1}|<1/2\\}.$
(2.5)

Here, $\eta\in[-\pi,\pi]$ is the Floquet parameter while
$\Lambda^{\varepsilon}(\eta)$ and $U^{\varepsilon}(\cdot;\eta)$, respectively,
are the new notations for the eigenvalues and eigenfunctions in the model
problem. Notice that $x\in\Pi^{\varepsilon}$ on the left of (2.1) but
$x\in\varpi^{\varepsilon}$ on the right. Basic properties of the FBG-transform
can be found in the above-cited publications.

Figure 5: The dispersion curves of the perturbed problem with the mirror
symmetry of the hole $\omega$ in the cases a) $H\in(1/\sqrt{8},1/2)$, b)
$H\in(1/2,1)$ but $H\neq 1/\sqrt{3}$ and c) $H\in(1,+\infty).$ Spectral gaps
are the projections of the shaded rectangles on the ordinate $\Lambda$-axis.

The variational statement of the problem (2.2)–(2.5) appeals to the integral
identity [19]

$\begin{array}[]{c}(\nabla_{x}U^{\varepsilon},\nabla_{x}V^{\varepsilon})_{\varpi^{\varepsilon}}=\Lambda^{\varepsilon}(U^{\varepsilon},V^{\varepsilon})_{\varpi^{\varepsilon}}\quad\forall
V^{\varepsilon}\in H^{1,\eta}_{per}(\varpi^{\varepsilon}),\end{array}$ (2.6)

where $H^{1,\eta}_{per}(\varpi^{\varepsilon})$ is the Sobolev space of
functions satisfying the stable quasi-periodicity conditions  (2.3) and (2.4).
In view of the compact embedding $H^{1}(\varpi^{\varepsilon})\subset
L^{2}(\varpi^{\varepsilon})$, the positive self-adjoint operator
$A^{\varepsilon}(\eta)$ in $L^{2}(\varpi^{\varepsilon})$ associated with the
problem (2.6), cf. [9, Section 10.2], has a discrete spectrum constituting the
unbounded monotone sequence of eigenvalues

$\begin{array}[]{c}0\leq\Lambda^{\varepsilon}_{1}(\eta)\leq\Lambda^{\varepsilon}_{2}(\eta)\leq\dots\leq\Lambda^{\varepsilon}_{p}(\eta)\leq\dots\to+\infty,\quad{\color[rgb]{0,0,0}\mbox{
as }p\to+\infty},\end{array}$ (2.7)

where their multiplicity is taken into account. Furthermore, the functions

$\begin{array}[]{c}[-\pi,\pi]\ni\eta\,\,\mapsto\,\,\Lambda^{\varepsilon}_{p}(\eta),\quad
p\in{\mathbb{N}},\end{array}$ (2.8)

are continuous and $2\pi$-periodic (see again any of the above-cited
references). Hence, the sets in (1.7)

$\begin{array}[]{c}\beta^{\varepsilon}_{p}=\\{\Lambda^{\varepsilon}_{p}(\eta):\,\eta\in[-\pi,\pi]\\}\subset\overline{{\mathbb{R}}_{+}}\end{array}$
(2.9)

are closed, connected, and finite segments. Indeed, formulas (1.7) and (2.9)
for the spectrum of the operator $A^{\varepsilon}(\eta)$ and the boundary-
value problem (1.4), (1.5) are well-known in the framework of the
Floquet–Bloch–Gelfand theory.

### 2.2 The limit problem and the limit dispersion curves

In Section 2.3 we will prove the relationship

$\begin{array}[]{c}\displaystyle\Lambda^{\varepsilon}_{p}(\eta)\,\,\to\Lambda^{0}_{p}(\eta)\,\,\mbox{\rm
as}\,\,\varepsilon\to+0\end{array}$ (2.10)

between entries of the sequence (2.7) and those of the sequence

$\begin{array}[]{c}0\leq\Lambda^{0}_{1}(\eta)\leq\Lambda^{0}_{2}(\eta)\leq\dots\leq\Lambda^{0}_{p}(\eta)\leq\dots\to+\infty,\quad{\color[rgb]{0,0,0}\mbox{
as }p\to+\infty},\end{array}$ (2.11)

which consists of eigenvalues of the limit problem in the rectangle

$\varpi^{0}=\\{x:\,|x_{1}|<1/2,\,x_{2}\in(0,H)\\}$

obtained from the periodicity cell (1.8) by filling all voids, cf. (3.2).
Above, the convention of repeated eigenvalues has been adopted, and the limit
problem is also referred to as homogenized problem. It involves the
differential equation

$\begin{array}[]{c}-\Delta{\color[rgb]{0,0,0}{{}_{x}}}U^{0}(x;\eta)=\Lambda^{0}(\eta)U^{0}(x;\eta),\,\,x\in\varpi^{0},\end{array}$
(2.12)

the Neumann conditions on the horizontal sides of the rectangle

$\begin{array}[]{c}\displaystyle\frac{\partial U^{0}}{\partial
x_{2}}(x_{1},0{\color[rgb]{0,0,0};\eta})=\frac{\partial U^{0}}{\partial
x_{2}}(x_{1},H{\color[rgb]{0,0,0};\eta})=0,\,\,x_{1}\in\Big{(}-\frac{1}{2},\frac{1}{2}\Big{)},\end{array}$
(2.13)

and the quasi-periodicity conditions on its vertical sides, cf. (2.3) and
(2.4),

$\begin{array}[]{c}\displaystyle
U^{0}\Big{(}\frac{1}{2},x_{2};\eta\Big{)}=e^{i\eta}U^{0}\Big{(}-\frac{1}{2},x_{2};\eta\Big{)},\\\
\\\ \displaystyle\frac{\partial U^{0}}{\partial
x_{1}}\Big{(}\frac{1}{2},x_{2};\eta\Big{)}=e^{i\eta}\frac{\partial
U^{0}}{\partial
x_{1}}\Big{(}-\frac{1}{2},x_{2};\eta\Big{)},\,\,x_{2}\in(0,H).\end{array}$
(2.14)

This problem has the following explicit eigenvalues and eigenfunctions

$\displaystyle\begin{array}[]{ll}\Lambda^{0}_{jk}(\eta)=(\eta+2\pi
j)^{2}+\pi^{2}\frac{k^{2}}{H^{2}},\\\ &\qquad
j\in{\mathbb{Z}},\,\,k\in{\mathbb{N}}_{0}={\mathbb{N}}\cup\\{0\\}.\\\
U^{0}_{jk}(x;\eta)=e^{i(\eta+2\pi j)x_{1}}\cos\Big{(}\pi
k\frac{x_{2}}{H}\Big{)},\end{array}$ (2.15)

It should be mentioned that renumeration of the eigenvalues in (2.15) is
needed to compose the monotone sequence (2.11).

Figure 6: The hypothetical dispersion curves of the perturbed problem without
the mirror symmetry of the hole $\omega$ in the cases a)
$H\in(1/\sqrt{8},1/2)$, b) $H\in(1/2,1)$ but $H\neq 1/\sqrt{3}$ and c)
$H\in(1,+\infty).$ Many more spectral gaps are opened than in Figure 5.

Graphs of several eigenvalues (2.15) of the problem (2.12)–(2.14), that is,
the dispersion curves, are drawn in Figure 4 a)–c), respectively, for the
following cases:

$\mbox{a)}\quad
H\in\Big{(}\frac{1}{\sqrt{8}},\frac{1}{2}\Big{)}\qquad\mbox{b)}\quad
H\in\Big{(}\frac{1}{2},1\Big{)}\qquad\mbox{c)}\quad H\in(1,+\infty).$

Figure 3 also displays dispersion curves of the limit problem in the cases
$H\in(0,1/3)$, $H=1/\sqrt{8}$ and $H=1/2$, respectively. Figures 3 and 4 show
the great variety of behaviors of the dispersion curves and, consequently, the
complexity to open spectral gaps depending on $H$.

As was mentioned in Section 1.2 and depicted in Figure 5, due to the
perturbation by holes of the periodicity cell, the dispersion curves for
problem (2.2)–(2.5) may separate at the truss nodes marked with the signs
$\square$ and $\circ$ in Figure 4 (see Section 7).

### 2.3 The convergence results

First, we obtain some estimates for the eigenvalues and provide an extension
for eigenfunctions over the whole $\varpi^{0}$ necessary to show the
convergence. The constants appearing throughout the section $c_{m}$ and
$C_{m}$, are independent of both variables $\varepsilon$ and $\eta$.

Let $\eta\in[-\pi,\pi]$ be fixed and let $U^{\varepsilon}_{m}(\cdot;\eta)\in
H^{1,\eta}_{per}(\varpi^{\varepsilon})$ be an eigenfunction of the problem
(2.6) corresponding to the eigenvalue $\Lambda^{\varepsilon}_{m}(\eta)$. The
minimax principle assures the estimate

$\begin{array}[]{c}\displaystyle\Lambda^{\varepsilon}_{m}(\eta)\leq
c_{m}\,\,\mbox{\rm for}\,\,\varepsilon\in(0,\varepsilon_{m}]\end{array}$
(2.16)

with some positive $\varepsilon_{m}$ and $c_{m}$.  Indeed, we write

$\Lambda^{\varepsilon}_{m}(\eta)=\min_{E_{m}^{\varepsilon}\subset
H^{1,\eta}_{per}(\varpi^{\varepsilon})}\max_{V\in E_{m}^{\varepsilon},V\neq
0}\frac{(\nabla_{x}V,\nabla_{x}V)_{\varpi^{\varepsilon}}}{(V,V)_{\varpi^{\varepsilon}}},$

where the minimum is computed over the set of subspaces $E_{m}^{\varepsilon}$
of $H^{1,\eta}_{per}(\varpi^{\varepsilon})$ with dimension $m$. To prove
(2.16), we take a particular $E_{m}^{\varepsilon}$ that we construct as
follows: we consider the eigenfunctions corresponding to the $m$ first
eigenvalues of the mixed eigenvalue problem in the rectangle
$(1/4,1/2)\times(0,H)$, with Neumann condition on the part of the boundary
$\\{1/2\\}\times(0,H)$, and Dirichlet condition on the rest of the boundary.
We extend these eigenfunctions by zero for $x\in[0,1/4]\times(0,H)$, and by
symmetry for $x\in[-1/2,0]\times(0,H)$. Finally, multiplying these
eigenfunctions by $e^{i\eta x_{1}}$ gives $E_{m}^{\varepsilon}$ and the right
hand side of (2.16) holds.

Let us construct an extension for the eigenfunctions to be uniformly bounded
in $H^{1}(\varpi^{0})$. Being normalized in $L^{2}(\varpi^{\varepsilon})$, on
account of (2.16), the eigenfunctions of (2.6) satisfy

$\|\nabla_{x}U^{\varepsilon}_{m};L^{2}(\varpi^{\varepsilon})\|^{2}=\Lambda^{\varepsilon}_{m}(\eta)\|U^{\varepsilon}_{m};L^{2}(\varpi^{\varepsilon})\|^{2}=\Lambda^{\varepsilon}_{m}(\eta)\leq
c_{m}\,\,\mbox{\rm for}\,\,\varepsilon\in(0,\varepsilon_{m}].$

Therefore, we can extend $U^{\varepsilon}_{m}(\cdot;\eta)$ over the holes
(3.2) and obtain a function ${\widehat{U}}^{\,\varepsilon}_{m}(\cdot;\eta)\in
H^{1,\eta}_{per}(\varpi^{0})$ such that for
$\varepsilon\in(0,\varepsilon_{m}]$,

$\begin{array}[]{c}\displaystyle{\widehat{U}}^{\,\varepsilon}_{m}(x;\eta)=U^{\varepsilon}_{m}(x;\eta)\,\,\mbox{
for }x\in\varpi^{\varepsilon}\quad\mbox{ and }\vspace{0.2cm}\\\
\|{\widehat{U}}^{\,\varepsilon}_{m}(\cdot;\eta);H^{1}(\varpi^{0})\|\leq
C\|U^{\varepsilon}_{m};H^{1}(\varpi^{\varepsilon})\|\leq C_{m},\end{array}$
(2.17)

see, for example, Section I.4.2 in [37] for  such an extension. In addition,
from (2.17) and the estimate

$\|V;L^{2}(\varpi^{0}\cap\\{|x_{1}|<\varepsilon\\})\|^{2}\leq
C\varepsilon\|V;H^{1}(\varpi^{0})\|^{2}\quad\forall V\in H^{1}(\varpi^{0})$

(see, e.g., Lemma 2.4 in [22]), they satisfy

$\begin{array}[]{c}\displaystyle\|{\widehat{U}}^{\,\varepsilon}_{m}(\cdot;\eta);L^{2}(\varpi^{0}\setminus\varpi^{\varepsilon})\|^{2}\leq
c\varepsilon\|U^{\varepsilon}_{m};H^{1}(\varpi^{\varepsilon})\|^{2}\leq
C_{m}\varepsilon.\end{array}$ (2.18)

Moreover, the following results state the spectral convergence for problem
(2.6),  as $\varepsilon\to 0$. As a matter of fact, Theorem 2.1 also shows the
stability of the limit of the spectrum of the perturbation problem (2.2)–(2.5)
under any perturbation of the Floquet-parameter $\eta$.

######  Theorem 2.1.

For each sequence $\\{(\varepsilon_{k},\eta_{k})\\}_{k=1}^{\infty}$ such that
$\varepsilon_{k}\to 0$ and $\eta_{k}\to\widehat{\eta}\in[-\pi,\pi]$ as
$k\to+\infty$, the eigenvalues $\Lambda_{m}^{\varepsilon_{k}}(\eta_{k})$ of
problem (2.2)–(2.5) when $(\varepsilon,\eta)\equiv(\varepsilon_{k},\eta_{k})$
converge, as $k\to+\infty,$ towards the eigenvalues of problem (2.12)–(2.14)
for $\eta\equiv\widehat{\eta}$ and there is conservation of the multiplicity.
Namely, for each $m=1,2,\cdots$, the convergence

$\Lambda^{\varepsilon_{k}}_{m}(\eta_{k})\to\Lambda^{0}_{m}(\widehat{\eta}),\quad\mbox{
as }k\to+\infty,$

holds, where $\Lambda^{0}_{m}(\widehat{\eta})$ is the m-th eigenvalue in the
sequence

$0\leq\Lambda^{0}_{1}(\widehat{\eta})\leq\Lambda^{0}_{2}(\widehat{\eta})\leq\dots\leq\Lambda^{0}_{m}(\widehat{\eta})\leq\dots\to+\infty,\quad\mbox{
as }m\to+\infty,$

of eigenvalues of (2.12)–(2.14) for $\eta\equiv\widehat{\eta}$, which are
counted according to their multiplicities. In addition, we can extract a
subsequence, still denoted by $\varepsilon_{k}$, such that the extension
$\\{{\widehat{U}}_{m}^{\,\varepsilon_{k}}\\}_{m=1}^{\infty}$ converge in
$L^{2}(\varpi^{0})$, as $\varepsilon_{k}\to 0$, towards the eigenfunctions of
(2.12)–(2.14) for $\eta\equiv\widehat{\eta}$, which form an orthonormal basis
of $L^{2}(\varpi^{0})$.

###### Proof.

For each $\varepsilon_{k}$ and $\eta_{k}$, we write the integral equation
satisfied by the eigenvalue $\Lambda_{m}^{\varepsilon_{k}}(\eta_{k})$ and
corresponding eigenfunction ${U}_{m}^{\,\varepsilon_{k}}(\cdot,\eta_{k})$:

$\begin{array}[]{c}(\nabla_{x}U_{m}^{\varepsilon_{k}},\nabla_{x}V^{\varepsilon_{k}})_{\varpi^{\varepsilon_{k}}}=\Lambda_{m}^{\varepsilon_{k}}(\eta_{k})(U_{m}^{\varepsilon_{k}},V^{\varepsilon_{k}})_{\varpi^{\varepsilon_{k}}}\quad\forall
V^{\varepsilon_{k}}\in
H^{1,\eta_{k}}_{per}(\varpi^{\varepsilon_{k}}),\end{array}$ (2.19)

Since the constants appearing (2.17) and (2.18) are independent of
$\eta\in[-\pi,\pi]$ and $\varepsilon$, the estimates hold $\varepsilon$ and
$\eta$ ranging in sequences of $\\{\varepsilon_{k}\\}_{k}$ and
$\\{\eta_{k}\\}_{k}$, in the statement of the theorem. Thus, we use an
extension of $U^{\varepsilon_{k}}_{m}(\cdot;\eta)$ over the holes (3.2)
denoted by ${\widehat{U}}^{\,\varepsilon_{k}}_{m}(\cdot;\eta)\in
H^{1,\eta_{k}}_{per}(\varpi^{0})$, which satisfies

$\begin{array}[]{c}\displaystyle\|{\widehat{U}}^{\,\varepsilon_{k}}_{m}(\cdot;\eta_{k});H^{1}(\varpi^{0})\|\leq
C_{m},\mbox{ \quad and \quad}\vspace{0.2cm}\\\
\displaystyle\|{\widehat{U}}^{\,\varepsilon_{k}}_{m}(\cdot;\eta_{k});L^{2}(\varpi^{0}\setminus\varpi^{\varepsilon_{k}})\|^{2}\leq
C_{m}\varepsilon_{k}\|{U}^{\,\varepsilon_{k}}_{m}(\cdot;\eta_{k});H^{1}(\varpi^{\varepsilon_{k}})\|^{2}\leq
C_{m}\varepsilon_{k};\end{array}$ (2.20)

for sufficiently small $\varepsilon_{k}$, with a constant $C_{m}$ independent
of $\varepsilon_{k}$ and $\eta_{k}$.

In view of (2.16) and (2.20), for each fixed $m\geq 1$ and for each
subsequence of $k$, we can extract a subsequence, still denoted by $k$, such
that

$\begin{array}[]{c}\displaystyle\Lambda^{\varepsilon_{k}}_{m}(\eta_{k})\rightarrow\mbox{\boldmath$\widehat{\Lambda}$}^{0}_{m},\quad\mbox{as
}k\to+\infty,\vspace{0.2cm}\\\
\\!\\!{\widehat{U}}^{\,\varepsilon_{k}}_{m}(\cdot;\eta_{k})\rightarrow{\bf\widehat{U}}^{0}_{m}\,\,\mbox{\rm
weakly\,\,in}\,\,H^{1}(\varpi^{0})\,\,\mbox{\rm
and\,\,strongly\,\,in}\,\,L^{2}(\varpi^{0}),\quad\mbox{as
}k\to+\infty,\end{array}$ (2.21)

for some real number $\mathbf{\widehat{\Lambda}}^{0}_{m}\geq 0$ and some
function ${\bf\widehat{U}}^{0}_{m}\in H^{1}(\varpi^{0})$ which we determine
below depending on $\widehat{\eta}$.

We take a test function $V\in C^{\infty}(\overline{\varpi^{0}})$ verifying the
periodicity condition at the lateral sides of $\varpi^{0}$ and consider
$V^{k}=Ve^{i\eta_{k}x_{1}}$ which satisfies the quasi-periodicity condition in
(2.14) with $\eta\equiv\eta_{k}$. For $V^{\varepsilon_{k}}\equiv V^{k}$, we
rewrite the integral identity (2.19) in the form

$(\nabla_{x}{\widehat{U}}^{\,\varepsilon_{k}}_{m},\nabla_{x}V^{k})_{\varpi^{0}}-\Lambda^{\varepsilon_{k}}_{m}(\eta_{k})({\widehat{U}}^{\,\varepsilon_{k}}_{m},V^{k})_{\varpi^{0}}=(\nabla_{x}{\widehat{U}}^{\,\varepsilon_{k}}_{m},\nabla_{x}V^{k})_{\varpi^{0}\setminus\varpi^{\varepsilon_{k}}}-\Lambda^{\varepsilon_{k}}_{m}(\eta)({\widehat{U}}^{\,\varepsilon_{k}}_{m},V^{k})_{\varpi^{0}\setminus\varpi^{\varepsilon_{k}}}.$
(2.22)

According to (2.20) and (2.16), the modulo of the right-hand side of (2.22)
does not exceed

$\begin{array}[]{c}\displaystyle(1+c_{m}\varepsilon_{k})\|{\widehat{U}}^{\,\varepsilon_{k}}_{m};H^{1}(\varpi^{0})\|\,\|V^{k};H^{1}(\varpi^{0}\setminus\varpi^{\varepsilon_{k}})\|\leq
c_{m}(V^{k})\mbox{\rm mes}_{2}(\varpi^{0}\setminus\varpi^{\varepsilon_{k}})\\\
\\\ \leq
c_{m}\max_{x\in\overline{\varpi^{0}}}(|V^{k}(x)|,|\partial_{x_{1}}V^{k}(x)|,|\partial_{x_{2}}V^{k}(x)|)\varepsilon_{k}\leq
C_{m}(V)(1+\eta_{k})\varepsilon_{k}.\end{array}$ (2.23)

Thus, the right hand side of (2.22) converges towards zero as $k\to+\infty$.
Let us analyze the left hand side in further detail.

In order to do this, let us consider the well-known change
$\widehat{U}_{m}^{\varepsilon_{k}}(\cdot;\eta_{k})=V_{m}^{\varepsilon_{k}}(\cdot;\eta_{k})e^{i\eta_{k}x_{1}}$
which converts the Laplacian into the differential operator

$-\Big{(}\frac{\partial}{\partial
x_{1}}+i\eta\Big{)}\Big{(}\frac{\partial}{\partial
x_{1}}+i\eta\Big{)}-\frac{\partial^{2}}{\partial x_{2}^{2}},$

while the $\eta_{k}$ quasi-periodicity condition for
$\widehat{U}_{m}^{\varepsilon_{k}}(\cdot;\eta_{k})$ becomes a periodicity
condition for $V_{m}^{\varepsilon_{k}}(\cdot;\eta_{k})$. Consequently,
$V_{m}^{\varepsilon_{k}}(\cdot;\eta_{k})\in H^{1}_{per}(\varpi^{0})$ and since
$\eta_{k}\to\widehat{\eta}$ as $k\to+\infty$ (equivalently,
$\varepsilon_{k}\to 0$), we also have the bound for $V_{m}^{\varepsilon_{k}}$
in $H^{1}_{per}(\varpi^{0})$ which holds uniformly in $\eta_{k}$ and
$\varepsilon_{k}$, and a convergence of $V_{m}^{\varepsilon_{k}}$ (by
subsequences, still denoted by $k$) towards a function $V^{0}_{m}\in
H^{1}_{per}(\varpi^{0})$ holds in the weak topology of $H^{1}(\varpi^{0})$.
Let us show that

$V_{m}^{0}={\bf\widehat{U}}_{m}^{0}e^{-i\widehat{\eta}x_{1}}.$ (2.24)

To do this, it suffices to show

$\|\widehat{U}_{m}^{\varepsilon_{k}}(\cdot;\eta_{k})e^{-i\eta_{k}x_{1}}-{\bf\widehat{U}}_{m}^{0}e^{-i\widehat{\eta}x_{1}};L^{2}(\varpi^{0})\|\to
0\quad\mbox{ as }k\to+\infty,$

and we check this by considering

$\begin{split}&\|\widehat{U}_{m}^{\varepsilon_{k}}(\cdot;\eta_{k})e^{-i\eta_{k}x_{1}}-{\bf\widehat{U}}_{m}^{0}e^{-i\widehat{\eta}x_{1}};L^{2}(\varpi^{0})\|\\\
&\quad\leq\|\big{(}\widehat{U}_{m}^{\varepsilon_{k}}(\cdot;\eta_{k})-{\bf\widehat{U}}_{m}^{0}\big{)}e^{-i\eta_{k}x_{1}};L^{2}(\varpi^{0})\|+\|{\bf\widehat{U}}_{m}^{0}\big{(}e^{-i\eta_{k}x_{1}}-e^{-i\widehat{\eta}x_{1}}\big{)};L^{2}(\varpi^{0})\|,\end{split}$

the convergence (2.21), the smoothness of the exponential function and the
convergence of $\eta_{k}$.

Introducing the change $V^{k}=Ve^{i\eta_{k}x_{1}}$ in (2.22), we write

$\begin{array}[]{l}\displaystyle(\nabla_{x}{\widehat{U}}^{\,\varepsilon_{k}}_{m},\nabla_{x}(Ve^{i\eta_{k}x_{1}}))_{\varpi^{0}}-\Lambda^{\varepsilon_{k}}_{m}(\eta_{k})({\widehat{U}}^{\,\varepsilon_{k}}_{m},Ve^{i\eta_{k}x_{1}})_{\varpi^{0}}\vspace{0.2cm}\\\
=(\nabla_{x}{\widehat{U}}^{\,\varepsilon_{k}}_{m},\nabla_{x}(Ve^{i\widehat{\eta}x_{1}}))_{\varpi^{0}}+(\nabla_{x}{\widehat{U}}^{\,\varepsilon_{k}}_{m},\nabla_{x}(V(e^{i\eta_{k}x_{1}}-e^{i\widehat{\eta}x_{1}})))_{\varpi^{0}}\vspace{0.2cm}\\\
\quad-\Lambda^{\varepsilon_{k}}_{m}(\eta_{k})({\widehat{U}}^{\,\varepsilon_{k}}_{m},Ve^{i\widehat{\eta}x_{1}})_{\varpi^{0}}-\Lambda^{\varepsilon_{k}}_{m}(\eta_{k})({\widehat{U}}^{\,\varepsilon_{k}}_{m},V(e^{i\eta_{k}x_{1}}-e^{i\widehat{\eta}x_{1}}))_{\varpi^{0}}\,.\end{array}$
(2.25)

Let us show

$Ve^{i\eta_{k}x_{1}}\to Ve^{i\widehat{\eta}x_{1}}\mbox{ in
}H^{1}(\varpi^{0})\quad\mbox{ as }k\to+\infty$ (2.26)

and therefore, from (2.21), also the convergence

$(\nabla_{x}{\widehat{U}}^{\,\varepsilon_{k}}_{m},\nabla_{x}(V(e^{i\eta_{k}x_{1}}-e^{i\widehat{\eta}x_{1}})))_{\varpi^{0}}-\Lambda^{\varepsilon_{k}}_{m}(\eta_{k})({\widehat{U}}^{\,\varepsilon_{k}}_{m},V(e^{i\eta_{k}x_{1}}-e^{i\widehat{\eta}x_{1}})))_{\varpi^{0}}\mathop{\displaystyle{\stackrel{{\scriptstyle\scriptstyle{k\to+\infty}}}{{\hbox
to40.0pt{\rightarrowfill}}}}}0,$

holds. Indeed, on account of the smoothness of $V$, we have

$\begin{split}&\|V(e^{i\eta_{k}x_{1}}-e^{i\widehat{\eta}x_{1}});H^{1}(\varpi^{0})\|^{2}\\\
&\,\,\leq
C(V)\Big{(}\|e^{i\eta_{k}x_{1}}-e^{i\widehat{\eta}x_{1}};L^{2}(\varpi^{0})\|^{2}+\|\eta_{k}e^{i\eta_{k}x_{1}}-\widehat{\eta}e^{i\widehat{\eta}x_{1}};L^{2}(\varpi^{0})\|^{2}\Big{)}\mathop{\displaystyle{\stackrel{{\scriptstyle\scriptstyle{k\to+\infty}}}{{\hbox
to40.0pt{\rightarrowfill}}}}}0,\end{split}$

for a certain positive constant $C(V)$, and this shows (2.26).

Then, taking limits in (2.25) as $k\to\infty$, on account of (2.21), (2.23),
(2.26) and (2.24), we obtain the integral identity

$(\nabla_{x}{\bf\widehat{U}}^{0}_{m},\nabla_{x}(Ve^{i\widehat{\eta}x_{1}}))_{\varpi^{0}}-\mbox{\boldmath$\widehat{\Lambda}$}^{0}_{m}({\bf\widehat{U}}^{0}_{m},Ve^{i\widehat{\eta}x_{1}})_{\varpi^{0}}=0\quad\forall
V\in C^{\infty}(\overline{\varpi^{0}})\cap H^{1}_{per}(\varpi^{0}),$

while, by a completion argument, we can write

$(\nabla_{x}{\bf\widehat{U}}^{0}_{m},\nabla_{x}(Ve^{i\widehat{\eta}x_{1}}))_{\varpi^{0}}-\mbox{\boldmath$\widehat{\Lambda}$}^{0}_{m}({\bf\widehat{U}}^{0}_{m},Ve^{i\widehat{\eta}x_{1}})_{\varpi^{0}}=0\quad\forall
V\in H^{1}_{per}(\varpi^{0}),$

or equivalently,

$(\nabla_{x}{\bf\widehat{U}}^{0}_{m},\nabla_{x}V)_{\varpi^{0}}-\mbox{\boldmath$\widehat{\Lambda}$}^{0}_{m}({\bf\widehat{U}}^{0}_{m},V)_{\varpi^{0}}=0\quad\forall
V\in H^{1,\widehat{\eta}}_{per}(\varpi^{0}).$ (2.27)

On account of (2.24), also ${\bf\widehat{U}}_{m}^{0}\in
H^{1,\widehat{\eta}}_{per}(\varpi^{0})$ and, consequently, (2.27) is nothing
but the weak formulation of (2.12)–(2.14) for $\eta\equiv\widehat{\eta}$.

Furthermore,

$1=\|U^{\varepsilon_{k}}_{m}(\cdot,\eta_{k});L^{2}(\varpi^{\varepsilon_{k}})\|^{2}=\|{\widehat{U}}^{\,\varepsilon_{k}}_{m}(\cdot,\eta_{k});L^{2}(\varpi^{0})\|^{2}-\|{\widehat{U}}^{\,\varepsilon_{k}}_{m}(\cdot,\eta_{k});L^{2}(\varpi^{0}\setminus\varpi^{\varepsilon_{k}})\|^{2}$

and taking limits as $k\to+\infty$, on account of (2.20) and (2.21), gives

$\|{\bf\widehat{U}}^{0}_{m};L^{2}(\varpi^{0})\|^{2}=1.$

This, together with (2.27), identifies
$(\mbox{\boldmath$\widehat{\Lambda}$}^{0}_{m},{\bf\widehat{U}}_{m}^{0})$ with
an eigenpair of (2.12)–(2.14) when $\eta\equiv\widehat{\eta}$.

Therefore, we conclude that ${\bf\widehat{\Lambda}}_{m}^{0}$ is an eigenvalue
with the corresponding eigenfunction ${\bf\widehat{U}}_{m}^{0}$ of the limit
problem (2.12)–(2.14) when $\eta\equiv\widehat{\eta}$, and we get a dependence
of ${\bf\widehat{\Lambda}}^{0}_{m}$ on $\widehat{\eta}$, so we write
${\bf\widehat{\Lambda}}^{0}_{m}:={\bf\widehat{\Lambda}}^{0}_{m}(\widehat{\eta})$.

Note that the extracted subsequences and limits may depend on $m$. However,
using a diagonalization argument, for each sequence of $k$, we can extract
another subsequence, still denoted by $k$, but independent of $m$, such that
(2.21) holds $\forall m\in{\mathbb{N}}$. Then, by the construction, we have
obtained an increasing sequence

$0\leq{\bf\widehat{\Lambda}}^{0}_{1}(\widehat{\eta})\leq{\bf\widehat{\Lambda}}^{0}_{2}(\widehat{\eta})\leq\dots\leq{\bf\widehat{\Lambda}}^{0}_{m}(\widehat{\eta})\leq\dots.$
(2.28)

In what follows we prove that the sequence
$\\{{\bf\widehat{\Lambda}}^{0}_{m}(\widehat{\eta})\\}_{m=1}^{\infty}$
converges towards infinity while the whole sequence coincides with that in
(2.11) when $\eta\equiv\widehat{\eta}$.

From the orthogonality of $U^{\varepsilon_{k}}_{m}(\cdot;\eta_{k})$ in
$L^{2}(\varpi^{\varepsilon_{k}})$, we write

$\displaystyle({\widehat{U}}^{\,\varepsilon_{k}}_{m}(\cdot;\eta_{k}),{\widehat{U}}^{\,\varepsilon_{k}}_{n}(\cdot;\eta_{k}))_{\varpi^{0}}=({\widehat{U}}^{\,\varepsilon_{k}}_{m}(\cdot;\eta_{k}),{\widehat{U}}^{\,\varepsilon_{k}}_{n}(\cdot;\eta_{k}))_{\varpi^{0}\setminus\varpi^{\varepsilon_{k}}}\quad\forall
m,n\in{\mathbb{N}},\,\,m\not=n$

and use (2.20) to get the orthogonality of the eigenfunctions
$\\{{\bf\widehat{U}}^{0}_{m}\\}_{m=1}^{\infty}$ in $L^{2}(\varpi^{0})$. This
shows that the sequence in (2.28) converges towards infinity as $m\to\infty$.

In order to show that with the above limits (2.28) we reach all the
eigenvalues in the entry (2.11) when $\eta\equiv\widehat{\eta}$, namely, that
$\Lambda^{0}_{m}(\widehat{\eta})=\mathbf{\widehat{\Lambda}}^{0}_{m}(\widehat{\eta})$,
it suffices to show that the set
$\\{{\bf\widehat{U}}^{0}_{m}\\}_{m=1}^{\infty}$ forms a basis of
$L^{2}(\varpi^{0})$. Indeed, this is a classical process of contradiction
(see, for instance, Section III.1 of [37] and Section III.9.1 of [2]). In this
way, we have proved that (2.10) holds for any $p\in\mathbb{N}$, where
$\\{{\bf\widehat{\Lambda}}^{0}_{p}(\widehat{\eta})\\}_{p=1}^{\infty}$ are the
set of eigenvalues of (2.12)–(2.14) when $\eta\equiv\widehat{\eta}$ and the
eigenfunctions $\\{{\bf\widehat{U}}^{0}_{p}\\}_{p=1}^{\infty}$ form an
orthonormal basis of $L^{2}(\varpi^{0})$. Consequently, the sequence (2.28)
coincides with (2.11), and the theorem is proved. ∎

######  Corollary 2.2.

For any $\eta\in[-\pi,\pi]$, the eigenvalues $\Lambda_{m}^{\varepsilon}(\eta)$
of problem (2.2)–(2.5) in the sequence (2.7) converge, as $\varepsilon\to+0,$
towards the eigenvalues of problem (2.12)–(2.14) in the sequence (2.11) and
there is conservation of the multiplicity. In addition, for each sequence, we
can extract a subsequence, still denoted by $\varepsilon$, such that the
extensions $\\{{\widehat{U}}_{m}^{\,\varepsilon}\\}_{m=1}^{\infty}$ converge
in $L^{2}(\varpi^{0})$, as $\varepsilon\to 0$, towards the eigenfunctions of
(2.12)–(2.14), which form an orthonormal basis of $L^{2}(\varpi^{0})$. Also,
for each eigenfunction $U_{p}^{0}$ of (2.12)–(2.14) associated with the
eigenvalue $\Lambda_{p}^{0}(\eta)$ of multiplicity $n_{p}$,
$\Lambda_{p}^{0}(\eta)=\Lambda_{p+1}^{0}(\eta)=\cdots=\Lambda_{p+n_{p}-1}^{0}(\eta)$
in (2.11), there is a linear combination $U^{\varepsilon}$ of eigenfunctions
corresponding to the eigenvalues
$\Lambda_{p}^{\varepsilon}(\eta),\,\Lambda_{p+1}^{\varepsilon}(\eta),\,\dots,\,\Lambda_{p+n_{p}-1}^{\varepsilon}(\eta)$
in (2.7), that converges towards $U_{p}^{0}$ in $L^{2}(\omega).$

###### Proof.

The first two assertions hold from Theorem 2.1 taking $\eta_{k}\equiv\eta$
fixed for all $k$ with minor modifications. Moreover, the last result is
obtained using the technique in Theorem III.1.7 in [37]. ∎

In addition to the bounds (2.16), we state the following lower bounds for the
first eigenvalues of problem (2.6).

######  Proposition 2.3.

Let $H\in(0,1)$. Let $\delta_{1}>0$ (and $<\pi$). Then, there exists a
positive constant $\varepsilon_{1}=\varepsilon_{1}(H,\delta_{1})$ such that
the entries $\Lambda^{\varepsilon}_{2}(\eta)$ and
$\Lambda^{\varepsilon}_{3}(\eta)$ of the eigenvalue sequence (2.7) meet the
estimates

$\displaystyle\Lambda_{2}^{\varepsilon}(\eta)>\pi^{2}+K_{1}\qquad$
$\displaystyle\mbox{for }\eta\in
I_{1}=[-\pi+\delta_{1},\pi-\delta_{1}],\,\varepsilon<\varepsilon_{1},$ (2.29)
$\displaystyle\Lambda_{3}^{\varepsilon}(\eta)>\pi^{2}+K_{2}\qquad$
$\displaystyle\mbox{for }\eta\in[-\pi,\pi],\,\varepsilon<\varepsilon_{1},$
(2.30)

where

$K_{1}=\min\left\\{2\pi\delta_{1},\frac{\pi^{2}(1-H^{2})}{2H^{2}}\right\\},\qquad
K_{2}=\min\left\\{2\pi^{2},\frac{2\pi^{2}(1-H^{2})}{3H^{2}}\right\\}.$ (2.31)

###### Proof.

We proceed by contradiction, denying (2.29). This implies that for any
$\varepsilon_{1}>0$ there exist $\varepsilon<\varepsilon_{1}$ and $\eta\in
I_{1}$ such that $\Lambda_{2}^{\varepsilon}(\eta)\leq\pi^{2}+K_{1}.$ It is
clear that we can take a sequence $\\{\varepsilon_{k}\\}_{k=1}^{\infty}$ such
that $\varepsilon_{k}\to 0$ as $k\to+\infty$, and an associated sequence
$\\{{\eta_{k}}\\}_{k=1}^{\infty}$ which is bounded from above and from below,
and satisfies

$\Lambda_{2}^{\varepsilon_{k}}(\eta_{k})\leq\pi^{2}+K_{1}.$ (2.32)

By subsequences, we can construct a sequence (still denoted by $k$) such that

$(\eta_{k},\varepsilon_{k})\to(\widehat{\eta},0)\hbox{ as }k\to+\infty,$

for certain $\widehat{\eta}\in I_{1}$. Let us show that this last assertion
leads us to a contradiction.

According to Theorem 2.1, taking limits in (2.32), we get
$\Lambda^{0}_{2}(\widehat{\eta})\leq\pi^{2}+K_{1}.$ Since
$\Lambda^{0}_{2}(\widehat{\eta})$ can only be $(2\pi-|\widehat{\eta}|)^{2}$ or
$(\pi/H)^{2}+|\widehat{\eta}|^{2}$ with $\widehat{\eta}\in I_{1}$, we obtain

$\min\left\\{(\pi+\delta_{1})^{2},\frac{\pi^{2}}{H^{2}}\right\\}\leq\Lambda^{0}_{2}(\widehat{\eta})\leq\pi^{2}+K_{1},$

and this contradicts the hypothesis on the chosen $K_{1}$. Consequently,
(2.29) is proved.

Analogously, we proceed by contradiction, denying (2.30). Thus, as in the
proof of (2.29), there exists $\widehat{\eta}\in[-\pi,\pi]$ such that
$\Lambda^{0}_{3}(\widehat{\eta})\leq\pi^{2}+K_{2}.$ Since
$\Lambda^{0}_{3}(\widehat{\eta})$ can be only $(2\pi+|\widehat{\eta}|)^{2}$ or
$(\pi/H)^{2}+|\widehat{\eta}|^{2}$ ($\widehat{\eta}\in[-\pi,\pi]),$

$\min\left\\{4\pi^{2},\frac{\pi^{2}}{H^{2}}\right\\}\leq\Lambda^{0}_{3}(\widehat{\eta})\leq\pi^{2}+K_{2},$

and this contradicts the hypothesis on the chosen $K_{2}$. ∎

######  Proposition 2.4.

Let $H\in(0,1/2)$. Let $\delta_{3}>0$ (and $<\pi$). Then, there exists a
positive constant $\varepsilon_{1}=\varepsilon_{1}(H,\delta_{3})$ such that
the entries $\Lambda^{\varepsilon}_{3}(\eta)$ and
$\Lambda^{\varepsilon}_{4}(\eta)$ of the eigenvalue sequence (2.7) meet the
estimates

$\displaystyle\Lambda_{3}^{\varepsilon}(\eta)>4\pi^{2}+K_{3}\qquad$
$\displaystyle\mbox{for }\eta\in
I_{3}=[-\pi,-\delta_{3}]\cup[\delta_{3},\pi],\,\varepsilon<\varepsilon_{1},$
(2.33) $\displaystyle\Lambda_{4}^{\varepsilon}(\eta)>4\pi^{2}+K_{4}\qquad$
$\displaystyle\mbox{for }\eta\in[-\pi,\pi],\,\varepsilon<\varepsilon_{1},$
(2.34)

where

$K_{3}=\min\left\\{4\pi\delta_{3},\frac{\pi^{2}(1-4H^{2})}{H^{2}}\right\\}>0,\qquad
K_{4}=\min\left\\{4\pi^{2},\frac{\pi^{2}(1-4H^{2})}{2H^{2}}\right\\}>0.$
(2.35)

###### Proof.

First we prove (2.33) with the same analysis as in Proposition 2.3. We proceed
by contradiction, denying (2.33). Thus, as in the proof of (2.29), there
exists $\widehat{\eta}\in I_{3}$ such that
$\Lambda^{0}_{3}(\widehat{\eta})\leq 4\pi^{2}+K_{3}.$ Since $H<1/2$,
$\Lambda^{0}_{3}(\widehat{\eta})$ can only be $(2\pi+|\widehat{\eta}|)^{2}$ or
$(\pi/H)^{2}+|\widehat{\eta}|^{2}$ ($\widehat{\eta}\in I_{3}$). Therefore, we
get

$\min\left\\{(2\pi+\delta_{3})^{2},\frac{\pi^{2}}{H^{2}}+\delta^{2}_{3}\right\\}\leq\Lambda^{0}_{3}(\widehat{\eta})\leq
4\pi^{2}+K_{3},$

and this contradicts the hypothesis on the chosen $K_{3}$.

Finally, we proceed by contradiction, denying (2.34). Thus, as in the proof of
(2.29), there exists $\widehat{\eta}\in[-\pi,\pi]$ such that
$\Lambda^{0}_{4}(\widehat{\eta})\leq 4\pi^{2}+K_{4}.$ Since $H<1/2$,
$\Lambda^{0}_{4}(\widehat{\eta})$ can only be $(4\pi-|\widehat{\eta}|)^{2}$ or
$(\pi/H)^{2}+|\widehat{\eta}|^{2}$ for $\widehat{\eta}\in[-\pi,\pi]$.
Therefore, we get

$\min\left\\{9\pi^{2},\frac{\pi^{2}}{H^{2}}\right\\}\leq\Lambda^{0}_{4}(\widehat{\eta})\leq
4\pi^{2}+K_{4},$

and this contradicts the hypothesis on the chosen $K_{4}$. ∎

## 3 The boundary layer phenomenon in the periodicity cell

The traditional results of the homogenization theory given in Corollary 2.2 do
not help to conclude on the splitting of band edges and in this section we
examine special solutions of a boundary-value problem in the strip $\Pi$ with
the only hole $\overline{\omega}$ of unit size. Let us define

$\Xi:=\Pi\setminus\overline{\omega}\,.$ (3.1)

Although we apply these solutions under the symmetry assumption (1.9), we only
use it in Section 3.4; see Figure 2.

### 3.1 The problems in $\Xi$ and their solvability

According to [32], near the perforation string

$\begin{array}[]{c}\displaystyle\omega^{\varepsilon}(0,0),\dots,\omega^{\varepsilon}(0,N-1)\subset\varpi^{0}=\\{x:\,{\color[rgb]{0,0,0}|x_{1}|<1/2},x_{2}\in(0,H)\\},\end{array}$
(3.2)

cf., (1.2), there appears a boundary layer which is described in the stretched
coordinates

$\begin{array}[]{c}\xi=(\xi_{1},\xi_{2})=\varepsilon^{-1}{\color[rgb]{0,0,0}(x_{1},x_{2}-\varepsilon
kH)}\end{array}$ (3.3)

by means of a family of special solutions to the Laplace equation

$\begin{array}[]{c}-\Delta_{\xi}W(\xi)=0,\,\,\xi\in{\color[rgb]{0,0,0}\Xi,}\end{array}$
(3.4)

or the Poisson equation

$\begin{array}[]{c}\displaystyle-\Delta_{\xi}W(\xi)=F(\xi),\,\,\xi\in{\color[rgb]{0,0,0}\Xi},\end{array}$
(3.5)

with the periodicity conditions

$\begin{array}[]{c}\displaystyle W(\xi_{1},H)=W(\xi_{1},0),\,\,\frac{\partial
W}{\partial\xi_{2}}(\xi_{1},H)=\frac{\partial
W}{\partial\xi_{2}}(\xi_{1},0),\,\,\xi_{1}\in{\mathbb{R}},\end{array}$ (3.6)

and the Neumann condition on the boundary of the hole $\overline{\omega}$
inside the strip (1.1), either homogeneous

$\begin{array}[]{c}\partial_{\nu(\xi)}W(\xi)=0,\,\,\xi\in\partial\omega,\end{array}$
(3.7)

or inhomogeneous

$\begin{array}[]{c}\displaystyle\partial_{\nu(\xi)}W(\xi)=G(\xi),\,\,\xi\in\partial\omega,\end{array}$
(3.8)

with particular functions $F(\xi)$ and $G(\xi)$. Here,
$\nu(\xi)=(\nu_{1}(\xi),\nu_{2}(\xi))$ is the outward (with respect to $\Xi$)
normal vector on $\partial\omega$ and, therefore, the inward one with respect
to $\omega$.

######  Remark 3.1.

The boundary condition (3.7) is directly inherited from the original condition
(1.5) on the boundary of the perforation string (3.2). For any
$\Lambda^{\varepsilon}\leq C$, we have

$\Delta_{x}+\Lambda^{\varepsilon}=\varepsilon^{-2}(\Delta_{\xi}+\varepsilon^{2}\Lambda^{\varepsilon})\,\,\mbox{\rm
with}\,\,\varepsilon^{2}\Lambda^{\varepsilon}\leq C\varepsilon^{2}$

and the main asymptotic part $\Delta_{\xi}$ of the above differential operator
is involved with the Laplace equation (3.4). The periodicity conditions (3.6)
have no relation to the original quasi-periodicity conditions (2.3), (2.4) but
are needed to support the standard asymptotic ansatz

$U^{\varepsilon}\approx w(x_{2})W(\varepsilon^{-1}x),$

where $w$ is a smooth function in $x_{2}\in[0,H]$ and $W$ is a function
$H$-periodic in $\xi_{2}=\varepsilon^{-1}x_{2}$.

We proceed with the variational formulation

$\begin{array}[]{c}\displaystyle(\nabla_{\xi}W,\nabla_{\xi}V)_{\Xi}=(F,V)_{\Xi}+(G,V)_{\partial\color[rgb]{0,0,0}\omega}\quad\forall
V\in{\cal H}^{1}_{per}(\Xi)\end{array}$ (3.9)

of the Poisson equation (3.5) with the periodicity (3.6) and boundary (3.8)
conditions. Here, ${\cal H}^{1}_{per}(\Xi)$ is the completion of the linear
space $C^{\infty}_{per}(\overline{\Xi})$ (infinitely differentiable
$H$-periodic in $\xi_{2}$ functions with compact supports) in the norm

$\begin{array}[]{c}\displaystyle\|W;{\cal
H}^{1}_{per}(\Xi)\|=\Big{(}\|\nabla_{\xi}W;L^{2}(\Xi)\|^{2}+\|W;L^{2}(\Xi(2R))\|^{2}\Big{)}^{1/2},\end{array}$
(3.10)

where $R$ is a fixed positive constant such that

$\begin{array}[]{c}\displaystyle\overline{\omega}\subset\Xi(R):=\\{\xi\in\Xi:\,|\xi_{1}|<R\\}.\end{array}$
(3.11)

Also, for convenience, we introduce here the cut-off functions, $\chi_{\pm}\in
C^{\infty}({\mathbb{R}})$,

$\begin{array}[]{c}\displaystyle\chi_{\pm}(t)=1\,\,\mbox{\rm for}\,\,\pm
t>2R\,\,\mbox{\rm and}\,\,\chi_{\pm}(t)=0\,\,\mbox{\rm for}\,\,\pm
t<R,\end{array}$ (3.12)

with a fixed $R>0$ satisfying (3.11).

######  Proposition 3.2.

Let the functions on the right-hand sides of the problem (3.5), (3.6), (3.8)
meet the inclusions

$\begin{array}[]{c}\displaystyle(1+\xi^{2}_{1})^{1/2}F\in L^{2}(\Xi),\quad
G\in L^{2}(\partial\omega)\end{array}$ (3.13)

and the orthogonality condition

$\begin{array}[]{c}\displaystyle\int\limits_{\Xi}F(\xi)d\xi+\int\limits_{\partial\omega}G(\xi){\color[rgb]{0,0,0}ds_{\xi}}=0.\end{array}$
(3.14)

Then the problem has a solution $W\in{\cal H}_{per}^{1}(\Xi)$ which is defined
up to an additive constant.

###### Proof.

We consider the perturbed equation

$\begin{array}[]{c}\displaystyle-\Delta_{\xi}W(\xi)+\mu
X_{R}(\xi)W(\xi)=F(\xi),\,\,\xi\in{\color[rgb]{0,0,0}\Xi,}\end{array}$ (3.15)

with the boundary conditions (3.6) and (3.8); here, $\mu>0$ is a parameter and
$X_{R}$ is the  characteristic function of the truncated domain $\Xi(2R)$,
i.e. $X_{R}(\xi)=1$ for $|\xi_{1}|<2R$ and $X_{R}(\xi)=0$ for $|\xi_{1}|>2R$.
The variational formulation of the problem (3.15), (3.8), (3.6) reads:

$\begin{array}[]{c}\displaystyle(\nabla_{\xi}W,\nabla_{\xi}V)_{\Xi}+\mu(W,V)_{\Xi(2R)}=(F,V)_{\Xi}+(G,V)_{\partial{\color[rgb]{0,0,0}\omega}}\quad\forall
V\in{\cal H}^{1}_{per}(\Xi).\end{array}$ (3.16)

In view of the one-dimensional Hardy inequality

$\int\limits_{R}^{+\infty}\Big{|}\frac{dZ}{d\xi_{1}}(\xi_{1})\Big{|}^{2}d\xi_{1}\geq\frac{1}{4}\int\limits_{R}^{+\infty}|Z(\xi_{1})|^{2}\,\frac{d\xi_{1}}{\xi_{1}^{2}}\quad\forall
Z\in C^{\infty}_{c}(R,+\infty),$

applied to the functions
$Z(\pm\xi_{1},\xi_{2})=\chi_{\pm}(\xi_{1})W(\xi_{1},\xi_{2})$ with
$\chi_{\pm}$ defined by (3.12), and integrated in $\xi_{2}\in(0,H)$, the norm
(3.10) is equivalent to the norm

$\begin{array}[]{c}\displaystyle\Big{(}\|\nabla_{\xi}W;L^{2}(\Xi)\|^{2}+\|(1+\xi_{1}^{2})^{-1/2}W;L^{2}(\Xi)\|^{2}\Big{)}^{1/2}.\end{array}$
(3.17)

Notice that the last Lebesgue norm in (3.10) is computed over a compact set
while the weighted Lebesgue norm in (3.17) involves the whole infinite domain
$\Xi$.

It is self evident that the left-hand side of the integral identity (3.16)
with $\mu>0$ can be taken as a scalar product in the Hilbert space ${\cal
H}^{1}_{per}(\Xi)$. Hence, according to the equivalency of the norms (3.10)
and (3.17), and, owing to (3.13), the right-hand side of (3.16) defines a
continuous functional in ${\cal H}^{1}_{per}(\Xi)$. Thus, the Riesz
representation theorem assures that the problem (3.16) with $\mu>0$ has a
unique solution $W\in{\cal H}^{1}_{per}(\Xi)$ in the case (3.13).

According to the above mentioned equivalence of norms, considering the space
${\cal H}_{per}^{1}(\Xi)$ with the norm (3.17), and the fact that the
embedding ${\cal H}_{per}^{1}(\Xi)\subset L^{2}(\Xi(2R))$ is compact, for any
fixed $\mu$, the spectral problem associated to (3.16)

$\begin{array}[]{c}\displaystyle(\nabla_{\xi}W,\nabla_{\xi}V)_{\Xi}+\mu(W,V)_{\Xi(2R)}=\nu(W,V)_{\Xi(2R)}\quad\forall
V\in{\cal H}^{1}_{per}(\Xi),\end{array}$ (3.18)

has a discrete spectrum with the corresponding eigenfunctions being orthogonal
both in ${\cal H}_{per}^{1}(\Xi)$ and $L^{2}(\Xi(2R))$. In addition, $\nu=\mu$
is an eigenvalue of (3.18) with the associated eigenspace of the constant
functions $\mathfrak{C}$. Thus, considering the decomposition ${\cal
H}^{1}_{per}(\Xi)=\mathfrak{C}\bigoplus\mathfrak{C}^{\bot}$ with
$\mathfrak{C}^{\bot}$ the subspace formed by the elements of ${\cal
H}^{1}_{per}(\Xi)$ which are orthogonal to the constants, by the Fredholm
alternative, problem (3.9) has a unique solution in $\mathfrak{C}^{\bot}$
provided that the functional on the right hand side of (3.9) is in the dual
space $(\mathfrak{C}^{\bot})^{*}$, namely, provided that it satisfies the
orthogonality condition (3.14). This concludes the proof of the proposition. ∎

### 3.2 Integral characteristics

First of all, we recall that, according to the general theory of elliptic
problems in domains with cylindrical outlets to infinity, see [33, Ch. 5] and
[24, Section 3, 5], the homogeneous problem (3.4), (3.6), (3.7) has just
two111$2=\frac{1}{2}(2\times 2)$ where the last $2$ is the number of outlets
to infinity in the domain $\Xi$ and the next to the last $2$ is the number of
linearly independent, $H$-periodic in $\xi_{2}$ and polynomial in $\xi_{1}$,
harmonics in the intact strip $\Pi$, namely $1$ and $\xi_{1}$ in our case.
This mnemonic rule works for many other problems in domains with cylindrical
and periodic outlets to infinity, see the review paper [24, Section 3, 5].
linearly independent solutions with the polynomial behavior at infinity.  It
is evident that the first solution is a constant, and we set

$W^{0}(\xi)=1.$ (3.19)

Let us seek the second solution to the problem (3.4), (3.6), (3.7) in the form

$\begin{array}[]{c}W^{1}(\xi)=\xi_{1}+W^{1}_{0}(\xi){\color[rgb]{0,0,0}+C}\end{array}$
(3.20)

where $C$ is a certain constant, cf. (3.25), and $W^{1}_{0}\in{\cal
H}^{1}_{per}(\Xi)$ satisfies the Laplace equation (3.4), the periodicity
conditions (3.6) and the inhomogeneous Neumann condition

$\begin{array}[]{c}\partial_{\nu(\xi)}W^{1}_{0}(\xi)=-\partial_{\nu(\xi)}\xi_{1}=-\nu_{1}(\xi),\,\,\xi\in\partial\omega.\end{array}$
(3.21)

######  Proposition 3.3.

There is a unique solution of problem (3.4), (3.6), (3.7) with the
decomposition

$\begin{array}[]{c}\displaystyle
W^{1}(\xi)=\sum\limits_{\pm}\chi_{\pm}(\xi_{1})(\xi_{1}\pm
m_{1}(\Xi))+\widetilde{W}^{1}(\xi)\end{array}$ (3.22)

where $\chi_{\pm}$ is defined by (3.11)–(3.12), $m_{1}(\Xi)$ is a constant,
and the remainder  $\widetilde{W}^{1}(\xi)$ and its derivatives get the
exponential decay $O(e^{-2\pi|\xi_{1}|/H})$ as $\xi_{1}\to\pm\infty$. The
quantity $m_{1}(\Xi)$ in (3.22) is given by

$\begin{array}[]{c}\displaystyle
m_{1}(\Xi)=\frac{1}{2H}\left(\|\nabla_{\xi}W^{1}_{0};L^{2}(\Xi)\|^{2}+|\omega|\right)>0,\end{array}$
(3.23)

where $|\omega|=\mbox{\rm mes}_{2}\,\omega$ and $W_{0}^{1}$ is a solution of
(3.4), (3.6) and (3.21) in the space ${\cal H}^{1}_{per}(\Xi)$. In addition,
any solution of the problem (3.4), (3.6), (3.7) with polynomial growth at
infinity is a linear combination $c_{0}W^{0}+c_{1}W^{1}$ with some
coefficients $c_{0},c_{1}$.

###### Proof.

In the case $F=0$ and $G(\xi)=-\partial_{\nu}\xi_{1}$ the equality (3.14) is
evidently fulfilled and, thus, the problem (3.4), (3.6), (3.21) in its
variational form (3.9) has a solution $W^{1}_{0}\in{\cal H}^{1}_{per}(\Xi)$
which is  uniquely defined up to an additive constant. Since the boundary
$\partial\omega$ is smooth, this solution is infinitely differentiable in
$\overline{\Xi}$ and the Fourier method, in particular, gives the
decomposition

$\begin{array}[]{c}W^{1}_{0}(\xi)=\sum\limits_{\pm}\chi_{\pm}(\xi_{1})C_{\pm}+\widetilde{W}^{1}_{0}(\xi)\end{array}$
(3.24)

with the exponentially decaying remainder
$\color[rgb]{0,0,0}\widetilde{W}^{1}_{0}$, and some constants $C_{\pm}$ which
can also depend on $R$, cf. (3.11). Setting

$C=-\frac{1}{2}(C_{+}+C_{-}),$ (3.25)

the function $W^{1}(\xi)=\xi_{1}+W^{1}_{0}(\xi)+C$ becomes the desired
solution (3.20) of the problem (3.4), (3.6), (3.7) admitting the
representation (3.22) with $m_{1}(\Xi)=\frac{1}{2}(C_{+}-C_{-})$.

To prove the relation (3.23), we apply the Green formula twice and write

$\begin{split}&-\int\limits_{\partial\omega}(\xi_{1}+W^{1}_{0}(\xi))\partial_{\nu(\xi)}\xi_{1}ds_{\xi}=-\int\limits_{\partial\omega}\xi_{1}\partial_{\nu(\xi)}\xi_{1}ds_{\xi}+\int\limits_{\partial\omega}W^{1}_{0}(\xi)\partial_{\nu(\xi)}W^{1}_{0}(\xi)ds_{\xi}\\\
&\quad=\int\limits_{\omega}|\nabla_{\xi}\xi_{1}|^{2}d\xi+\|\nabla_{\xi}W^{1}_{0};L^{2}(\Xi)\|^{2}=|\omega|+\|\nabla_{\xi}W^{1}_{0};L^{2}(\Xi)\|^{2},\end{split}$

where we have used equalities (3.21) and (3.9). Similarly, we write

$\displaystyle\displaystyle\int\limits_{\partial\omega}(\xi_{1}+W^{1}_{0}(\xi))\partial_{\nu(\xi)}\xi_{1}ds_{\xi}=\int\limits_{\partial\omega}\big{(}(\xi_{1}+W^{1}_{0}(\xi))\partial_{\nu(\xi)}\xi_{1}-\xi_{1}\partial_{\nu(\xi)}(\xi_{1}+W^{1}_{0}(\xi))\big{)}ds_{\xi}$
$\displaystyle\displaystyle\quad=\lim\limits_{T\to+\infty}\sum\limits_{\pm}\mp\int\limits_{0}^{H}\Big{(}(\xi_{1}+W^{1}_{0}(\xi))\frac{\partial\xi_{1}}{\partial\xi_{1}}-\xi_{1}\frac{\partial}{\partial\xi_{1}}(\xi_{1}+W^{1}_{0}(\xi))\Big{)}\bigg{|}_{\xi_{1}=\pm
T}d\xi_{2}$
$\displaystyle\displaystyle\quad=\lim\limits_{T\to+\infty}\sum\limits_{\pm}\mp\int\limits_{0}^{H}W^{1}_{0}(\pm
T,\xi_{2})d\xi_{2}=-2Hm_{1}(\Xi),$

and the relationship (3.23) follows immediately. ∎

######  Remark 3.4.

The quantity (3.23) is an integral characteristics of the Neumann hole
$\overline{\omega}$ in the strip $\Pi$ of width $H$ with the periodicity
conditions at the lateral sides. This characteristics looks quite similar to
the classical virtual mass tensor in the exterior Neumann problem, although
it is a scalar, cf., [38, Appendix G]. For any set $\overline{\omega}$ of the
positive area $\mbox{\rm mes}_{2}\,\omega$, we have $m_{1}(\Xi)>0$. At the
same time, in the case of a crack $\Upsilon=\overline{\omega}$ along the
$\xi_{1}$-axis we observe that $\partial_{\nu(\xi)}\xi_{1}=0$ on
$\partial\omega$, $W^{1}_{0}(\xi)=C^{1}_{0}$ and $\mbox{\rm
mes}_{2}\,\Upsilon=0$, therefore, $m_{1}(\Pi\setminus\Upsilon)=0$. However,
the smoothness assumption on the boundary in Section 1.1 excludes cracks from
our present consideration.

### 3.3 Other special solutions

It proves necessary to introduce here two solutions of boundary value problems
in $\Xi.$ First, let us introduce $W^{2}$ a solution of the problem (3.4),
(3.6) and the inhomogeneous Neumann condition (3.21) with the replacement
$1\mapsto 2$, namely

$\begin{array}[]{c}\partial_{\nu(\xi)}W^{2}(\xi)=-\partial_{\nu(\xi)}\xi_{2}=-\nu_{2}(\xi),\,\,\xi\in\partial\omega.\end{array}$
(3.26)

The compatibility condition (3.14) is again fulfilled so that the problem
(3.4), (3.6), (3.26) has a bounded solution which is  uniquely defined up to
an additive constant and, therefore, is fixed uniquely in the form

$\begin{array}[]{c}\displaystyle
W^{2}(\xi)=\sum\limits_{\pm}\pm\chi_{\pm}(\xi_{1})m_{2}(\Xi)+\widetilde{W}^{2}(\xi)\end{array}$
(3.27)

where $m_{2}(\Xi)$ is a constant, and the remainder $\widetilde{W}^{2}(\xi)$
and its derivatives get the exponential decay as $\xi_{1}\to\pm\infty$.

In contrast to the quantity (3.23) the coefficient $m_{2}(\Xi)$ in (3.27) can
get arbitrary sign (see Section 3.4). Notice that the following integral
representation is valid,  cf. (3.20) and (3.21):

$\int\limits_{\partial\omega}W^{1}(\xi)\partial_{\nu(\xi)}\xi_{2}\,ds_{\xi}=\int\limits_{\partial\omega}\big{(}W^{2}(\xi)\partial_{\nu(\xi)}W^{1}(\xi)-W^{1}(\xi)\partial_{\nu(\xi)}W^{2}(\xi)\big{)}\,ds_{\xi}\\\
=\lim\limits_{T\to+\infty}\sum\limits_{\mp}\\!\mp\int\limits_{0}^{H}\\!\Big{(}W^{2}(\pm
T,\xi_{2})\frac{\partial W^{1}}{\partial\xi_{1}}(\pm T,\xi_{2})-W^{1}(\pm
T,\xi_{2})\frac{\partial W^{2}}{\partial\xi_{1}}(\pm
T,\xi_{2})\Big{)}\,d\xi_{2}=-2Hm_{2}(\Xi).$ (3.28)

Finally, we introduce a solution $W^{3}$ to the Poisson equation

$\begin{array}[]{c}\displaystyle-\Delta_{\xi}W^{3}(\xi)=1,\,\,\xi\in\Xi,\end{array}$
(3.29)

with the boundary conditions (3.6), (3.7) which can be found in the form

$\begin{array}[]{c}\displaystyle
W^{3}(\xi)=-\frac{1}{2}\xi^{2}_{1}+\sum\limits_{\pm}\pm\chi_{\pm}(\xi_{1})\bigg{(}\frac{|\omega|}{2H}\xi_{1}+m_{3}(\Xi)\bigg{)}+{\color[rgb]{0,0,0}\widetilde{W}^{\,3}}(\xi)\end{array}$
(3.30)

where $m_{3}(\Xi)$ is a constant, and the remainder $\widetilde{W}^{3}(\xi)$
and its derivatives get the exponential decay as $\xi_{1}\to\pm\infty$. To
show this, we accept the representation
$W^{3}(\xi)=W^{3}_{0}(\xi)-\xi^{2}_{1}/2$ and observe that $W^{3}_{0}$ is a
solution of the problem (3.4), (3.6) with the Neumann condition

$\partial_{\nu(\xi)}W^{3}_{0}(\xi)=\nu_{1}(\xi)\xi_{1},\,\,\xi\in\partial\omega.$

Thus, the argument in the proof of Proposition 3.3 to get $W_{0}^{1}$ and
$W^{1}$ (cf. (3.20)) gives us a solution with the linear growth as
$\xi_{1}\rightarrow\pm\infty$, and we can provide the decomposition

$W^{3}_{0}(\xi)=\sum\limits_{\pm}\pm{\color[rgb]{0,0,0}\chi_{\pm}(\xi_{1})}\Big{(}C^{3}_{1}\xi_{1}+C^{3}_{0}({\color[rgb]{0,0,0}\omega})\Big{)}+{\color[rgb]{0,0,0}\widetilde{W}^{\,3}}(\xi),$

for certain coefficients $C_{1}^{3}$ and $C^{3}_{0}(\omega)$.

To compute the coefficient $C_{1}^{3}$, we apply the Green formula twice as
follows:

$-|\omega|=\int\limits_{\partial\omega}\frac{1}{2}\frac{\partial}{\partial\nu(\xi)}\xi_{1}^{2}ds_{\xi}=\int\limits_{\partial\omega}\partial_{\nu(\xi)}W^{3}_{0}(\xi)ds_{\xi}=\lim\limits_{T\rightarrow+\infty}\sum\limits_{\pm}\mp\int\limits_{0}^{H}\frac{\partial}{\partial\xi_{1}}W^{3}_{0}(\pm
T,\xi_{2})d\xi_{2}=-2HC^{3}_{1}.$

In contrast to $C^{3}_{1}$, the coefficient $C^{3}_{0}$ depends on the shape
of $\omega$ but we will not use it in the sequel, and we avoid introducing
here its computation.

### 3.4 The symmetry assumption and its consequences

As pointed out in Section 1.1 we can describe the band-gap structure of the
low-frequency range of the spectrum (1.7) only in the case of the mirror
symmetry of the hole. Therefore, we will justify the derived asymptotics under
the supposition (1.9), cf. Section 4. First of all, we realize that

$\begin{array}[]{c}\displaystyle m_{2}(\Xi)=0,\end{array}$ (3.31)

so that all asymptotic expansions will simplify. This is a consequence of the
fact that the boundary layer terms have the following important properties.

######  Lemma 3.5.

Under the assumption (1.9), the functions $W^{1}$, $W^{3}$ and $W^{2}$,
respectively, are even and odd in the variable $\xi_{2}-H/2$ and, hence,

$\begin{array}[]{c}\displaystyle\frac{\partial
W^{j}}{\partial\xi_{2}}(\xi_{1},0)=\frac{\partial
W^{j}}{\partial\xi_{2}}(\xi_{1},H)=0,{\color[rgb]{0,0,0}\,\,\,\xi_{1}\in\mathbb{R},}\,\,j=1,3,\vspace{0.2cm}\\\
W^{2}(\xi_{1},0)=W^{2}(\xi_{1},H)=0{\color[rgb]{0,0,0},\,\,\,\xi_{1}\in\mathbb{R}}.\end{array}$
(3.32)

The results in Lemma 3.5 are a consequence of the definition of functions
$W^{i}$, $i=1,2,3$, and the uniqueness of the solutions of the problems that
they satisfy in the way stated throughout the section. Equation (3.31) follows
from the evenness of $W^{1}$ and formula (3.28).

## 4 Formal asymptotic analysis of simple eigenvalues

In this section, by means of matched asymptotic expansions, we construct a
corrector improving the first approximation (2.10). In particular, we provide
the complete analysis of the first correction term of the eigenpairs of
(2.2)–(2.5) in the case where the limit eigenvalue is simple (see Remark 4.1
for multiple eigenvalues). The asymptotic structures here constructed will
give us a reason to introduce the symmetry assumption (1.9), see Section 4.5
and Remark 4.2.

### 4.1 Asymptotic ansätze

Let us fix the Floquet parameter $\eta\in[-\pi,\pi]$ such that the eigenvalue
$\Lambda^{0}_{p}(\eta)$ of the problem (2.12)–(2.14) is simple. In other
words, only one dispersion curve crosses the point
$(\eta,\Lambda^{0}_{p}(\eta))$. Let us fix $U^{0}_{p}(\cdot;\eta)$ a
corresponding eigenfunction (see (4.24)). We employ the method of matched
asymptotic expansions, see e.g. [36, 21], in the interpretation [28, 30], to
obtain corrector terms for $\Lambda^{0}_{p}(\eta)$ and
$U^{0}_{p}(\cdot;\eta)$.

Let us accept the simplest asymptotic ansätze

$\begin{array}[]{c}\displaystyle\Lambda^{\varepsilon}_{p}(\eta)=\Lambda^{0}_{p}(\eta)+\varepsilon\Lambda^{\prime}_{p}(\eta)+\varepsilon^{2}\Lambda^{\prime\prime}_{p}(\eta)+\dots,\end{array}$
(4.1) $\begin{array}[]{c}\displaystyle
U^{\varepsilon}_{p}(x;\eta)=U^{0}_{p}(x;\eta)+\varepsilon
U^{\prime}_{p}(x;\eta)+\varepsilon^{2}U^{\prime\prime}_{p}(x;\eta)+\dots,\vspace{0.2cm}\\\
\end{array}$ (4.2)

We regard (4.2) as the outer expansion, which fits in
$\varpi^{0}\setminus\varsigma$ at a distance from the vertical mid-line
$\varsigma=\\{x:\,x_{1}=0,x_{2}\in(0,H)\\}$. We have excluded the line segment
$\varsigma$ in the equation (4.2) because of the perforation string (3.2)
which provokes the boundary layer phenomenon. Here, and in what follows, dots
stand for higher-order terms which are inessential in our formal asymptotic
analysis.

Note that, although we will not determine the second order terms
$\Lambda^{\prime\prime}_{p}(\eta)$ and $U^{\prime\prime}_{p}(x;\eta)$, they
are involved with the asymptotic procedure. Also, we emphasize that the main
term $U^{0}_{p}$ in (4.2) is a smooth function in $\varpi^{0}$ but the
correction terms may present jumps through $\varsigma$.

Inserting these ansätze into the equations (2.2)–(2.5) and extracting terms of
order $\varepsilon$ readily yield the following restrictions for  the first
order terms $\Lambda^{\prime}_{p}(\eta)$ and $U^{\prime}_{p}(x;\eta)$: the
differential equation

$\begin{array}[]{c}-\Delta_{x}U_{p}^{\prime}(x;\eta)-\Lambda_{p}^{0}(\eta)U^{\prime}_{p}(x;\eta)=\Lambda_{p}^{\prime}(\eta)U_{p}^{0}(x;\eta),\,\,x\in\varpi^{0}\setminus\varsigma,\end{array}$
(4.3)

the quasi-periodicity conditions (2.14) at the vertical sides, the Neumann
conditions on the punctured horizontal sides

$\begin{array}[]{c}\displaystyle\frac{\partial U^{\prime}_{p}}{\partial
x_{2}}(x_{1},0;\eta)=\frac{\partial U^{\prime}_{p}}{\partial
x_{2}}(x_{1},H;\eta)=0,\,\,x_{1}\in\Big{(}-\frac{1}{2},0\Big{)}\cup\Big{(}0,\frac{1}{2}\Big{)},\end{array}$
(4.4)

and some transmission conditions on $\varsigma$ that we determine by the
matching procedure (cf. (4.12) and (4.23)). This is the aim of Section 4.2 and
4.3 below, while $\Lambda_{p}^{\prime}(\eta)$ is determined in Section 4.4.

In order to do this, we introduce the inner expansion

$\begin{array}[]{c}\displaystyle
U^{\varepsilon}_{p}(x;\eta)=w^{0}_{p}(x_{2};\eta)+\varepsilon
w^{\prime}_{p}(\xi,x_{2};\eta)+\varepsilon^{2}w^{\prime\prime}_{p}(\xi,x_{2};\eta)+\dots,\end{array}$
(4.5)

where we have assumed that the main term $w^{0}_{p}$ is constant in $\xi$, cf.
(3.3), while the functions arising in further terms, $w^{\prime}_{p}$ and
$w^{\prime\prime}_{p}$, depend on $\xi$ and satisfy a periodicity condition in
the $\xi_{2}$-direction, namely, conditions (3.6).

### 4.2 The first transmission condition

The Taylor formula implies

$U^{0}_{p}(x;\eta)+\varepsilon
U^{\prime}_{p}(x;\eta)+\varepsilon^{2}U^{\prime\prime}_{p}(x;\eta)=U^{0}_{p}(0,x_{2};\eta)+\varepsilon\Big{(}U^{\prime}_{p}(\pm
0,x_{2};\eta)+\xi_{1}\frac{\partial U^{0}_{p}}{\partial
x_{1}}(0,x_{2};\eta)\Big{)}\\\ +\varepsilon^{2}\Big{(}U^{\prime\prime}_{p}(\pm
0,x_{2};\eta)+\xi_{1}\frac{\partial U^{\prime}_{p}}{\partial x_{1}}(\pm
0,x_{2};\eta)+\frac{\xi_{1}^{2}}{2}\frac{\partial^{2}U^{0}_{p}}{\partial
x^{2}_{1}}(0,x_{2};\eta)\Big{)}+\dots.$ (4.6)

Hence, comparing terms of order $1$ in (4.5) and (4.6) leads us to the formula

$\begin{array}[]{c}\displaystyle
w^{0}_{p}(x_{2};\eta)=U^{0}_{p}(0,x_{2};\eta).\end{array}$ (4.7)

In addition, taking derivatives with respect to $\xi$ in equations (2.2) and
(2.5), inserting (4.5) in (2.2) and (2.5), and extracting the terms of order
$\varepsilon$, we obtain that the first order term in the inner expansion
(4.5) satisfies the equation (3.4), with periodicity conditions (3.6) and the
inhomogeneous Neumann condition

$\begin{array}[]{c}\displaystyle\partial_{\nu(\xi)}w^{\prime}_{p}(\xi,x_{2};\eta)=-\frac{\partial
U^{0}_{p}}{\partial
x_{2}}(0,x_{2};\eta)\partial_{\nu(\xi)}\xi_{2},\,\,\xi\in\partial\omega,\end{array}$
(4.8)

which takes into account the discrepancy in (3.7) of the main term (4.7) due
to its dependence on the slow variable $x_{2}$. Indeed, we have used the
formula for the directional derivative:

$\frac{\partial
V}{\partial\nu(x)}(\xi,x_{2})=\frac{1}{\varepsilon}\frac{\partial
V}{\partial\nu(\xi)}(\xi,x_{2})+\nu_{2}(\xi)\frac{\partial V}{\partial
x_{2}}(\xi,x_{2}).$

Furthermore, the matching of the outer and inner expansions at the first order
prescribes the following behavior at infinity for $w^{\prime}_{p}$

$\begin{array}[]{c}\displaystyle
w^{\prime}_{p}(\xi,x_{2};\eta)\sim\xi_{1}\frac{\partial U^{0}_{p}}{\partial
x_{1}}(0,x_{2};\eta)+U^{\prime}_{p}(\pm 0,x_{2};\eta)\quad\,\,\mbox{ as
}\quad\xi_{1}\to\pm\infty,\end{array}$ (4.9)

cf. (4.7) and the factor of $\varepsilon$ on the right-hand side of (4.6).

The solution of the problem (3.4), (3.6), (4.8), (4.9) is nothing but  a
linear combination

$w^{\prime}_{p}(\xi,x_{2};\eta)=\frac{\partial U^{0}_{p}}{\partial
x_{1}}(0,x_{2};\eta)W^{1}(\xi)+\frac{\partial U^{0}_{p}}{\partial
x_{2}}(0,x_{2};\eta)W^{2}(\xi)+C^{\prime}_{p}(x_{2};\eta)W^{0}$ (4.10)

of the solutions of the problems on $\Xi$ introduced in Section 3.2 and 3.3,
see (3.1), where the factor $C^{\prime}_{p}(x_{2};\eta)$ is related to (4.9),
not determined yet. However, it does not influence our further analysis.

Using the decompositions (3.22), (3.27) and recalling (3.19), we find the
following expressions in (4.9):

$U^{\prime}_{p}(\pm 0,x_{2};\eta)=\pm\Big{(}\frac{\partial U^{0}_{p}}{\partial
x_{1}}(0,x_{2};\eta)m_{1}(\Xi)+\frac{\partial U^{0}_{p}}{\partial
x_{2}}(0,x_{2};\eta)m_{2}(\Xi)\Big{)}\\!+\\!C^{\prime}_{p}(x_{2};\eta).$
(4.11)

Although the traces (4.11) are not yet fixed, we compute the jump of
$U^{\prime}_{p}$ through $\varsigma$,

$[U^{\prime}_{p}]_{0}(x_{2};\eta)=U^{\prime}_{p}(+0,x_{2};\eta)-U^{\prime}_{p}(-0,x_{2};\eta),$

that is,

$\begin{array}[]{c}\displaystyle[U^{\prime}_{p}]_{0}(x_{2};\eta)=2\frac{\partial
U^{0}_{p}}{\partial x_{1}}(0,x_{2};\eta)m_{1}(\Xi)+2\frac{\partial
U^{0}_{p}}{\partial x_{2}}(0,x_{2};\eta)m_{2}(\Xi),\quad
x_{2}\in(0,H).\end{array}$ (4.12)

### 4.3 The second transmission condition

To proceed, we have to deal with the third term of the inner expansion (4.5)
which after inserting into the problem (2.2)–(2.5) and extracting terms of
order $\varepsilon^{2}$ leads to the problem

$\begin{array}[]{c}\displaystyle-\Delta_{\xi}w_{p}^{\prime\prime}(\xi,x_{2};\eta)=f_{p}^{\prime\prime}(\xi,x_{2};\eta),\,\,\xi\in\Xi,\\\
\\\
\displaystyle\partial_{\nu(\xi)}w_{p}^{\prime\prime}(\xi,x_{2};\eta)=g_{p}^{\prime\prime}(\xi,x_{2};\eta),\,\,\xi\in\partial\omega,\end{array}$
(4.13)

with the periodicity conditions (3.6). According to (4.7), (2.12) and (4.10),
the right-hand sides of (4.13) are given by

$\begin{split}f_{p}^{\prime\prime}(\xi,x_{2};\eta)&=\Delta_{x}w_{p}^{0}(\xi,x_{2};\eta)+\Lambda^{0}_{p}(\eta)w_{p}^{0}(\xi,x_{2};\eta)+2\frac{\partial^{2}w_{p}^{\prime}}{\partial
x_{2}\partial\xi_{2}}(\xi,x_{2};\eta)\\\
&=-\frac{\partial^{2}U^{0}_{p}}{\partial
x^{2}_{1}}(0,x_{2};\eta)+2\frac{\partial^{2}U^{0}_{p}}{\partial x_{1}\partial
x_{2}}(0,x_{2};\eta)\frac{\partial
W^{1}}{\partial\xi_{2}}(\xi)+2\frac{\partial^{2}U^{0}_{p}}{\partial
x_{2}^{2}}(0,x_{2};\eta)\frac{\partial
W^{2}}{\partial\xi_{2}}(\xi),\end{split}$ (4.14)
$\begin{split}g_{p}^{\prime\prime}(\xi,x_{2};\eta)&=-\nu_{2}(\xi)\frac{\partial
w_{p}^{\prime}}{\partial x_{2}}(\xi,x_{2};\eta)\\\
&=-\nu_{2}(\xi)\frac{\partial^{2}U^{0}_{p}}{\partial x_{1}\partial
x_{2}}(0,x_{2};\eta)W^{1}(\xi)-\nu_{2}(\xi)\frac{\partial^{2}U^{0}_{p}}{\partial
x^{2}_{2}}(0,x_{2};\eta)W^{2}(\xi)-\nu_{2}(\xi)\frac{\partial
C^{\prime}_{p}}{\partial x_{2}}(x_{2};\eta).\end{split}$

Furthermore, the matching procedure and the Taylor formula (4.6), up to the
order $\varepsilon^{2}$, establish the following behavior at infinity:

$\displaystyle
w_{p}^{\prime\prime}(\xi,x_{2};\\!\eta)\sim\frac{\xi_{1}^{2}}{2}\frac{\partial^{2}U^{0}_{p}}{\partial
x^{2}_{1}}(0,x_{2};\\!\eta)+\xi_{1}\frac{\partial U^{\prime}_{p}}{\partial
x_{1}}(\pm 0,x_{2};\\!\eta)+U^{\prime\prime}_{p}(\pm
0,x_{2};\\!\eta)\quad\hbox{ as }\quad\xi_{1}\rightarrow\pm\infty.$ (4.15)

We observe that, owing to (3.22) and (3.27), the derivatives $\partial
W^{q}/\partial\xi_{2}$ decay exponentially at infinity while the first term on
the right-hand side (4.14) is constant in $\xi$. Thus, a solution of the
problem (4.13), (3.6), (4.15) admits the quadratic growth as
$\xi_{1}\rightarrow\pm\infty$, and we set

$\begin{array}[]{c}\displaystyle
w_{p}^{\prime\prime}(\xi,x_{2};\eta)=-\frac{\partial^{2}U^{0}_{p}}{\partial
x^{2}_{1}}(0,x_{2};\eta)W^{3}(\xi)+\widehat{w}^{\,\prime\prime}_{p}(\xi,x_{2};\eta),\end{array}$
(4.16)

where $W^{3}$ is given by (3.30). The remaining part
$\widehat{w}^{\,\prime\prime}_{p}$ verifies the problem

$\begin{array}[]{c}\displaystyle-\Delta_{\xi}\widehat{w}^{\,\prime\prime}_{p}(\xi,x_{2};\eta)=\widehat{f}^{\,\prime\prime}_{p}(\xi,x_{2};\eta),\,\,\xi\in\Xi,\\\
\\\
\displaystyle\partial_{\nu(\xi)}\widehat{w}^{\,\prime\prime}_{p}(\xi,x_{2};\eta)=g_{p}^{\prime\prime}(\xi,x_{2};\eta),\,\,\xi\in\partial\omega,\end{array}$
(4.17)

with the periodicity condition (3.6), where

$\widehat{f}^{\,\prime\prime}_{p}(\xi,x_{2};\eta)=f^{\prime\prime}_{p}(\xi,x_{2};\eta)+\frac{\partial^{2}U^{0}_{p}}{\partial
x^{2}_{1}}(0,x_{2};\eta)\in L^{2}(\Xi),$

and gets an exponential decay at infinity. A solution of such a problem exists
in the form

$\\!\\!\widehat{w}^{\,\prime\prime}_{p}(\xi,x_{2};\eta)=\sum\limits_{\pm}\pm\chi_{\pm}(\xi_{1})(\widehat{C}_{1}(x_{2};\eta)\xi_{1}+\widehat{C}_{0}(x_{2};\eta))+\widetilde{w}^{\,\prime\prime}_{p}(\xi,x_{2};\eta)$
(4.18)

for certain coefficients $\widehat{C}_{0}$ and $\widehat{C}_{1}$, and a
remainder $\widetilde{w}^{\,\prime\prime}_{p}$ which gets the exponential
decay as $\xi_{1}\to\pm\infty.$ To derive the second transmission condition
for $U^{\prime}_{p}$ arising in (4.3), it suffices to compute the coefficient
$\widehat{C}_{1}$ because the other coefficient $\widehat{C}_{0}$ proves to be
of no further use.

Indeed, by applying the Green formula in (4.17), we readily obtain

$\int\limits_{\Xi}\widehat{f}^{\,\prime\prime}_{p}(\xi,x_{2};\eta)d\xi+\int\limits_{\partial\omega}g^{\prime\prime}_{p}(\xi,x_{2};\eta)ds_{\xi}=-\lim\limits_{T\rightarrow+\infty}\sum\limits_{\pm}\pm\int\limits_{0}^{H}\frac{\partial\widehat{w}^{\,\prime\prime}_{p}}{\partial\xi_{1}}(\pm
T,\xi_{2},x_{2};\eta)d\xi_{2}={\color[rgb]{0,0,0}-}2H\widehat{C}_{1}(x_{2};\eta).$
(4.19)

Let us to process the left-hand side. First, we take $V=\xi_{2}$ and $W=W^{q}$
with $q=1,2$ in formula (3.9), and we get

$\int\limits_{\Xi}\frac{\partial
W^{q}}{\partial\xi_{2}}(\xi,x_{2})d\xi-\int\limits_{\partial\omega}\nu_{2}(\xi)W^{q}(\xi,x_{2})ds_{\xi}=0,\quad
q=1,2.$

Using these formulas in the definitions of $\widehat{f}^{\,\prime\prime}_{p}$
and $g^{\prime\prime}_{p}$, we have

$\begin{split}\int\limits_{\Xi}\widehat{f}^{\,\prime\prime}_{p}(\xi,x_{2};\eta)d\xi+\int\limits_{\partial\omega}g^{\prime\prime}_{p}(\xi,x_{2};\eta)ds_{\xi}=&\frac{\partial^{2}U^{0}_{p}}{\partial
x_{1}\partial
x_{2}}(0,x_{2};\eta)\\!\int\limits_{\partial\omega}\nu_{2}(\xi)W^{1}(\xi)ds_{\xi}\\\
&+\frac{\partial^{2}U^{0}_{p}}{\partial
x^{2}_{2}}(0,x_{2};\eta)\\!\int\limits_{\partial\omega}\nu_{2}(\xi)W^{2}(\xi)ds_{\xi}-\frac{\partial
C^{\prime}_{p}}{\partial
x_{2}}(x_{2};\eta)\\!\int\limits_{\partial\omega}\nu_{2}(\xi)ds_{\xi}.\end{split}$

Now, let us note that by the Green formula it follows

$\int\limits_{\partial\omega}\nu_{2}(\xi)ds_{\xi}=\int\limits_{\partial\omega}\partial_{\nu(\xi)}\xi_{2}ds_{\xi}=0,$

which cancels the term containing the derivative of
$C^{\prime}_{p}(x_{2};\eta).$ Besides, from (3.26) and (3.9), we obtain

$-\int\limits_{\partial\omega}\nu_{2}(\xi)W^{2}(\xi){\color[rgb]{0,0,0}ds_{\xi}}=\int\limits_{\partial\omega}W^{2}(\xi)\partial_{\nu(\xi)}W^{2}(\xi){\color[rgb]{0,0,0}ds_{\xi}}=\|\nabla_{\xi}W^{2};L^{2}(\Xi)\|^{2}=:M(\Xi)>0.$
(4.20)

Finally, considering (3.28) and using (4.19)–(4.20), we get

$\begin{array}[]{c}\displaystyle-2H\widehat{C}_{1}(x_{2};\eta)={\color[rgb]{0,0,0}2H}\frac{\partial^{2}U^{0}_{p}}{\partial
x_{1}\partial
x_{2}}(0,x_{2};\eta)m_{2}(\Xi)-\frac{\partial^{2}U^{0}_{p}}{\partial
x^{2}_{2}}(0,x_{2};\eta)M(\Xi).\end{array}$ (4.21)

Gathering (4.15), (4.16), (4.18), (4.21) and (3.30) we conclude that

$\frac{\partial U^{\prime}_{p}}{\partial x_{1}}(\pm
0,x_{2};\eta)={\color[rgb]{0,0,0}\mp}m_{2}(\Xi)\frac{\partial^{2}U^{0}_{p}}{\partial
x_{1}\partial
x_{2}}(0,x_{2};\eta)\mp\frac{|\omega|}{2H}\frac{\partial^{2}U^{0}_{p}}{\partial
x^{2}_{1}}(0,x_{2};\eta){\color[rgb]{0,0,0}\pm\frac{M(\Xi)}{2H}}\frac{\partial^{2}U^{0}_{p}}{\partial
x^{2}_{2}}(0,x_{2};\eta).$ (4.22)

Thus, we obtain the jump through $\varsigma$ for the normal derivative of
$U^{\prime}_{p}$

$\Big{[}\frac{\partial U^{\prime}_{p}}{\partial
x_{1}}\Big{]}_{0}(x_{2};\eta)={\color[rgb]{0,0,0}-}2m_{2}(\Xi)\frac{\partial^{2}U^{0}_{p}}{\partial
x_{1}\partial
x_{2}}(0,x_{2};\eta)-\frac{|\omega|}{H}\frac{\partial^{2}U^{0}_{p}}{\partial
x^{2}_{1}}(0,x_{2};\eta){\color[rgb]{0,0,0}+\frac{M(\Xi)}{H}}\frac{\partial^{2}U^{0}_{p}}{\partial
x^{2}_{2}}(0,x_{2};\eta).$ (4.23)

This completes the problem for correction terms $\Lambda^{\prime}_{p}(\eta)$
and $U^{\prime}_{p}(\cdot;\eta)$ in the ansätze (4.1) and (4.2). Namely, they
are the unknowns of the problem (4.3), (4.4), (2.14), (4.12) and (4.23). The
existence and uniqueness of both terms is provided below.

### 4.4 Computing the correction term in the eigenvalue asymptotics

Since, by our assumption, the eigenvalue $\Lambda_{p}^{0}(\eta)$ is simple,
the solution of problem (4.3), (4.4), (2.14), (4.12), (4.23) has only one
compatibility condition. Indeed, it must satisfy the orthogonality condition,
in the sense of the Green formula, of the right-hand side of (4.3) to the
eigenfunction

$\begin{array}[]{c}\displaystyle
U^{0}_{p}(x;\eta):=U^{0}_{jk}(x;\eta)=e^{i(\eta+2\pi j)x_{1}}\cos\Big{(}\pi
k\frac{x_{2}}{H}\Big{)},\end{array}$ (4.24)

see (2.15). This determines completely $\Lambda_{p}^{\prime}(\eta)$ as we show
below (cf. (4.25)).

First, we observe that, by (4.24),

$\|U^{0}_{p};L^{2}(\varpi^{0})\|^{2}=\|U^{0}_{p};L^{2}(\varsigma)\|^{2}=\frac{1}{2}(1+\delta_{k,0})H,$

where $\delta_{k,l}$ denotes the Kronecker symbol. Then, we multiply (4.3) by
the conjugate of $U^{0}_{p}$ and integrate over $\varpi^{0}\setminus\varsigma$
to get

$\frac{1}{2}(1+\delta_{k,0})H\Lambda^{\prime}_{p}(\eta)=-\int\limits_{\varpi^{0}}\left(\Delta_{x}U^{\prime}_{p}(x;\eta)+\Lambda^{0}_{p}{\color[rgb]{0,0,0}(\eta)}U^{\prime}_{p}(x;\eta)\right)\overline{U^{0}_{p}(x;\eta)}dx.$

Because of (4.4), (2.14) and (4.24), the Green formula yields

$\begin{split}\frac{1}{2}(1+\delta_{k,0})H\Lambda^{\prime}_{p}(\eta)&=\int_{0}^{H}\Big{(}\overline{U^{0}_{p}(x;\eta)}\frac{\partial
U_{p}^{\prime}}{\partial
x_{1}}(x;\eta)-U^{\prime}_{p}(x;\eta)\overline{\frac{\partial
U_{p}^{0}}{\partial
x_{1}}(x;\eta)}\Big{)}\bigg{|}_{x_{1}=-0}^{+0}\\!\\!dx_{2}\\\
&=\int_{0}^{H}\Big{(}\overline{U^{0}_{p}(0,x_{2};\eta)}\Big{[}\frac{\partial
U_{p}^{\prime}}{\partial
x_{1}}\Big{]}_{0}(x_{2};\eta)-[U^{\prime}_{p}]_{0}(x_{2};\eta)\overline{\frac{\partial
U_{p}^{0}}{\partial x_{1}}(0,x_{2};\eta)}\Big{)}dx_{2}.\end{split}$

Now, taking into account the jump conditions (4.12) and (4.23), and using
(4.24),

$\int_{0}^{H}\cos^{2}\Big{(}\pi
k\frac{x_{2}}{H}\Big{)}dx_{2}=\\!\frac{H}{2}(1+\delta_{k,0}){\color[rgb]{0,0,0}\quad\mbox{and}\quad}\int_{0}^{H}\cos\Big{(}\pi
k\frac{x_{2}}{H}\Big{)}\sin\Big{(}\pi k\frac{x_{2}}{H}\Big{)}dx_{2}=0,$

we have

$\frac{1}{2}(1+\delta_{k,0})H\Lambda^{\prime}_{p}(\eta)=\\!\frac{H}{2}(1+\delta_{k,0})\bigg{(}\\!\frac{|\omega|}{H}(\eta+2\pi
j)^{2}{\color[rgb]{0,0,0}-\frac{M(\Xi)}{H}}\Big{(}\\!\frac{\pi
k}{H}\\!\Big{)}^{\\!2}\\!-2(\eta+2\pi j)^{2}m_{1}(\Xi)\\!\bigg{)}.$

As a result, we obtain the relationship

$\Lambda^{\prime}_{p}(\eta)={\color[rgb]{0,0,0}-}2\bigg{(}\frac{\pi^{2}k^{2}}{H^{2}}{\color[rgb]{0,0,0}\frac{M(\Xi)}{2H}}+(\eta+2\pi
j)^{2}\Big{(}m_{1}(\Xi)-\frac{|\omega|}{2H}\Big{)}\bigg{)}.$ (4.25)

Notice that, according to (4.20) and (3.23), the right-hand side of (4.25) is
negative. Also, the process determines uniquely the terms
$\Lambda_{p}^{\prime}$ and $U_{p}^{\prime}$ in the asymptotic series (4.1) and
(4.2).

######  Remark 4.1.

Assuming that $(\eta,\Lambda_{p}^{0}(\eta))$ is a crossing point of two
dispersion curves does not affect the formal computations in Sections 4.1–4.3,
$U_{p}^{0}(\cdot,\eta)$ being any of the corresponding eigenfunctions in
(4.24) with $k=0$. Also, in Section 4.4, when determining the second term of
the asymptotic expansions $\Lambda_{p}^{\prime}(\eta)$ and
$U_{p}^{\prime}(\cdot,\eta)$, there is no contradiction since the
corresponding eigenfunctions only depend on $x_{1}$, namely, rewriting
computations we obtain

$\Lambda^{\prime}_{p}(\eta)=-2(\eta+2\pi
j)^{2}\Big{(}m_{1}(\Xi)-\frac{|\omega|}{2H}\Big{)}$

while there are two associated solutions $U_{p}^{\prime}(\cdot,\eta)$ one for
each eigenfunction of $\Lambda_{p}^{0}(\eta)$.

### 4.5 On the symmetry assumption

The first term (4.7) of the inner expansion (4.5) meets the Neumann condition
(2.5) at the sides $x_{2}=0$ and $x_{2}=H$ of the periodicity cell
$\varpi^{\varepsilon}$ because $U^{0}_{p}$ does. Let us examine the second
term (4.10) which satisfies

$\begin{split}\\!\\!\\!\frac{\partial w^{\prime}_{p}}{\partial
x_{2}}\Big{(}\frac{x}{\varepsilon},x_{2};\eta\Big{)}\bigg{|}_{x_{2}=0}=&\frac{1}{\varepsilon}\bigg{(}\frac{\partial
W^{1}}{\partial\xi_{2}}\Big{(}\frac{x_{1}}{\varepsilon},0\Big{)}\frac{\partial
U^{0}_{p}}{\partial x_{1}}({\color[rgb]{0,0,0}0},0;\eta)+\frac{\partial
W^{2}}{\partial\xi_{2}}\Big{(}\frac{x_{1}}{\varepsilon},0\Big{)}\frac{\partial
U^{0}_{p}}{\partial x_{2}}({\color[rgb]{0,0,0}0},0;\eta)\bigg{)}\\\
&+\frac{\partial^{2}U^{0}_{p}}{\partial x_{1}\partial
x_{2}}({\color[rgb]{0,0,0}0},0;\eta)W^{1}\Big{(}\frac{x_{1}}{\varepsilon},0\Big{)}+\frac{\partial^{2}U^{0}_{p}}{\partial
x^{2}_{2}}({\color[rgb]{0,0,0}0},0;\eta)W^{2}\Big{(}\frac{x_{1}}{\varepsilon},0\Big{)}{\color[rgb]{0,0,0}+\frac{\partial
C^{\prime}_{p}}{\partial x_{2}}(0;\eta)}.\end{split}$ (4.26)

A similar formula is valid at $x_{2}=H$. Using (2.13) the second and third
terms on the right-hand side of (4.26) vanish.  Similarly, the last term
vanishes because, by construction (cf. (4.10), (4.9) and (4.4)), it satisfies

$\frac{\partial C^{\prime}_{p}}{\partial x_{2}}(0;\eta)=\frac{\partial
C^{\prime}_{p}}{\partial x_{2}}(H;\eta)=0,$

but the other addends do so only if

$\begin{array}[]{c}\displaystyle\frac{\partial
W^{1}}{\partial\xi_{2}}(\xi_{1},0)=0,\,\,W^{2}(\xi_{1},0)=0,\quad\xi_{1}\in{\mathbb{R}},\end{array}$
(4.27)

cf. also (4.24). There is no reason for (4.27) to be fulfilled for any
asymmetric hole $\omega$ but, owing to Lemma 3.5, the assumption (1.9) gives
us the relations (3.32) and, therefore, (4.27). Furthermore, all terms on the
right-hand side of (4.26) vanish.

######  Remark 4.2.

If the relation (4.27) is denied, the inner expansion (4.10) leaves in the
Neumann condition (2.5) discrepancies of order $1$ which are localized in the
vicinity of the points $(0,0)$, $(0,H)$ and decay exponentially at a distant
from them. To compensate, a new boundary layer is needed involving solutions
to the Neumann problems for the Laplace operator in the half-planes with semi-
infinite families of holes, that is, in

${\mathbb{R}}^{2}_{+}\setminus\bigcup\limits_{k=0}^{\infty}\overline{\omega(1,k)}\,\,\mbox{\rm
and}\,\,{\mathbb{R}}^{2}_{-}\setminus\bigcup\limits_{k=0}^{\infty}\overline{\omega(1,-k)},$

cf. (1.3). Asymptotics at infinity of solutions to elliptic boundary-value
problems in angular domains with periodic boundaries have been investigated in
[23, 25]. However, such a two-dimensional boundary layer seriously complicates
the asymptotic procedure and we postpone the research in the case of more
general perforation for another paper.

## 5 Some bounds for convergence rates

In this section, we obtain some important complementary results on the
approximation (2.10). In particular, we get some estimates which establish the
closeness of eigenvalues $\Lambda^{\varepsilon}(\eta)$ of problem (2.2)–(2.5)
and the first three dispersion curves of the homogenized problem (see Theorem
5.1). As a consequence, we can identify the first eigenvalue
$\Lambda_{1}^{\varepsilon}(\eta)$ at a certain distance from the nodes
$(\eta_{\mbox{\tiny$\square$}},\Lambda_{\mbox{\tiny$\square$}})=(\pm\pi,\pi^{2})$
where the question of their splitting does not appear at all (see Corollary
5.2). For this first eigenvalue, we get a uniform bound for the convergence
rate. The analysis of this section does not take into account the multiplicity
of the eigenvalues of the limit problem.

Let us summarize the results of the section:

######  Theorem 5.1.

There exist $\Lambda_{\star}^{\varepsilon}(\eta)$ and
$\Lambda_{\pm}^{\varepsilon}(\eta)$ eigenvalues of the problem (2.2)–(2.5)
which satisfy

$\displaystyle|\Lambda_{\star}^{\varepsilon}(\eta)-\Lambda_{1}^{0}(\eta)|\leq
C_{0}\varepsilon$ $\displaystyle\quad\mbox{for
}\eta\in[-\pi,\pi],\,0<\varepsilon<\varepsilon_{0},$ (5.1)
$\displaystyle|\Lambda_{\pm}^{\varepsilon}(\eta)-\Lambda_{\pm}^{0}(\eta)|\leq
C_{0}\varepsilon$ $\displaystyle\quad\mbox{for
}\eta\in[-\pi,\pi],\,0<\varepsilon<\varepsilon_{0},$ (5.2)

where $\Lambda_{1}^{0}(\eta)=\eta^{2}$ and $\Lambda_{\pm}^{0}(\eta)=(\eta\pm
2\pi)^{2}$ are eigenvalues of problem (2.12)–(2.14), and the positive
constants $\varepsilon_{0}$ and $C_{0}$ are independent of $\varepsilon$ and
$\eta$.

######  Corollary 5.2.

Let $H\in(0,1).$ Fixed $\delta\in(0,\pi)$, there exists
$\varepsilon_{0}=\varepsilon_{0}(H)$ such that the eigenvalue
$\Lambda_{1}^{\varepsilon}(\eta)$ of problem (2.2)–(2.5) in the sequence
(2.7), and the eigenvalue $\Lambda_{1}^{0}(\eta)$ of problem (2.12)–(2.14) in
the sequence (2.11) satisfy

$|\Lambda_{1}^{\varepsilon}(\eta)-\Lambda_{1}^{0}(\eta)|\leq
C_{0}\varepsilon\qquad\mbox{for }\eta\in[-\pi+\delta,\pi-\delta]\mbox{ and
}0<\varepsilon<\varepsilon_{0},$

where the positive constant $C_{0}$ is independent of the parameters
$\varepsilon$ and $\eta$.

The proofs of these results are in Section 5.4 and use the lemma on almost
eigenvalues which we introduce in Section 5.1. They rely on the construction
of approximations to eigenvalues and eigenfunctions which is done in Sections
5.2 and 5.3.

### 5.1 The abstract setting

We first reformulate the spectral problem (2.2)–(2.5) in terms of operators on
Hilbert spaces, cf. (5.5). In the space
$H^{1,\eta}_{per}(\varpi^{\varepsilon})$ we consider the scalar product

$\begin{array}[]{c}\displaystyle\langle
U^{\varepsilon},V^{\varepsilon}\rangle_{\varepsilon\eta}=(\nabla_{x}U^{\varepsilon},\nabla_{x}V^{\varepsilon})_{\varpi^{\varepsilon}}+(U^{\varepsilon},V^{\varepsilon})_{\varpi^{\varepsilon}}\end{array}$
(5.3)

and the positive, compact and symmetric operator ${\cal
B}^{\varepsilon}(\eta)$,

$\begin{array}[]{c}\displaystyle\langle{\cal
B}^{\varepsilon}(\eta)U^{\varepsilon},V^{\varepsilon}\rangle_{\varepsilon\eta}=(U^{\varepsilon},V^{\varepsilon})_{\varpi^{\varepsilon}}\quad\forall
U^{\varepsilon},V^{\varepsilon}\in
H^{1,\eta}_{per}(\varpi^{\varepsilon}).\end{array}$ (5.4)

The space $H^{1,\eta}_{per}(\varpi^{\varepsilon})$ equipped with the scalar
product (5.3) is denoted by ${\cal H}^{\varepsilon}(\eta)$ and
$\|U^{\varepsilon};{\cal H}^{\varepsilon}(\eta)\|$ denotes the norm generated
by (5.3).

Comparing (5.3), (5.4) with (2.6), we see that the variational formulation of
the problem (2.2)–(2.5) is equivalent to the equation

$\begin{array}[]{c}\displaystyle{\cal
B}^{\varepsilon}(\eta)U^{\varepsilon}(\eta)=M^{\varepsilon}(\eta)U^{\varepsilon}(\eta)\,\,\mbox{\rm
in}\,\,{\cal H}^{\varepsilon}(\eta)\end{array}$ (5.5)

with the new spectral parameter

$\begin{array}[]{c}\displaystyle
M^{\varepsilon}(\eta)=(1+\Lambda^{\varepsilon}(\eta))^{-1}.\end{array}$ (5.6)

The following result (a lemma on almost eigenvalues, cf. [44]) is a
consequence of the spectral decomposition of resolvent, cf. [9, Ch. 6].

######  Lemma 5.3.

Let $M^{\varepsilon}_{as}(\eta)\in{\mathbb{R}}$ and
$U^{\varepsilon}_{as}(\eta)\in{\cal H}^{\varepsilon}(\eta)\setminus\\{0\\}$
verify the relationship

$\|{\cal
B}^{\varepsilon}(\eta)U^{\varepsilon}_{as}(\eta)-M^{\varepsilon}_{as}(\eta)U^{\varepsilon}_{as}(\eta);{\cal
H}^{\varepsilon}(\eta)\|=\delta_{\varepsilon}\|U^{\varepsilon}_{as}(\eta);{\cal
H}^{\varepsilon}(\eta)\|.$ (5.7)

Then, there exists an eigenvalue $M^{\varepsilon}(\eta)$ of the operator
${\cal B}^{\varepsilon}(\eta)$ such that

$|M^{\varepsilon}(\eta)-M^{\varepsilon}_{as}(\eta)|\leq\delta_{\varepsilon}.$

In Sections 5.2 and 5.3 below, we provide $M^{\varepsilon}_{as}(\eta)$ and
$U^{\varepsilon}_{as}(\eta)$ and obtain a bound for the rest
$\delta_{\varepsilon}$ in (5.7).

### 5.2 Approximate eigenvalue and eigenfunction

Let $\Lambda^{0}_{\pm}(\eta)=({\color[rgb]{0,0,0}\eta}\pm 2\pi)^{2}$ be
eigenvalues in (2.15) corresponding to a fixed Floquet parameter
$\eta\in[-\pi,\pi]$. According to (5.6) we take

$\begin{array}[]{c}\displaystyle
M_{\pm}^{0}(\eta)=(1+\Lambda^{0}_{\pm}(\eta))^{-1}\end{array}$ (5.8)

as an approximate eigenvalue ($\pm$ respectively), and

$U^{\varepsilon}_{\pm}(x;\eta)=X^{\varepsilon}(x_{1})U^{0}_{\pm}(x_{1};\eta)+(1-X^{\varepsilon}(x_{1}))\Big{(}U^{0}_{\pm}(0;\eta)+x_{1}\frac{\partial
U^{0}_{\pm}}{\partial
x_{1}}(0;\eta)\Big{)}+\varepsilon\chi_{0}(x_{1})\frac{\partial
U^{0}_{\pm}}{\partial
x_{1}}(0;\eta)W^{1}_{0}\Big{(}\frac{x}{\varepsilon}\Big{)},$ (5.9)

as an approximate eigenfunction constructed from the asymptotic expansions in
Section 4 (cf. (4.2), (4.5), (4.7) and (4.10) which holds for
$\eta\in[-\pi,\pi]$). $W^{1}_{0}$ is the bounded harmonics in $\Xi$, see
(3.20) and (3.24),

$\begin{array}[]{c}\displaystyle
U^{0}_{\pm}(x_{1};\eta)=e^{i(\eta\pm{\color[rgb]{0,0,0}2}\pi)x_{1}},\end{array}$
(5.10) $\displaystyle
X^{\varepsilon}(x_{1})=1\\!-\\!\chi_{+}(x_{1}/\varepsilon)\\!-\\!\chi_{-}(x_{1}/\varepsilon),\mbox{
i.e. }X^{\varepsilon}(x_{1})=1\,\,\mbox{\rm for }|x_{1}|\geq
2R\varepsilon,\,\,X^{\varepsilon}(x_{1})=0\,\,\mbox{\rm for }|x_{1}|\leq
R\varepsilon,\vspace{0.15cm}$ $\displaystyle\displaystyle\chi_{0}\in
C^{\infty}({\mathbb{R}}),\quad\chi_{0}(x_{1})=1\,\,\mbox{\rm for }|x_{1}|\leq
1/6,\,\,\chi_{0}(x_{1})=0\,\,\mbox{\rm for }|x_{1}|\geq 1/3,$ (5.11)

where the even smooth cut-off functions $\chi_{\pm}$ are defined by (3.12).
Notice that,  for $0<|\eta|<\pi$, $\Lambda^{0}_{\pm}(\eta)$ is a simple
eigenvalue so that it corresponds to the only eigenfunction (5.10), see (2.15)
with $j=\pm 1$ and $k=0$, so that the sign plus or minus is fixed in these
formulas.

### 5.3 Estimating the discrepancy

The function (5.9) satisfies the Neumann condition (2.5) as well as the quasi-
periodicity conditions (2.3), (2.4). To conclude these  assertions, we recall
(3.32)  and (3.21), and observe that $X^{\varepsilon}(x_{1})=1$ and
$\chi_{0}(x_{1})=0$ near the points $x_{1}=\pm 1/2$.

In order to apply Lemma 5.3, we multiply (5.7) by
$\|U^{\varepsilon}_{as}(\eta);{\cal H}^{\varepsilon}(\eta)\|^{-1}$ and obtain
the relation

$\begin{array}[]{rl}\delta^{\varepsilon}_{\pm}(\eta):=\\!&\\!\|U_{\pm}^{\varepsilon};{\cal
H}^{\varepsilon}(\eta)\|^{-1}\|{\cal
B}^{\varepsilon}(\eta)U_{\pm}^{\varepsilon}-M^{0}_{\pm}(\eta)U_{\pm}^{\varepsilon};{\cal
H}^{\varepsilon}(\eta)\|\vspace{0.1cm}\\\
=\\!&\\!\|U_{\pm}^{\varepsilon};{\cal
H}^{\varepsilon}(\eta)\|^{-1}M^{0}_{\pm}(\eta)\,\sup|(\nabla_{x}U^{\varepsilon}_{\pm},\nabla_{x}V^{\varepsilon})_{\varpi^{\varepsilon}}-\Lambda^{0}_{\pm}(\eta)(U^{\varepsilon}_{\pm},V^{\varepsilon})_{\varpi^{\varepsilon}}|\vspace{0.1cm}\\\
=\\!&\\!\|U_{\pm}^{\varepsilon};{\cal
H}^{\varepsilon}(\eta)\|^{-1}M^{0}_{\pm}(\eta)\,\sup|(\Delta_{x}U^{\varepsilon}_{\pm}+\Lambda^{0}_{\pm}(\eta))U^{\varepsilon}_{\pm},V^{\varepsilon})_{\varpi^{\varepsilon}}|.\end{array}$
(5.12)

Here, the supreme is computed over all function $V^{\varepsilon}\in{\cal
H}^{\varepsilon}(\eta)$ with unit norm and this calculation takes into account
definitions (5.3), (5.4), (5.8) and the Green formula together with the
Neumann and quasi-periodicity conditions for $U^{\varepsilon}_{\pm}$ and the
Neumann and periodicity conditions for $W_{0}^{1}$, (3.6) and (3.21),
respectively. Let us show the estimate

$\delta^{\varepsilon}_{\pm}(\eta)\leq c\varepsilon\quad\mbox{for
}\varepsilon\leq\varepsilon_{0},$ (5.13)

with some constants $c$ and $\varepsilon_{0}$ independent of $\eta$.

Indeed, by (5.9), we write

$\Delta_{x}U_{\pm}^{\varepsilon}(x;\eta)+\Lambda^{0}_{\pm}(\eta){\color[rgb]{0,0,0}U_{\pm}^{\varepsilon}}(x;\eta)=:\sum\limits_{j=1}^{{\color[rgb]{0,0,0}6}}S^{\varepsilon}_{j,\pm}(x;\eta)$
(5.14)

with

$\displaystyle
S^{\varepsilon}_{1,\pm}(x;\eta)=X^{\varepsilon}(x_{1})(\Delta_{x}U^{0}_{\pm}(x_{1};\psi)+\Lambda^{0}_{\pm}(\eta)U^{0}_{\pm}(x_{1};\psi)),$
$\displaystyle
S^{\varepsilon}_{2,\pm}(x;\eta)=[\Delta_{x},X^{\varepsilon}(x_{1})]\Big{(}U^{0}_{\pm}(x_{1};\eta)-U^{0}_{\pm}(0;\eta)-x_{1}\frac{\partial
U^{0}_{\pm}}{\partial x_{1}}(0;\eta)\Big{)},$ $\displaystyle
S^{\varepsilon}_{3,\pm}(x;\eta)=\frac{1}{\varepsilon}\frac{\partial
U^{0}_{\pm}}{\partial
x_{1}}(0;\eta)\chi_{0}(x_{1})\Delta_{\xi}W^{1}_{0}(\xi),$ $\displaystyle
S^{\varepsilon}_{4,\pm}(x;\eta)=\varepsilon\Lambda^{0}_{\pm}(\eta)\chi_{0}(x_{1})\frac{\partial
U^{0}_{\pm}}{\partial x_{1}}(0;\eta)W^{1}_{0}(\xi),$ $\displaystyle
S^{\varepsilon}_{5,\pm}(x;\eta)=\varepsilon\frac{\partial
U^{0}_{\pm}}{\partial
x_{1}}(0;\eta)[\Delta_{x},\chi_{0}(x_{1})]W^{1}_{0}(\xi),$ $\displaystyle
S^{\varepsilon}_{6,\pm}(x;\eta)=\Lambda^{0}_{\pm}(\eta)(1-X^{\varepsilon}(x_{1}))\Big{(}U^{0}_{\pm}(0;\eta)+x_{1}\frac{\partial
U^{0}_{\pm}}{\partial x_{1}}(0;\eta)\Big{)};$

here, $[\Delta_{x},\chi]$ stands for the commutator, i.e.
$[\Delta_{x},\chi]U:=\Delta_{x}(\chi U)-\chi\Delta_{x}U$. Note that
$[\Delta_{x},\chi]U=2\nabla_{x}\chi\cdot\nabla_{x}U+U\Delta_{x}\chi$ and
$[\Delta_{x},1-X^{\varepsilon}(x_{1})]=-[\Delta_{x},X^{\varepsilon}(x_{1})]$.

We have $S^{\varepsilon}_{1,\pm}(x;\eta)=0$ and
$S^{\varepsilon}_{3,\pm}(x;\eta)=0$ because of the equations (2.2) for
$U^{0}_{\pm}$ and (3.4) for $W^{1}_{0}$. Since $W^{1}_{0}$ admits the
representation (3.24) and, therefore, is bounded together with its derivative,
we conclude that

$\displaystyle\\!\\!\\!\\!|(S^{\varepsilon}_{4,\pm},V^{\varepsilon})_{\varpi^{\varepsilon}}|\leq
c\,\varepsilon\sup\limits_{\xi\in\Xi}|W^{1}_{0}(\xi)|\,\|V^{\varepsilon};L^{2}(\varpi^{\varepsilon})\|\leq
C\,\varepsilon{\color[rgb]{0,0,0}\,\|V^{\varepsilon};L^{2}(\varpi^{\varepsilon})\|}.\quad$
(5.15)

Moreover, since the coefficients in the commutator
$[\Delta_{x},\chi_{0}({\color[rgb]{0,0,0}x_{1}})]$ do not depend on
$\varepsilon$ and have their supports in the union of the rectangles
$\Upsilon^{0}_{\pm}=\\{x:\,\pm x_{1}\in[1/6,1/3],x_{2}\in[0,H]\\}$, and
$\nabla_{\xi}W_{0}^{1}$ has an exponential decay, we have

$|(S^{\varepsilon}_{5,\pm},V^{\varepsilon})_{\varpi^{\varepsilon}}|\leq\,c\,\varepsilon\sup\limits_{\xi\in\Xi}(|W^{1}_{0}(\xi)|+|\xi_{1}||\nabla_{\xi_{1}}W^{1}_{0}(\xi)|)\,\|V^{\varepsilon};L^{2}(\varpi^{\varepsilon})\|\leq\,C\,\varepsilon\,\|V^{\varepsilon};L^{2}(\varpi^{\varepsilon})\|.$
(5.16)

On the other hand, owing to (5.11), the support of $S^{\varepsilon}_{2,\pm}$
belongs to the union of the thin rectangles
${\color[rgb]{0,0,0}\Upsilon^{\varepsilon}_{\pm}}=\\{x:\pm
x_{1}\in[R\varepsilon,2R\varepsilon],x_{2}\in[0,H]\\}$ and the coefficient of
the derivative and the free coefficient in the commutator

$[\Delta_{x},X^{\varepsilon}(x_{1})](\cdot)=2\frac{\partial
X^{\varepsilon}}{\partial x_{1}}(x_{1})\frac{\partial(\cdot)}{\partial
x_{1}}+\Delta_{x}X^{\varepsilon}(x_{1})(\cdot)$

are of order $\varepsilon^{-1}$ and $\varepsilon^{-2}$ respectively. Besides,
the inequality

$\|V^{\varepsilon};L^{2}(\Upsilon^{\varepsilon}_{\pm})\|\leq
c\varepsilon^{1/2}\|V^{\varepsilon};{\color[rgb]{0,0,0}{\mathcal{H}}^{\varepsilon}(\eta)}\|$

is valid, see for example the proof of (2.17) and (2.18). Thus, based on the
Taylor formula for $U^{0}_{\pm}$, we see that

$|(S^{\varepsilon}_{2,\pm},V^{\varepsilon})_{\varpi^{\varepsilon}}|\leq\,c\sum\limits_{\pm}|\Upsilon^{\varepsilon}_{\pm}|^{1/2}\max\limits_{x\in\Upsilon^{\varepsilon}_{\pm}}\bigg{|}\frac{\partial^{2}U^{0}_{\pm}}{\partial
x_{1}^{2}}(x_{1};{\color[rgb]{0,0,0}\eta})\bigg{|}\,\|V^{\varepsilon};L^{2}(\Upsilon^{\varepsilon}_{\pm})\|\leq\,c\varepsilon\,{\color[rgb]{0,0,0}\|V^{\varepsilon};{\mathcal{H}}^{\varepsilon}(\eta)\|}.$
(5.17)

Similarly, since the support of $S_{6,\pm}^{\varepsilon}$ is included in
$\Theta^{\varepsilon}=[-2R\varepsilon,2R\varepsilon]\times[0,H]$, we have

$\begin{array}[]{c}\displaystyle|(S^{\varepsilon}_{6,\pm},V^{\varepsilon})_{\varpi^{\varepsilon}}|\leq
C|\Theta^{\varepsilon}|^{1/2}\|V^{\varepsilon};\\!L^{2}(\Theta^{\varepsilon})\|\leq
C\,\varepsilon\,\|V^{\varepsilon};{\mathcal{H}}^{\varepsilon}(\eta)\|.\end{array}$
(5.18)

Finally, by definition of $U_{\pm}^{\varepsilon}$ (see (5.9) and (5.10)), it
can be proved that

$\|U_{\pm}^{\varepsilon};{\cal
H}^{\varepsilon}(\eta)\|^{2}\mathop{\displaystyle{\stackrel{{\scriptstyle\scriptstyle{\varepsilon\to
0}}}{{\hbox
to40.0pt{\rightarrowfill}}}}}\|U_{\pm}^{0};L^{2}(\varpi^{0})\|^{2}+\|\nabla_{x}U_{\pm}^{0};L^{2}(\varpi^{0})\|^{2}=(1+\Lambda_{\pm}^{0}(\eta))H.$
(5.19)

Based on the representation (5.12), (5.14), the estimates (5.15), (5.16),
(5.17) and (5.18), and the convergence (5.19), we arrive at (5.13).

### 5.4 Asymptotics of the eigenvalues

Considering the estimate (5.13), Lemma 5.3 gives us an eigenvalue
$M^{\varepsilon}_{\pm}(\eta)$ of the operator ${\cal B}^{\varepsilon}(\eta)$
such that

$\begin{array}[]{c}\displaystyle|M^{\varepsilon}_{\pm}(\eta)-M^{0}_{\pm}(\eta)|\leq
c\varepsilon\end{array}$ (5.20)

where the factor $c$ is independent of $\eta$. Recalling (5.6), we derive from
(5.20) that

$\displaystyle|\Lambda^{\varepsilon}_{\pm}(\eta)-\Lambda^{0}_{\pm}(\eta)|\leq
c\varepsilon(1+\Lambda^{0}_{\pm}(\eta))(1+\Lambda^{\varepsilon}_{\pm}(\eta)),$
(5.21)

and, hence

$(1+\Lambda^{\varepsilon}_{\pm}(\eta))(1-c\varepsilon(1+\Lambda^{0}_{\pm}(\eta))\leq
1+\Lambda^{0}_{\pm}(\eta).$

Let us set

$\varepsilon_{0}:=\frac{1}{2c(1+4\pi^{2})}.$

Then, for $\varepsilon<\varepsilon_{0}$ and $\eta\in[-\pi,\pi]$, we have
$(1-c\varepsilon(1+\Lambda^{0}_{\pm}(\eta))\\!>\\!1/2$ and therefore

$\begin{array}[]{c}\displaystyle|\Lambda^{\varepsilon}_{\pm}(\eta)-\Lambda^{0}_{\pm}(\eta)|\leq
2c\varepsilon(1+\Lambda^{0}_{\pm}(\eta))^{2}\leq
2c\varepsilon(1+9\pi^{2})^{2}=:C_{0}\varepsilon.\end{array}$ (5.22)

This ends the proof of (5.2).

In a similar way, replacing $\Lambda_{\pm}^{0}(\eta)$ and
$U_{\pm}^{0}(x_{1};\eta)$ by $\Lambda_{1}^{0}(\eta)=\eta^{2}$ and
$U_{1}^{0}(x_{1};\eta)=e^{i\eta x_{1}}$ respectively in (5.8) and (5.9), we
obtain some constants $\varepsilon_{0},C_{0}>0$ and certain eigenvalues
$\Lambda_{\star}^{\varepsilon}(\eta)$ of (2.2)–(2.5) which satisfy (5.1).
Thus, the proof of Theorem 5.1 is completed.

Finally, on account of (2.29), there cannot be more than one eigenvalue
$\Lambda_{p}^{\varepsilon}(\eta)$ in the box
$[-\pi+\delta,\pi-\delta]\times[0,\pi^{2}+K_{1}]$ for any $\delta>0$ and
$K_{1}$ defined by (2.31), and hence we can identify the eigenvalue
$\Lambda_{\star}^{\varepsilon}(\eta)$ given in Proposition 5.1 with the first
eigenvalue $\Lambda_{1}^{\varepsilon}(\eta)$ at a distance $\delta$ from
$\eta_{\mbox{\tiny$\square$}}=\pm\pi$ and Corollary 5.2 holds.

## 6 Asymptotic analysis near nodes

The main difference between the asymptotic analysis in the previous and the
next sections is that in what follows the limit eigenvalue under consideration
is always multiple and gives rise to a node of the dispersion curves in Figure
4 a)–b). Furthermore, examining the splitting of the band edges and the
opening of spectral gaps requires much more precise asymptotic formulas for
the eigenvalues in (2.7) which are valid in a neighborhood of a certain value
of the Floquet parameter $\eta$. This seriously complicates the asymptotic
analysis as well as the justification procedure. In fact, the asymptotic
analysis is somehow double, since it takes into account the small parameter
and the small neighborhood of the nodes
$(\eta_{\mbox{\small{$\circ$}}},\Lambda_{\mbox{\small{$\circ$}}})=(0,4\pi^{2})$
and
$(\eta_{\mbox{\tiny$\square$}},\Lambda_{\mbox{\tiny$\square$}})=(\pm\pi,\pi^{2}).$
In Sections 6.1–6.3, we perform all the computations for the node
$(0,4\pi^{2})$ while, for the sake of brevity, we sketch the main changes for
the nodes $(\pm\pi,\pi^{2}),$ cf. Section 6.4. Section 6.1 contains the
asymptotic analysis based on asymptotic expansions while Sections 6.2–6.3
contain a justification scheme for the abstract formulation in Section 5.1.

### 6.1 The node
$(\eta_{\mbox{\small{$\circ$}}},\Lambda_{\mbox{\small{$\circ$}}})=(0,4\pi^{2})$
for $H\in(0,1/2)$

This node marked with $\circ$ occurs in Figure 4 a) (cf. also Figure 3) under
the assumption $H\in(0,1/2)$ as the intersection point of the two (plus and
minus) limit dispersion curves

$\begin{array}[]{c}\displaystyle\Lambda_{\pm}^{0}(\eta)=(\eta\pm
2\pi)^{2},\,\,\eta\in[-\pi,\pi].\end{array}$ (6.1)

The problem (2.12)–(2.14) with $\eta=0$ has the eigenvalue
$\Lambda^{0}:=\Lambda_{2}^{0}(0)=\Lambda_{3}^{0}(0)=4\pi^{2}$ of multiplicity
$2$ with the eigenfunctions

$\begin{array}[]{c}\displaystyle U^{0}_{\pm}(x)=e^{\pm 2\pi
ix_{1}}.\end{array}$ (6.2)

To investigate the perturbed dispersion curves (2.8) with $p=2,3$ near the
point
$(\eta_{\mbox{\small{$\circ$}}},\Lambda_{\mbox{\small{$\circ$}}})=(0,4\pi^{2})$,
we use the idea in [27] by introducing the rapid Floquet variable

$\begin{array}[]{c}\displaystyle\psi=\varepsilon^{-1}\eta\end{array}$ (6.3)

in a neighborhood of $\eta=0$, and perform the asymptotic ansatz for the
eigenvalues

$\begin{array}[]{c}\displaystyle\Lambda^{\varepsilon}_{p}(\eta)=\Lambda^{0}+\varepsilon\Lambda^{\prime}(\psi)+\varepsilon^{2}\Lambda^{\prime\prime}(\psi)+\dots\end{array}$
(6.4)

with $p=2,3$ as in Figure 5 a). To shorten the notation, we do not display the
index $p$ in the terms of the anzätze.

We assume the outer expansion for the corresponding eigenfunction

$\begin{array}[]{c}\displaystyle
U^{\varepsilon}(x;\\!\eta)=U^{0}(x;\\!\psi)+\varepsilon
U^{\prime}(x;\\!\psi)+\varepsilon^{2}U^{\prime\prime}(x;\\!\psi)+\dots\end{array}$
(6.5)

to be valid in $\varpi^{0}\setminus\varsigma$, where

$\begin{array}[]{c}\displaystyle U^{0}(x;\psi)=a_{+}(\psi)e^{+2\pi
ix_{1}}+a_{-}(\psi)e^{-2\pi ix_{1}},\end{array}$ (6.6)

$\psi$ is a parameter, $\psi=O(1)$, and $a(\psi)=(a_{+}(\psi),a_{-}(\psi))$ is
a column vector in ${\mathbb{C}}^{2}$ to be determined together with the
correction terms $\Lambda^{\prime}(\psi)$ and $U^{\prime}(\cdot;\psi)$ in the
ansätze (6.4) and (6.5), respectively. We follow the technique developed in
Sections 4.1–4.4 and we only outline the main differences. As in Section 4,
the terms $\Lambda^{\prime\prime}(\psi)$ in (6.4) and
$U^{\prime\prime}(x;\psi)$ in (6.5) are not of further use.

We look for an inner expansion in the vicinity of the transversal perforation
string (3.2)

$\begin{array}[]{c}\displaystyle
U^{\varepsilon}(x;\eta)=w^{0}(x_{2};\psi)+\varepsilon
w^{\prime}(\xi;\psi)+\varepsilon^{2}w^{\prime\prime}(\xi;\psi)+\dots,\end{array}$
(6.7)

where we have assumed that the main term $w^{0}$ does not depend on
$\xi=\varepsilon^{-1}x$ while the functions arising in further terms,
$w^{\prime}$ and $w^{\prime\prime}$ satisfy a periodicity condition in the
$\xi_{2}$-direction. Following the scheme in Section 4, the immediate result
of the matching procedure at the first order is

$\begin{array}[]{c}\displaystyle
w^{0}(x_{2};\psi)=U^{0}(0;\psi)=(a_{+}(\psi)+a_{-}(\psi))W^{0}=a_{+}(\psi)+a_{-}(\psi),\end{array}$
(6.8)

cf. (4.7) and (3.19). Since the main term (6.7) is independent of the
transversal variable, the dependence on $x_{2}$ (not on $\xi_{2}$!) disappear
in all terms and we will write the argument $x_{1}$ instead of $x$ on the
right-hand side of (6.5) and omit $x_{2}$ on the right-hand side of (6.7).

We continue with the matching procedure at the second order taking into
account the Taylor expansion for (6.5), cf. (4.6). The Taylor formula applied
to (6.6) gives

$\displaystyle U^{0}(x;\psi)=a_{+}(\psi)+a_{-}(\psi)+2\pi
ix_{1}(a_{+}(\psi)-a_{-}(\psi))-{\color[rgb]{0,0,0}2}\pi^{2}x_{1}^{2}(a_{+}(\psi)+a_{-}(\psi))+O(|x_{1}|^{3}),$
(6.9)

where $O(|x_{1}|^{3})$ depends on $\psi$, and, recalling the solution (3.20)
of the problem (3.4), (3.6), (3.7), cf. (4.10), we set

$\begin{array}[]{c}\displaystyle w^{\prime}(\xi;\psi)=2\pi
i(a_{+}(\psi)-a_{-}(\psi))W^{1}(\xi)+a^{\prime}(\psi)W^{0}\end{array}$ (6.10)

with some factor $a^{\prime}(\psi)$ which can be fixed arbitrarily at the
present stage of our analysis. In contrast to (4.10) the solution $W^{2}$ is
absent in (6.10). Thus, the first jump condition for the correction term in
(6.5) is (cf. (4.6), (4.12) and (6.9)):

$\begin{array}[]{c}\displaystyle[U^{\prime}]_{0}(\psi)=2\frac{\partial
U^{0}}{\partial x_{1}}(0;\\!\psi)m_{1}(\Xi)=4\pi
i(a_{+}(\psi)-a_{-}(\psi))m_{1}(\Xi).\end{array}$ (6.11)

This formula coincides with (4.12) because $\partial_{x_{2}}U^{0}\\!=\\!0,$
and it is independent of $x_{2}$.

The matching procedure at level $\varepsilon^{2}$, in the same way as in
Section 4.3, gives

$w^{\prime\prime}(\xi;\psi)=4\pi^{2}(a_{+}(\psi)+a_{-}(\psi))W^{3}(\xi)+a^{\prime\prime}(\psi)W^{0}+\widetilde{w}^{\,\prime\prime}(\xi;\psi)$

where $W^{3}$ is the solution (3.30) of the problem (3.29), (3.6), (3.7),
$a^{\prime\prime}(\psi)$ is some factor which can be fixed arbitrarily at the
present stage of our analysis and the remainder
$\widetilde{w}^{\,\prime\prime}$ gets the exponential decay as
$\xi_{1}\to\pm\infty$ (cf. (4.16), (4.18) and (4.21)). Besides, the second
jump condition (4.23) now takes the simplified form (cf. (4.23) and (6.9))

$\begin{array}[]{c}\displaystyle\Big{[}\frac{\partial U^{\prime}}{\partial
x_{1}}\Big{]}_{0}{\color[rgb]{0,0,0}(\psi)}={\color[rgb]{0,0,0}-}\frac{|\omega|}{H}\frac{\partial^{2}U^{0}}{\partial
x^{2}_{1}}({\color[rgb]{0,0,0}0};\psi)={\color[rgb]{0,0,0}4}\pi^{2}(a_{+}(\psi)+a_{-}(\psi))\frac{|\omega|}{H}.\end{array}$
(6.12)

Other restrictions on $U^{\prime}$ are readily inherited from (2.2), (2.5) and
(6.4), cf. Section 4.1:

$\begin{array}[]{c}-\Delta_{x}U^{\prime}(x;\psi)-\Lambda^{0}U^{\prime}(x;\psi)=\Lambda^{\prime}(\psi)U^{0}(x;\psi),\,\,x\in\varpi^{0}\setminus\varsigma,\\\
\\\ \displaystyle\frac{\partial U^{\prime}}{\partial
x_{2}}(x_{1},0;\psi)=\frac{\partial U^{\prime}}{\partial
x_{2}}(x_{1},H;\psi)=0,\,\,x_{1}\in\Big{(}-\frac{1}{2},0\Big{)}\cup\Big{(}0,\frac{1}{2}\Big{)}.\end{array}$
(6.13)

In the quasi-periodicity conditions it is also necessary to take into account
the fast Floquet parameter (6.3) and the Taylor formula

$\begin{array}[]{c}e^{i\eta}=e^{i\varepsilon\psi}=1+i\varepsilon\psi+O(\varepsilon^{2}).\end{array}$
(6.14)

In this way, inserting (6.5) into (2.3), (2.4), collecting terms of order
$\varepsilon$ and using (6.6) yield

$\begin{split}\displaystyle&U^{\prime}\Big{(}\frac{1}{2},x_{2};\psi\Big{)}\\!-\\!U^{\prime}\Big{(}\\!\\!-\frac{1}{2},x_{2};\psi\Big{)}\\!=\\!i\psi
U^{0}\Big{(}\\!\\!-\frac{1}{2},x_{2};\psi\Big{)}\\!=\\!-i\psi(a_{+}(\psi)+a_{-}(\psi)),\vspace{0.3cm}\\\
&\displaystyle\frac{\partial U^{\prime}}{\partial
x_{1}}\Big{(}\frac{1}{2},x_{2};\psi\Big{)}\\!-\\!\frac{\partial
U^{\prime}}{\partial
x_{1}}\Big{(}\\!\\!-\frac{1}{2},x_{2};\psi\Big{)}\\!=\\!i\psi\frac{\partial
U^{0}}{\partial
x_{1}}\Big{(}\\!\\!-\frac{1}{2},x_{2};\psi\Big{)}\\!=\\!2\pi\psi(a_{+}(\psi)\\!-\\!a_{-}(\psi)).\end{split}$
(6.15)

The problem (6.11), (6.12), (6.13), (6.15) has two compatibility conditions
which can be derived by multiplying the partial differential equations in
(6.13) by the eigenfunctions (6.2) and applying the Green formula on
$\varpi^{0}\setminus\varsigma$. Thus, we have

$\begin{split}\Lambda^{\prime}(\psi)Ha_{\pm}(\psi)&=-\int\limits_{\varpi^{0}}\overline{e^{\pm
2\pi
ix_{1}}}\left(\Delta_{x}U^{\prime}(x;\psi)+\Lambda^{0}U^{\prime}(x;\psi)\right)dx\\\
&=\int_{0}^{H}\\!\\!\Big{(}\frac{\partial U^{\prime}}{\partial
x_{1}}(x;\psi)\pm 2\pi
iU^{\prime}(x;\psi)\Big{)}\bigg{|}_{x_{1}=-\frac{1}{2}}^{\frac{1}{2}}\\!dx_{2}+\\!\int_{0}^{H}\\!\\!\Big{[}\frac{\partial
U^{\prime}}{\partial x_{1}}(x;\psi)\pm 2\pi
iU^{\prime}(x;\psi)\Big{]}_{0}\\!dx_{2}.\end{split}$ (6.16)

Notice that the factor $H$ on the left-hand side is due to the formula

$\begin{array}[]{c}\displaystyle\|e^{\pm 2i\pi
x_{1}};L^{2}(\varpi^{0})\|=H^{1/2}.\end{array}$

Using the inhomogeneous data in (6.11), (6.12) and (6.15), we observe that the
integrands are constants and reduce (6.16) to the system of two linear
algebraic equations with the spectral parameter $\Lambda^{\prime}(\psi)$:

$\begin{array}[]{l}\displaystyle\Lambda^{\prime}(\psi)a_{+}(\psi)=\left(4\pi^{2}\frac{|\omega|}{H}-8\pi^{2}m_{1}(\Xi)+4\pi\psi\right)a_{+}(\psi)+\left(4\pi^{2}\frac{|\omega|}{H}+8\pi^{2}m_{1}(\Xi)\right)a_{-}(\psi),\vspace{0.3cm}\\\
\displaystyle\Lambda^{\prime}(\psi)a_{-}(\psi)=\left(\\!4\pi^{2}\frac{|\omega|}{H}+8\pi^{2}m_{1}(\Xi)\right)a_{+}(\psi)+\left(4\pi^{2}\frac{|\omega|}{H}-8\pi^{2}m_{1}(\Xi)-4\pi\psi\\!\right)a_{-}(\psi).\end{array}$
(6.17)

The two eigenvalues of this system are

$\begin{array}[]{c}\displaystyle{\color[rgb]{0,0,0}\Lambda^{\prime}_{\pm}}(\psi)=4\pi\left(2\pi{\color[rgb]{0,0,0}\Big{(}\frac{|\omega|}{2H}-m_{1}(\Xi)\Big{)}}\pm\sqrt{4\pi^{2}\Big{(}m_{1}(\Xi){\color[rgb]{0,0,0}+}\frac{|\omega|}{2H}\Big{)}^{2}+\psi^{2}}\right).\end{array}$
(6.18)

In particular, we have ${\color[rgb]{0,0,0}\Lambda_{-}^{\prime}}(\psi)<0$ and
${\color[rgb]{0,0,0}\Lambda_{+}^{\prime}}(\psi)>0$ because

$\Lambda^{\prime}_{-}(\psi)\leq-16\pi^{2}m_{1}(\Xi)\quad\hbox{ and
}\quad\Lambda^{\prime}_{+}(\psi)\geq 8\pi^{2}\frac{|\omega|}{H},$ (6.19)

and $m_{1}(\Xi)>0$, see (3.23) and Remark 4.1 to compare.

In addition, the corresponding eigenvectors
$a^{\pm}(\psi)=(a^{\pm}_{+}(\psi),a^{\pm}_{-}(\psi))$ can also be easily
computed. Finally, the compatibility conditions in the problem (6.13), (6.11),
(6.12), (6.15) are satisfied and it has a solution $U^{\prime}(x;\psi)$ which
is defined up to a linear combination of the eigenfunctions (6.2) but, in the
sequel, it can be fixed orthogonal to them and therefore become unique. This
condition determines all the terms in the asymptotic ansätze (6.4), (6.5) and
(6.7).

According to (6.19) we have
$\Lambda^{\prime}_{+}(\psi)>\Lambda^{\prime}_{-}(\psi)$, so that the eigenpair
$\\{\Lambda^{\prime}_{-}(\psi),a^{-}(\psi)\\}$ can be related to the eigenpair
$\\{\Lambda^{\varepsilon}_{2}(\eta),U^{\varepsilon}_{2}(x;\psi)\\}$ of the
problem (2.2)–(2.5) while $\\{\Lambda^{\prime}_{+}(\psi),a^{+}(\psi)\\}$ does
to $\\{\Lambda^{\varepsilon}_{3}(\eta),U^{\varepsilon}_{3}(x;\psi)\\}$.

Now, we formulate our result on the splitting edges of the second and third
limit spectral bands giving rise to the open gap $\gamma_{2}^{\varepsilon}$
(cf. Figure 5 a)). Its proof is in Sections 6.2–6.3.

######  Theorem 6.1.

Let $H\in(0,1/2)$ and $\psi_{0}>0$. Then, there exist positive
$\varepsilon_{0}=\varepsilon_{0}(H{\color[rgb]{0,0,0},\psi_{0}})$ and
$C=C(H{\color[rgb]{0,0,0},\psi_{0}})$ such that, for
$\varepsilon\in(0,\varepsilon_{0}]$, the entries
$\Lambda^{\varepsilon}_{2}(\eta)$ and $\Lambda^{\varepsilon}_{3}(\eta)$ of the
eigenvalue sequence (2.7) with $\eta=\varepsilon\psi$, $|\psi|\leq\psi_{0}$,
meet the estimates

$\begin{array}[]{c}\displaystyle|\Lambda^{\varepsilon}_{3}(\varepsilon\psi)-4\pi^{2}-\varepsilon{\color[rgb]{0,0,0}\Lambda_{+}^{\prime}}(\psi)|\leq
C\varepsilon^{2},\\\ \\\
|\Lambda^{\varepsilon}_{2}(\varepsilon\psi)-4\pi^{2}-\varepsilon{\color[rgb]{0,0,0}\Lambda_{-}^{\prime}}(\psi)|\leq
C\varepsilon^{2},\end{array}$

where the quantities ${\color[rgb]{0,0,0}\Lambda_{\pm}^{\prime}}(\psi)$ are
given by (6.18).

### 6.2 Approximate eigenvalues and eigenfunctions

Recalling Section 5.1, based on calculations in Section 6.1, we set

$\begin{array}[]{c}\displaystyle
M^{1,\varepsilon}_{\pm}(\psi)=(1+4\pi^{2}+\varepsilon\Lambda^{\prime}_{\pm}(\psi))^{-1}\end{array}$
(6.20)

where $\Lambda^{\prime}_{\pm}(\psi)$ are taken from (6.18). Similarly to
(5.9), based on the asymptotic formulas in Section 6.1, we define the
approximate eigenfunction

$\displaystyle U^{\varepsilon}_{\pm}(x;\psi)=\displaystyle
X^{\varepsilon}(x_{1})\big{(}U^{0}_{\pm}(x_{1};\psi)+\varepsilon
U^{\prime}_{\pm}(x_{1};\psi)\big{)}+\varepsilon\chi_{0}(x_{1})\Big{(}\widetilde{w}^{\prime}_{\pm}(\frac{x}{\varepsilon};\psi)+\varepsilon\widehat{w}^{\,\prime\prime}_{\pm}(\frac{x}{\varepsilon};\psi)\Big{)}+\varepsilon^{2}R^{\varepsilon}_{\pm}(x;\psi)\vspace{0.1cm}\\\
\displaystyle+(1-X^{\varepsilon}(x_{1}))\bigg{(}U^{0}_{\pm}(0;\psi)+x_{1}\frac{\partial
U^{0}_{\pm}}{\partial
x_{1}}(0;\psi)+\frac{x_{1}^{2}}{2}\frac{\partial^{2}U^{0}_{\pm}}{\partial
x_{1}^{2}}(0;\psi)+\frac{\varepsilon}{2}\sum\limits_{\tau=\pm}\Big{(}U^{\prime}_{\pm}(\tau
0;\psi)+x_{1}\frac{\partial U^{\prime}_{\pm}}{\partial x_{1}}(\tau
0;\psi)\Big{)}\bigg{)}.$ (6.21)

Let us describe the terms arising in (6.21).

The cut-off functions are defined in (5.11). The main term $U^{0}_{\pm}$ is
the linear combination

$U_{\pm}^{0}(x;\psi)=a_{+}^{\pm}(\psi)e^{+2\pi
ix_{1}}+a_{-}^{\pm}(\psi)e^{-2\pi ix_{1}},$

where, for each sign $\pm$, the coefficient column vector
$a^{\pm}(\psi)=(a^{\pm}_{+}(\psi),a^{\pm}_{-}(\psi))$ is the eigenvector of
the system (6.17) with
$\Lambda^{\prime}(\psi)={\color[rgb]{0,0,0}\Lambda^{\prime}_{\pm}(\psi)}$ and
$U^{\prime}_{\pm}$ is a solution of the problem (6.13), (6.15), (6.11),
(6.12), the compatibility conditions of which are fulfilled due to (6.17).
Both, $U^{0}_{\pm}$ and $U^{\prime}_{\pm}$, depend on the variable $x_{1}$
only. We fix the main term by prescribing the normalization condition

$\begin{array}[]{c}\displaystyle|a^{\pm}(\psi)|=1\,\,\mbox{\rm which
implies}\,\,\|U^{0}_{\pm};H^{2}(\varpi^{0})\|=C_{0}>0.\end{array}$ (6.22)

Then, the solution of the problem (6.13), (6.15), (6.11), (6.12) meets the
estimate

$\begin{array}[]{c}\displaystyle\|U^{\prime}_{\pm};C^{2}(\overline{\varpi^{0}\cap\\{x_{1}>0}\\}\|+\|U^{\prime}_{\pm};C^{2}(\overline{\varpi^{0}\cap\\{x_{1}<0}\\}\|\leq
C_{1}(1+|\psi|)\end{array}$ (6.23)

due to the factor $\psi$ on the right-hand sides of (6.15) and (6.18).

The boundary layer terms $\widetilde{w}^{\prime}_{\pm}$ and
$\widehat{w}^{\,\prime\prime}_{\pm}$ take the form

$\begin{array}[]{c}\displaystyle{\color[rgb]{0,0,0}\widetilde{w}^{\prime}_{\pm}(\xi;\psi)=\frac{\partial
U^{0}_{\pm}}{\partial x_{1}}(0;\psi)\widetilde{W}^{\,1}_{0}(\xi)=2\pi
i(a^{\pm}_{+}(\psi)-a^{\pm}_{-}(\psi))\widetilde{W}^{\,1}_{0}(\xi)}\end{array}$
(6.24)

and

$\begin{array}[]{c}\displaystyle\widehat{w}^{\,\prime\prime}_{\pm}(\xi;\psi)={\color[rgb]{0,0,0}-}\frac{\partial^{2}U^{0}_{\pm}}{\partial
x^{2}_{1}}(0;\psi)\widehat{W}^{\,3}(\xi)+\frac{1}{2}\sum\limits_{\tau=\pm}\frac{\partial
U^{\prime}_{\pm}}{\partial x_{1}}(\tau
0;\psi)\widetilde{W}^{\,1}_{0}(\xi)\end{array}$ (6.25)

where $\widetilde{W}_{0}^{\,1}$ is the exponentially decaying remainder in the
decomposition (3.24) while $\widehat{W}^{\,3}$ is a bounded part of the
solution (3.30) of the problem (3.29), (3.6), (3.7), that is,

$\widehat{W}^{\,3}(\xi)=W^{3}(\xi)+\frac{\xi_{1}^{2}}{2}-\sum\limits_{\tau=\pm}\tau\chi_{\tau}(\xi_{1})\frac{|\omega|}{2H}\xi_{1}.$
(6.26)

Recalling Proposition 3.3 and the relation (6.22), we write

$\|e^{\sigma|\xi_{1}|}{\color[rgb]{0,0,0}\widetilde{w}^{\,\prime}_{\pm}};H^{2}(\Xi)\|\leq
C_{3}\,\,\mbox{\rm with any}\,\,\sigma\in\Big{(}0,\frac{2\pi}{H}\Big{)}.$
(6.27)

Finally in (6.21), we fix $R^{\varepsilon}_{\pm}$ to get
$U^{\varepsilon}_{\pm}\in{\cal H}^{\varepsilon}(\eta)\setminus\\{0\\}$. First,
we take functions $R^{\varepsilon}_{\pm}\in H^{2}(\varpi^{0})$ such that they
have support in $[1/4,1/2]\times[0,H]$ and satisfy the boundary conditions

$\begin{split}&\frac{\partial R^{\varepsilon}_{\pm}}{\partial
x_{2}}(x_{1},0;\psi)=\frac{\partial R^{\varepsilon}_{\pm}}{\partial
x_{2}}(x_{1},H;\psi)=0,\,\,x_{1}\in\Big{(}-\frac{1}{2},\frac{1}{2}\Big{)},\\\
&R^{\varepsilon}_{\pm}\Big{(}\frac{1}{2},x_{2};\psi\Big{)}=\varepsilon^{-2}\big{(}e^{i\varepsilon\psi}-1-i\varepsilon\psi\big{)}U^{0}_{\pm}\Big{(}-\frac{1}{2};\psi\Big{)}-\varepsilon^{-1}\big{(}e^{i\varepsilon\psi}-1\big{)}U^{\prime}_{\pm}\Big{(}-\frac{1}{2};\psi\Big{)},\,\,x_{2}\in(0,H),\\\
&\frac{\partial R^{\varepsilon}_{\pm}}{\partial
x_{1}}\Big{(}\frac{1}{2},x_{2};\psi\Big{)}=\varepsilon^{-2}\big{(}e^{i\varepsilon\psi}-1-i\varepsilon\psi\big{)}\frac{\partial
U^{0}_{\pm}}{\partial
x_{1}}\Big{(}-\frac{1}{2};\psi\Big{)}-\varepsilon^{-1}\big{(}e^{i\varepsilon\psi}-1\big{)}\frac{\partial
U^{\prime}_{\pm}}{\partial
x_{1}}\Big{(}-\frac{1}{2};\psi\Big{)},\,\,x_{2}\in(0,H).\end{split}$ (6.28)

Applying the Taylor formula to $e^{i\varepsilon\psi}$, and taking into account
(6.22) and (6.23), we find a function $R_{\pm}^{\varepsilon}$ such that, in
addition to (6.28), satisfies

$\begin{array}[]{c}\displaystyle\|R^{\varepsilon}_{\pm};H^{2}(\varpi^{0})\|\leq
C_{2}|\psi|(1+|\psi|).\end{array}$ (6.29)

Owing to the relations (6.28), the approximate eigenfunction (6.21) meets the
quasi-periodicity conditions (2.3), (2.4) with $\eta=\varepsilon\psi$. Thus,
$U_{\pm}^{\varepsilon}$ falls into ${\cal
H}^{\varepsilon}(\eta)\setminus\\{0\\}$.

Note that the function (6.21) satisfies the Neumann boundary condition on the
lateral sides $x_{2}=0$ and $x_{2}=H$ of the periodicity cell because of
(6.28) for $R^{\varepsilon}_{\pm}$ and (3.32) for (6.24) and (6.25). At the
boundaries of the holes $\varpi^{\varepsilon}(0,k)$ with $k=0,\dots,N-1$, by
definition (5.11) of the cut-off functions, (6.24), (6.25) and (3.3), we
obtain for $x\in\partial\varpi^{\varepsilon}(0,k)$:

$\begin{split}\partial_{\nu(x)}U^{\varepsilon}_{\pm}(x;\psi)=&\bigg{(}\frac{\partial
U^{0}_{\pm}}{\partial
x_{1}}(0;\psi)+\frac{\varepsilon}{2}\sum\limits_{\pm}\frac{\partial
U^{\prime}_{\pm}}{\partial x_{1}}(\tau
0;\psi)\\!\bigg{)}\Big{(}\nu_{1}(\xi)+\partial_{\nu(\xi)}\widetilde{W}^{1}_{0}(\xi)\Big{)}\\\
&+\varepsilon\frac{\partial^{2}U^{0}_{\pm}}{\partial
x^{2}_{1}}(0;\psi)\Big{(}\nu_{1}(\xi)\xi_{1}-\partial_{\nu(\xi)}\widehat{W}^{3}(\xi)\\!\Big{)}.\end{split}$

Now, by (3.21) and (3.24), we have that
$\partial_{\nu(\xi)}\widetilde{W}^{1}_{0}(\xi)=-\nu_{1}(\xi)$. Moreover, since
$\widehat{W}^{3}$ is defined by (6.26) with $W^{3}$ satisfying (3.7), we get
$\partial_{\nu(\xi)}\widehat{W}^{3}(\xi)=\xi_{1}\nu_{1}(\xi)$. Therefore, for
$x\in\partial\varpi^{\varepsilon}(0,k)$, we get
$\partial_{\nu(x)}U^{\varepsilon}_{\pm}(x;\psi)=0$.

### 6.3 Estimating the discrepancy

We take the value (6.20) and the function (6.21) to be the almost eigenvalue
and eigenfunction respectively and follow the analysis of Section 5.3. To make
it easier the analysis, we also keep the same notations.

Let us proceed to apply Lemma 5.3. Considering (6.20), we have

$\begin{split}\delta_{\pm}^{\varepsilon}(\psi):\\!&\\!=\|U^{\varepsilon}_{\pm};{{\mathcal{H}}}^{\varepsilon}(\varepsilon\psi)\|^{-1}\|{{\mathcal{B}}}^{\varepsilon}(\varepsilon\psi)U^{\varepsilon}_{\pm}-M^{1,\varepsilon}_{\pm}(\psi)U^{\varepsilon}_{\pm};{{\mathcal{H}}}^{\varepsilon}(\varepsilon\psi)\|\\\
\\!&\\!=\|U^{\varepsilon}_{\pm};{{\mathcal{H}}}^{\varepsilon}(\varepsilon\psi)\|^{-1}M^{1,\varepsilon}_{\pm}(\psi)\,\sup|(\Delta_{x}U^{\varepsilon}_{\pm}\\!+\\!(4\pi^{2}\\!+\\!\varepsilon\Lambda^{\prime}_{\pm}(\psi))U^{\varepsilon}_{\pm},V^{\varepsilon})_{\varpi^{\varepsilon}}|.\end{split}$
(6.30)

The supreme is computed over all function $V^{\varepsilon}\in{\cal
H}^{\varepsilon}(\varepsilon\psi)$ with unit norm and this calculation takes
into account definitions (5.3), (5.4) and the Green formula together with the
Neumann and quasi-periodicity conditions for $U^{\varepsilon}_{\pm}$. For any
fixed $\psi_{0}>0$, let us show the estimate

$\delta_{\pm}^{\varepsilon}(\psi)\leq
c(\psi_{0})\varepsilon^{2}\qquad\mbox{for
}|\psi|<\psi_{0},\,\varepsilon\leq\varepsilon_{0}$ (6.31)

with $c(\psi_{0})$ and $\varepsilon_{0}=\varepsilon_{0}(\psi_{0})$ some
constants independent of $\psi$ but they depend on $\psi_{0}.$

Indeed, we write

$\Delta_{x}U^{\varepsilon}_{\pm}(x;\psi)+\big{(}4\pi^{2}+\varepsilon\Lambda^{\prime}_{\pm}(\psi)\big{)}U^{\varepsilon}_{\pm}(x;\psi)=:\sum\limits_{j=1}^{10}S^{\varepsilon}_{j,\pm}(x;\psi),$
(6.32)

where

$\displaystyle
S^{\varepsilon}_{1,\pm}(x;\psi)=X^{\varepsilon}(x_{1})(\Delta_{x}U^{0}_{\pm}(x_{1};\psi)+4\pi^{2}U^{0}_{\pm}(x_{1};\psi)),$
$\displaystyle S^{\varepsilon}_{2,\pm}(x;\psi)=\varepsilon
X^{\varepsilon}(x_{1})\big{(}\Delta_{x}U^{\prime}_{\pm}(x_{1};\psi)+4\pi^{2}U^{\prime}_{\pm}(x_{1};\psi)+\Lambda^{\prime}_{\pm}(\psi)U^{0}_{\pm}(x_{1};\psi)\big{)},$
$\displaystyle
S^{\varepsilon}_{3,\pm}(x;\psi)=\varepsilon^{2}\left(X^{\varepsilon}(x_{1})\Lambda^{\prime}_{\pm}(\psi)U^{\prime}_{\pm}(x_{1};\psi)+\big{(}\Delta_{x}+4\pi^{2}+\varepsilon\Lambda^{\prime}_{\pm}(\psi)\big{)}R^{\varepsilon}_{\pm}(x;\psi)\right),$
$\displaystyle
S^{\varepsilon}_{4,\pm}(x;\psi)=\big{(}4\pi^{2}+\varepsilon\Lambda^{\prime}_{\pm}(\psi)\big{)}(1-X^{\varepsilon}(x_{1}))\left(U^{0}_{\pm}(0;\psi)+x_{1}\frac{\partial
U^{0}_{\pm}}{\partial
x_{1}}(0;\psi)+\frac{x_{1}^{2}}{2}\frac{\partial^{2}U^{0}_{\pm}}{\partial
x_{1}^{2}}(0;\psi)\right)$ $\displaystyle\hskip
71.13188pt+(1-X^{\varepsilon}(x_{1}))\frac{\partial^{2}U^{0}_{\pm}}{\partial
x_{1}^{2}}(0;\psi),$ $\displaystyle
S^{\varepsilon}_{5,\pm}(x;\psi)=\big{(}4\pi^{2}+\varepsilon\Lambda^{\prime}_{\pm}(\psi)\big{)}(1-X^{\varepsilon}(x_{1}))\frac{\varepsilon}{2}\sum\limits_{\tau=\pm}\Big{(}U^{\prime}_{\pm}(\tau
0;\psi)+x_{1}\frac{\partial U^{\prime}_{\pm}}{\partial x_{1}}(\tau
0;\psi)\Big{)},$ $\displaystyle
S^{\varepsilon}_{6,\pm}(x;\psi)=[\Delta_{x},X^{\varepsilon}(x_{1})]\bigg{(}U^{0}_{\pm}(x_{1};\psi)-U^{0}_{\pm}(0;\psi)-x_{1}\frac{\partial
U^{0}_{\pm}}{\partial
x_{1}}(0;\psi)-\frac{x_{1}^{2}}{2}\frac{\partial^{2}U^{0}_{\pm}}{\partial
x_{1}^{2}}(0;\psi)\bigg{)},$ $\displaystyle
S^{\varepsilon}_{7,\pm}(x;\psi)=\varepsilon[\Delta_{x},X^{\varepsilon}(x_{1})]\bigg{(}U^{\prime}_{\pm}(x_{1};\psi)-\frac{1}{2}\sum\limits_{\tau=\pm}\bigg{(}U^{\prime}_{\pm}(\tau
0;\psi)+x_{1}\frac{\partial U^{\prime}_{\pm}}{\partial x_{1}}(\tau
0;\psi)\bigg{)}\bigg{)},$ $\displaystyle
S^{\varepsilon}_{8,\pm}(x;\psi)=\chi_{0}(x_{1})\Big{(}\varepsilon^{-1}\Delta_{\xi}\widetilde{w}^{\prime}_{\pm}(\xi;\psi)+\Delta_{\xi}\widehat{w}^{\,\prime\prime}_{\pm}(\xi;\psi)\Big{)},$
$\displaystyle