# Bloch Spectra for High Contrast Elastic Media

Robert Lipton Department of Mathematics, Louisiana State University, Baton
Rouge, LA 70803, USA, https://orcid.org/0000-0002-1382-3204<EMAIL_ADDRESS>Ruchira Perera *Corresponding Author, Department of Mathematics, Lockett Hall
343, Louisiana State University, Baton Rouge, LA 70803, USA,
https://orcid.org/0000-0001-9621-7359<EMAIL_ADDRESS>
###### Abstract

Analytic representation formulas and power series are developed to describe
the band structure inside periodic elastic crystals made from high contrast
inclusions. We use source free modes associated with structural spectra to
represent the solution operator of the Lamé system inside phononic crystals.
Convergent power series for the Bloch wave spectrum are obtained using the
representation formulas. An explicit bound on the convergence radius is given
through the structural spectra of the inclusion array and the Dirichlet
spectra of the inclusions. Sufficient conditions for the separation of
spectral branches of the dispersion relation for any fixed quasi-momentum are
identified. A condition is found that is sufficient for the emergence of band
gaps.

Keywords: Phononics, Bloch waves, Spectrum, High contrast, Bragg scattering.

##  1 Introduction

High contrast periodic elastic crystals have been studied both theoretically
and experimentally and have been shown to exhibit unique dispersive
properties. One can split high contrast crystals into two classes based on the
length scale of the crystal structure relative to the wavelength. When crystal
geometry is on the same length scale as the elastic wave the dispersion is due
to Bragg scattering and the patterned material is referred to as a phononic
crystal, [16, 36, 24, 37, 39, 2, 27]. Alternatively if the wave length lies
above the crystal period a sub-wavelength resonance can be induced and this
becomes the principal effect that controls wave dispersion. Crystals of this
type are referred to as phononic metamaterials, [3, 33, 38, 10, 40]. In this
article we address the former “multiple scattering,” problem when the wave
length is on the scale of the heterogeneities. We consider periodic arrays
with low wave velocity inclusions embedded in a high wave velocity medium
(often referred to as the matrix). Such crystals exhibit novel dispersion and
are known to exhibit band gaps [17].

In this article new rigorous and explicit analytic representation formulas and
power series are developed to describe wave dispersion inside phononic
crystals. These results apply to wave propagation inside phononic crystals
made from high contrast inclusions. The explicit formulas are used to
investigate the propagation band structure of the crystal as a function of the
inclusion geometry. The phononic elastic crystal is a composite of two
materials each with different density and elasticity. The propagation of a
Bloch wave $h(x)$ at frequency $\omega$ inside the elastic crystal is
described by the differential equation,

$-\nabla\cdot(\mathbf{C}(x)\mathcal{E}h(x))=\omega^{2}\rho(x)h(x),\;x\in\mathbb{R}^{d},\;d=2,3.$
(1.1)

Here $\mathbf{C}(x),x\in\mathbb{R}^{d}$, is the fourth rank tensor that
represents the local elastic constants of the material and $\rho(x)$ is the
local density. The crystal is taken to be infinite in extent and with out loss
of generality the unit period cell is the cube $Y=(0,1]^{d}$. The Bloch wave
$h(x)$ inside the crystal satisfies the $\alpha$ quasi-periodicity condition
$h(x+p)=h(x)e^{i\alpha\cdot p}$, where $\alpha$ is the quasi-momentum in the
first Brillouin zone $Y^{*}=(-\pi,\pi]^{d}$. The piece wise constant elastic
tensor and density are periodic and satisfy $\mathbf{C}(x)=\mathbf{C}(x+p)$
and $\rho(x)=\rho(x+p)$ with $p\in\mathbb{Z}^{d},d=2,3$. The crystal is
composed of a periodic array of isolated inclusions $D$ surrounded by a second
phase. The array of inclusions is described by the set
$\Omega=\cup_{m\in\mathbb{Z}^{d}}(D+m)$, and the connected phase is described
by $\mathbb{R}^{d}\setminus\Omega$. In this treatment the boundary of the
inclusion is taken to be $C^{\infty}$ smooth. The low velocity inclusions are
embedded in a high velocity matrix, i.e., $\rho^{1}>\rho^{2}$ and $k>1$. The
piece wise constant density and the piece wise constant elasticity tensor for
the medium are written

$\displaystyle\mathbf{C}(x)$
$\displaystyle=\mathbf{C}^{1}\chi_{\Omega}(x)+\mathbf{C}^{2}(1-\chi_{\Omega}(x))$
(1.2) $\displaystyle\rho(x)$
$\displaystyle=\rho^{1}\chi_{\Omega}(x)+\rho^{2}(1-\chi_{\Omega}(x))$

where $\rho^{1}$, $\rho^{2}$ are constant densities and the elasticity tensor
$\mathbf{C}^{1}:=C_{ijkl}:=\lambda_{1}\delta_{ij}\delta_{kl}+\mu_{1}(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$
is isotropic and specified by Lamé constants $(\lambda_{1},\mu_{1})$ and
$\mathbf{C}^{1}$ satisfies

$0<\gamma|\zeta|^{2}\leq\mathbf{C}^{1}\zeta:\zeta\leq\beta|\zeta|^{2},$ (1.3)

for all $\zeta\in Sym^{d}=\\{\zeta\in\mathbb{R}^{d\times
d}:\zeta=\zeta^{T}\\}$. The elastic moduli describing $\mathbf{C}^{2}$ are
given by $\lambda_{2}=k\lambda_{1}$ and $\mu_{2}=k\mu_{1}$, $1\leq k<\infty$,
where $k$ represents the contrast between the two elastic materials. The
symmetric gradient of the elastic displacement $u$ is denoted by
$\mathcal{E}(u)$, given by,

$\mathcal{E}(u)=\frac{1}{2}(\nabla u+\nabla u^{t}),$

where the superscript $t$ denotes the matrix transpose. The corresponding co-
normal derivative on $\partial\Omega$ is

$\partial_{n}u:=(\mathbf{C}^{1}\mathcal{E}(u))n$ (1.4)

where $n$ is the outward unit normal vector to $\partial\Omega$ and the Lamé
operator $\mathcal{L}$ on $\mathbb{R}^{d}$, $d=2,3$ is defined to be

$\mathcal{L}u:=\nabla\cdot\mathbf{C}^{1}\mathcal{E}(u)=\mu_{1}\Delta
u+(\lambda_{1}+\mu_{1})\nabla(\nabla\cdot u).$ (1.5)

Here we prove the results for the $d=3$ case. Our approach also applies to the
$d=2$ case, however the specifics differ and this will be reported in a
separate publication.

In this paper we investigate the band structure as a function of the elastic
contrast $k$ between the two materials, inclusion shape and placement inside
the period cell. It is known that frequency band gaps open up for elastic
crystals for sufficiently high contrast, see [17]. For each $\alpha\in Y^{*}$
the Bloch eigenvalues $\omega^{2}$ are of finite multiplicity and denoted by
$\xi_{j}(k,\alpha),\;j\in\mathbb{N}.$ We develop explicit series expansions in
the contrast $k$ for each branch of the dispersion relation

$\xi_{j}(k,\alpha)=\omega^{2},\;j\in\mathbb{N}$ (1.6)

that are valid for $k$ in a neighborhood of infinity. The radii of convergence
and convergence rate for the series are found to depend explicitly on the
inclusion shape and placement within the period cell, see sections 7 and 9.
Conditions sufficient for the separation of spectral branches of the
dispersion relation for any fixed quasi-momentum are found, see section 7. We
characterize the high contrast limit of the Bloch spectra and give sufficient
conditions for the emergence of band gaps, see section 10. When the inclusion
is symmetric a new spectral interlacing property is found that is identical to
that seen for scalar problems in acoustics, see [18]. The approach taken here
is distinct from other approaches and as noted earlier is not asymptotic,
instead it uses shape and configurational information contained in the
structural spectra of the periodic array of inclusions. The structural spectra
is identified here for the elastic problem and is a family of eigenvalues
$\\{\tau_{i}(\alpha)\\}_{i=1}^{\infty}$, $\alpha\in Y^{\ast}$ associated with
eigenvalue problems that encode the geometry of the crystal, see Definition
2.8. For fixed $\alpha\in Y^{\ast}$ the eigenvalues
$\\{\tau_{i}(\alpha)\\}_{i=1}^{\infty}$, are referred to as the quasi-periodic
spectra of the crystal, see (2.39) of section 2. We identify this spectrum
with the spectrum of the well known Neumann Poincaré operator [19], [4], [5]
constructed in the quasi periodic setting, see Lemma 2.5.

$Y\setminus D$$D$ Figure 1: Inclusion geometry inside a period Cell.

To proceed we complexify the problem and consider $k\in\mathbb{C}$. Now
$\mathbf{C}(x)$ takes on complex values inside $Y\setminus D$ and divergence
form operator is no longer uniformly elliptic. Our approach develops an
explicit representation formula for
$-\nabla\cdot(\mathbf{C}(x)\mathcal{E}(u))$ that holds for complex values of
$k$. We identify the subset $z=\frac{1}{k}\in\Omega_{0}$ of $\mathbb{C}$ where
this operator is invertible. The explicit formula shows that the solution
operator $(-\nabla\cdot(\mathbf{C}^{1}\chi_{D}+\mathbf{C}^{2}\chi_{Y\setminus
D}))\mathcal{E})^{-1}$ may be regarded more generally as a meromorhic operator
valued function of $z$ for $z\in\Omega_{0}=\mathbb{C}\setminus S^{\alpha},$
see section 4 and Lemma 4.1 . Here the set $S^{\alpha}$ consists of poles
lying on the negative real axis with three accumulation points bounded away
from $z=0$. These poles are in one to one relation to with the structural
spectra $\\{\tau_{i}(\alpha)\\}_{i=1}^{\infty}$, $\alpha\in Y^{\ast}$. The
interval of the negative real axis containing $S^{\alpha}$ can be bounded
uniformly for $\alpha\in Y^{\ast}$ for a wide class of inclusion shapes and
placements inside the unit period cell, see Sections 8 and 9. For the problem
treated here we expand about $z=0$ and the set $S^{\alpha}$ is used to bound
the radius of convergence for the power series. The spectral representation
for $-\nabla\cdot(\mathbf{C}^{1}\chi_{D}+\mathbf{C}^{2}\chi_{Y\setminus
D}))\mathcal{E}$ follows from the existence of a complete set of orthonormal
set of quasi-periodic functions associated with the quasi-periodic resonances
of the crystal, i.e., quasi periodic functions $v$ and real eigenvalues $\xi$
for which

$-\nabla\cdot(\mathbf{C}^{1}\chi_{D})\mathcal{E}(v)=-\xi\nabla\cdot\mathbf{C}^{1}\mathcal{E}(v).$
(1.7)

These resonances are shown to be connected to the spectra of elastostatic
Neumann-Poincaré operators associated with quasi periodic double layer
potentials. For $\alpha=0$ and for a single sphere in $\mathbb{R}^{3}$ these
correspond to the elastostatic eigenvalues identified in [15]. Both
elastostatic Neumann-Poincaré (N-P) operators and associated elastosatic
resonances have been the focus of theoretical investigations [4]. Unlike
scalar problems [19], these investigations have shown that N-P operator is not
compact even for smooth domains. Instead the seminal work [4] shows that the
N-P operator is polynomially compact. These results have been applied in
analysis of cloaking by anomalous localized resonance for the elastostatic
system [5]. The explicit spectral representation for the operator
$(-\nabla\cdot(\mathbf{C}^{1}\chi_{D}+\mathbf{C}^{2}\chi_{Y\setminus
D})\mathcal{E})$ developed here is crucial for elucidating the interaction
between the contrast $k$ and the quasi-periodic resonances of the crystal, see
(3.2), (2.54), and (2.55). The spectral representation is applied to
analytically continue the band structure
$\xi_{j}(k,\alpha)=\omega^{2},\;j\in\mathbb{N},\;\alpha\in Y^{*}$ for $k$ onto
$\mathbb{C},$ see Theorem 3.1. Application of the contour integral formula for
spectral projections [21], [22] and [23] delivers an analytic representation
formula for the band structure, see section 4. We apply perturbation theory in
section 4 together with a calculation provided in section 13 to find explicit
formula for the radii of convergence for the power series $\xi_{j}(k,\alpha)$
about $1/k=0$. The formula shows that the radius of convergence and separation
between different branches of the dispersion relation are determined by: 1)
the distance of the origin to the nearest pole $z^{*}$ of
$(-\nabla\cdot(\mathbf{C}^{1}\chi_{D}+\mathbf{C}^{2}\chi_{Y\setminus
D}))\mathcal{E})^{-1},$ and 2) the separation between distinct eigenvalues in
the $z=1/k\rightarrow 0$ limit, see Theorems 7.1 and 7.2. These theorems
provide conditions on the contrast guaranteeing the separation of spectral
bands that depend explicitly upon $z^{*},j\in\mathbb{N}$ and $\alpha\in
Y^{*}.$ Error estimates for series truncated after $N$ terms follows directly
from the formulation.

Next we apply these results and develop bounds on the convergence radii for a
wide class of inclusions called buffered geometries. A buffered geometry is
described by a randomly placed inclusion inside the unit period cell with a
finite distance of separation between inclusion and cell boundary, see section
8. For these geometries we demonstrate that the poles of
$(-\nabla\cdot(\mathbf{C}^{1}\chi_{D}+\mathbf{C}^{2}\chi_{Y\setminus
D}))\mathcal{E})^{-1}$ associated with the quasi-periodic spectra are bounded
away from the origin uniformly for $\alpha\in Y^{*}$. The quasi-periodic
spectra $\\{\tau_{i}(\alpha)\\}_{i\in\mathbb{N}}$ associated with a buffered
geometry is shown to lie inside the interval
$-1/2<\tau^{-}\leq\tau_{i}(\alpha)\leq 1/2,$ for every $\alpha\in Y^{*},$ see
Theorem 8.1 and Corollary 8.3. The lower bound $\tau^{-}$ is independent of
$\alpha\in Y^{*}$ and depends explicitly on the geometry of the inclusions.
This control insures that the associated poles of
$(-\nabla\cdot(\mathbf{C}^{1}\chi_{D}+\mathbf{C}^{2}\chi_{Y\setminus
D}))\mathcal{E})^{-1}$ are uniformly bounded away from the origin and provides
an explicit nonzero radius of convergence for the power series representation
for the band structure $\xi_{j}(k,\alpha)=\omega^{2}$ for each
$j\in\mathbb{N}$ and $\alpha\in Y^{*}$, see Theorems 7.1 and 7.2. In section 9
we apply these observations to periodic assemblages of buffered spheres. Here
a buffered sphere is characterized by a period containing a randomly placed
sphere within the interior of the unit call. The term buffer referrers to the
distance between the boundary of the sphere to the boundary of the cell. For
this case we recover explicit formulas for the radii of convergence of the
power series expansion for $\xi_{j}(k,\alpha)$ and explicit conditions for the
separation of spectral bands in terms of the distance between sphere boundary
and cell boundary. It is important to emphasize that the results on separation
of spectra and convergence of power series are not asymptotic results but are
valid for an explicitly delineated regime of finite contrast.

Earlier work on effective properties for periodic and random media [13], [20],
[30], show that the effective elasticity for a composite medium is an analytic
function of the contrast. The effective elasticity function is seen to be
nonzero and analytic off the negative real axis and is determined by its
singularities and zeros. Estimates for effective properties are obtained from
partial knowledge of the singularities and zeros. The work [6] develops power
series solutions to bound the poles and zeros of the effective elasticity
function. This provides bounds on the effective elasticity function for the
class of inclusion geometries discussed here. Asymptotic expansions for Bloch
eigenvalues are developed and applied to the high contrast setting for two
dimensional elasticity in [2]. The expansions are in terms of the contrast and
developed using a boundary integral perturbation approach based on the
generalized Rouché’s theorem. In that work the high contrast band structure is
identified and a criterion for band gap opening is given in 2 dimensions. The
criterion sufficient for band gap opening in three dimensions given here is
consistent with the one presented in [1]. However in the present context it is
shown to follow from the Lipschitz continuity of Bloch eigenvalues with
respect to quasi-momentum at fixed contrast. Recently the appropriate
structural spectrum for the Helmholtz operator has been identified and used to
quantitatively capture the photonic band structure for wave propagation
problems in high contrast media for TE electromagnetic modes in [28]. This
knowledge is used to establish explicit formulas for both photonic pass band
and band gap frequency intervals as functions of the inclusion geometry in
[29].

Last we point out that earlier related work using different methods provide
explicit power series representations for the spectra of two dimensional
photonic metamaterials. The work of [34] develops a convergent power series
representation for the spectra of metamaterial crystals made from high
contrast frequency dependent rods. Explicit of radii of convergence are
obtained. The work of [35] demonstrates the existence of convergent power
series expansions for the spectrum of metamaterials containing high contrast
positive or negative dielectric inclusions. It is shown that the power series
converge for sufficiently small contrast. The work of [11] provides a power
series representation of spectra for metamaterials made from periodic
configurations containing both high contrast dielectric rods and frequency
dependent dielectric rods. The existence of traveling waves with phase
velocity opposite to the direction of the Poynting vector is rigorously shown
to follow from Maxwell’s equations.

The paper is organized as follows: In the next section we introduce the
Hilbert space formulation of the problem and the variational formulation of
the quasi-static resonance problem. The completeness of the eigenfunctions
associated with the quasi-static spectrum is established and a spectral
representation for the operator
$(-\nabla\cdot(\mathbf{C}^{1}\chi_{D}+\mathbf{C}^{2}\chi_{Y\setminus
D}))\mathcal{E})$ is obtained. These results are collected and used to
continue the frequency band structure into the complex plane, see Theorem 3.1
section 3. Spectral perturbation theory [23] is applied to recover the power
series expansion for Bloch spectra in section 2.11. The leading order spectral
theory is developed for quasi-periodic $\alpha\neq 0$ and periodic $\alpha=0$
problems in sections 5 and 6. The main theorems on radius of convergence and
convergence rates are given by Theorems 7.1, 7.2, and 7.4 are presented in
section 7. The class of buffered inclusions is introduced in section 8 and the
explicit radii of convergence for a random suspension of disks is presented in
section 9. The structure of high contrast limit spectra is given in section
10. Explicit formulas for each term of the power series expansion is recovered
and expressed in terms of layer potentials in section 11. In section 12 the
explicit formula for the first order correction in the power series is
presented in the form of the Dirichlet energy of the solution of a
transmission boundary value problem. This formula follows from the layer
potential representation for the first term and consistent with the first
order correction obtained in the work of [2] for two dimensions. The explicit
formulas for the convergence radii are derived in section 13 as well as hands
on proofs of Theorems 7.1, 7.2 and the error estimates for the series
approximation.

##  2 Hilbert space setting, quasi-periodic resonances and representation
formulas

We denote the space of all $\alpha$ quasi-periodic complex vector valued
functions belonging to $L^{2}_{loc}(\mathbb{R}^{3})^{3}$ by
$L^{2}_{\\#}(\alpha,Y)^{3}$ and the inner product is denoted by

$(u,v)=\int_{Y}u\cdot\overline{v}\;dx.$ (2.1)

For $\alpha\neq 0$ the eigenfunctions $h$ for (1.1) belong to the space

$H^{1}_{\\#}(\alpha,Y)^{3}=\\{h\in{H^{1}_{loc}}(\mathbb{R}^{3})^{3}:h\;\text{is}\;\alpha\;\text{quasi-
periodic}\\}.$ (2.2)

This space does not contain the space of rigid motions hence the kernel of the
symmetric gradient is zero and the space $H^{1}_{\\#}(\alpha,Y)^{3}$ is a
Hilbert space under the inner product

$\langle
u,v\rangle=\int_{Y}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(v)}\;dx.$
(2.3)

The periodic eigenfunctions of $\eqref{I1}$ associated with nonzero
eigenvalues belong to the space

$H^{1}_{\\#}(0,Y)^{3}=\left\\{h\in{H^{1}_{loc}}(\mathbb{R}^{3})^{3}:h\;\text{is}\;\text{periodic},\;\int_{Y}\rho\,h\,dx=0\right\\}.$
(2.4)

Note here that $H^{1}_{\\#}(0,Y)^{3}$ does not contain the space of rigid
motions so it is also a Hilbert space with the inner product $\langle
u,v\rangle$ defined by (2.3).

In what follows we write $\rho=\rho(x)$ and for any $k\in\mathbb{C}$, the weak
formulation of the eigenvalue problem (1.1) for $h$ and $\omega^{2}$ is given
by

$B_{k}(u,v)=\omega^{2}(\rho u,v)\quad\text{for all}\quad v\in
H^{1}_{\\#}(\alpha,Y)^{3}$ (2.5)

where $B_{k}:H^{1}_{\\#}(\alpha,Y)^{3}\times
H^{1}_{\\#}(\alpha,Y)^{3}\mapsto\mathbb{C}$ is the sesquilinear form given by

$\displaystyle B_{k}(u,v)$
$\displaystyle=\int_{Y}{\mathbf{C}(x)\mathcal{E}(u):\overline{\mathcal{E}(v)}\;dx}$
(2.6) $\displaystyle=k\int_{Y\setminus
D}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(v)}\;dx+\int_{D}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(v)}\;dx.$

Let $T_{k}^{\alpha}:H^{1}_{\\#}(\alpha,Y)^{3}\mapsto
H^{1}_{\\#}(\alpha,Y)^{3}$ be the associated linear operator such that

$\langle T_{k}^{\alpha}u,v\rangle=B_{k}(u,v)\;\text{for all}\;v\in
H^{1}_{\\#}(\alpha,Y)^{3}.$ (2.7)

Hence from (2.5) the eigenvalue problem becomes finding the pair
$\omega^{2},u$ such that $\omega^{2}>0$ and $u\in H^{1}_{\\#}(\alpha,Y)^{3}$
for which

$\langle T_{k}^{\alpha}u,v\rangle=\omega^{2}(\rho u,v).$ (2.8)

Now we identify the operator associated with this eigenvalue problem. Let
$F(v):H^{1}_{\\#}(\alpha,Y)^{3}\mapsto\mathbb{C}$ be the linear functional
such that $F(v)=(\rho u,v)$ for fixed $u\in H^{1}_{\\#}(\alpha,Y)^{3}$. Then
by the Riesz Representation Theorem, there is a unique $z_{\rho u}\in
H^{1}_{\\#}(\alpha,Y)^{3}$ such that

$\langle z_{u\rho},v\rangle=F(v)=(\rho u,v)\;\text{for all}\;v\in
H^{1}_{\\#}(\alpha,Y)^{3}.$ (2.9)

Let $-\mathcal{L}_{\alpha}$ be the Lamé operator associated with the bilinear
form $\langle\cdot,\cdot\rangle$ defined on $H^{1}_{\\#}(\alpha,Y)^{3}.$ so

$z_{\rho u}=-\mathcal{L}_{\alpha}^{-1}\rho u,$

and

$\langle T_{k}^{\alpha}u,v\rangle=\omega^{2}\langle z_{\rho
u},v\rangle=\omega^{2}\langle-\mathcal{L}_{\alpha}^{-1}\rho
u,v\rangle\;\text{for all}\;v\in H^{1}_{\\#}(\alpha,Y)^{3},$

or equivalently

$T_{k}^{\alpha}u=-\omega^{2}\mathcal{L}_{\alpha}^{-1}\rho u\;\;\text{as
elements of}\;H^{1}_{\\#}(\alpha,Y)^{3}.$

Now we aim to find the $k$ values for a given $\alpha$ such that
$(T_{k}^{\alpha})^{-1}$ exists and write

$\frac{1}{\omega^{2}}u=(T_{k}^{\alpha})^{-1}(-\mathcal{L}_{\alpha})^{-1}\rho
u.$ (2.10)

This is equivalent to solving the original eigenvalue problem (1.1) on writing

$-\nabla\cdot(\mathbf{C}(x)\mathcal{E}u)=-\mathcal{L}_{\alpha}T_{k}^{\alpha}u$
(2.11)

and noting

$\mathcal{L}_{\alpha}T_{k}^{\alpha}u=\omega^{2}\rho u\;\text{as elements
of}\;L^{2}_{\\#}(\alpha,Y)^{3}.$ (2.12)

Thus we will consider the operator
$(T_{k}^{\alpha})^{-1}(-\mathcal{L}_{\alpha})^{-1}\rho$ of $\eqref{a-1}$ and
show that for a given subset of $k$ in $\mathbb{C}$ it is a bounded operator
from $L^{2}_{\\#}(\alpha,Y)^{3}$ to $H^{1}_{\\#}(\alpha,Y)^{3}$. In order to
accomplish this we express $(T_{k}^{\alpha})$ explicitly and discern the
values $k$ in $\mathbb{C}$ for which $(T_{k}^{\alpha})$ is invertible.

We start by decomposing $H^{1}_{\\#}(\alpha,Y)^{3}$ into invariant subspaces
of source free modes associated with a quasi-periodic resonance spectra. This
decomposition provides the explicit spectral representation for the operator
$(T_{k}^{\alpha})$, see Theorem 2.9. Consider the quasi-periodic case given by
$\alpha\in Y^{*}\setminus\\{0\\}$. Set $W^{\alpha}_{1}=\\{u\in
H^{1}_{\\#}(\alpha,Y)^{3}:\mathcal{E}(u)=0\;\text{in}\;D\\}$ and
$W^{\alpha}_{2}=\\{u\in
H^{1}_{\\#}(\alpha,Y)^{3}:\mathcal{E}(u)=0\;\text{in}\;Y\setminus D\\}$. One
checks that these spaces are orthogonal in the $\langle\cdot,\cdot\rangle$
inner product. We define $W^{\alpha}_{3}:=(W^{\alpha}_{1}\oplus
W^{\alpha}_{2})^{\perp}$ and

$H^{1}_{\\#}(\alpha,Y)^{3}=W^{\alpha}_{1}\oplus W^{\alpha}_{2}\oplus
W^{\alpha}_{3}.$ (2.13)

Now consider $\alpha=0$ and decompose $H^{1}_{\\#}(0,Y)^{3}$. Set
$W^{0}_{1}=\\{u\in H^{1}_{\\#}(0,Y)^{3}:\mathcal{E}(u)=0\;\text{in}\;D\\}$ and
$W^{0}_{2}=\\{u\in
H^{1}_{\\#}(0,Y)^{3}:\mathcal{E}(u)=0\;\text{in}\;Y\setminus D\\}$. One checks
that these spaces are orthogonal in the $\langle\cdot,\cdot\rangle$ inner
product. Set $\langle\rho\rangle=\int_{Y}\,\rho\,dx$, then one also has the
equivalent representation of $W^{0}_{2}$ given by

######  Lemma 2.1.

The subspace $W^{0}_{2}$ of $H^{1}_{\\#}(0,Y)^{3}$ has the representation

$W^{0}_{2}=\\{u=\tilde{u}-\langle\rho\rangle^{-1}\int_{D}\rho^{1}\,\tilde{u}\,dx\;1_{Y}\;|\;\tilde{u}\in\tilde{H}^{1}_{0}(D)^{3}\\}$
(2.14)

where $\tilde{H}^{1}_{0}(D)^{3}$ is the subspace of $H_{0}^{1}(Y)^{3}$ given
by all $H^{1}_{0}(D)^{3}$ functions extended by zero into $Y\setminus D$ and
and $1_{Y}$ is the indicator function of $Y$.

This Lemma is proved in the Appendix. Clearly $W^{0}_{1}$ and $W^{0}_{2}$ are
orthogonal subspaces of $H^{1}_{\\#}(0,Y)^{3}$ and define
$W^{0}_{3}:=(W^{0}_{1}\oplus W^{0}_{2})^{\perp}$ and

$H^{1}_{\\#}(0,Y)^{3}=W^{0}_{1}\oplus W^{0}_{2}\oplus W^{0}_{3}.$ (2.15)

With these definitions in hand we write $W^{\alpha}_{1}$, $W^{\alpha}_{2}$,
$W^{\alpha}_{3}$ for all $\alpha\in Y^{\ast}$. To set up the spectral analysis
we observe that orthogonality and integration by parts shows that for $u\in
W^{\alpha}_{3}$,

$\mathcal{L}u=0$ (2.16)

separately in $D$ and $Y\setminus D$, for all $\alpha\in Y^{\ast}$ and this
implies that elements of $W^{\alpha}_{3}$ can be represented in terms of
single layer potentials supported on $\partial D$. We introduce the
3-dimensional $\alpha$-quasi-periodic Green’s function

$\mathbf{G}^{\alpha}(x,y)=\frac{1}{\mu_{1}}\sum_{n\in\mathbb{Z}^{3}}e^{i(2\pi
n+\alpha)\cdot(x-y)}\Big{(}\frac{-\delta_{ij}}{|2\pi
n+\alpha|^{2}}+\frac{\lambda_{1}+\mu_{1}}{\lambda_{1}+2\mu_{1}}\frac{(2\pi
n+\alpha)_{i}(2\pi n+\alpha)_{j}}{|2\pi n+\alpha|^{4}}\Big{)},$ (2.17)

and for $\alpha=0$ the periodic Green’s function by

$\mathbf{G}^{0}(x,y)=\frac{1}{\mu_{1}}\sum_{n\in\mathbb{Z}^{3}\setminus\\{0\\}}e^{i(2\pi
n)\cdot(x-y)}\Big{(}\frac{-\delta_{ij}}{|2\pi
n|^{2}}+\frac{\lambda_{1}+\mu_{1}}{\lambda_{1}+2\mu_{1}}\frac{4\pi^{2}n_{i}n_{j}}{|2\pi
n|^{4}}\Big{)}\;\text{for}\;\alpha=0.$ (2.18)

Let $H^{1/2}(\partial D)^{3}$ be the fractional Sobolev space on $\partial D$
with dual $(H^{1/2}(\partial D)^{3})^{*}=H^{-1/2}(\partial D)^{3}$. For
$\phi\in H^{-1/2}(\partial D)^{3}$, and $\alpha\in Y^{*}$ define the single
layer potential $\mathcal{S}^{\alpha}_{D}[\phi](x)$ associated with the Lamé
system

$\mathcal{S}^{\alpha}_{D}[\phi](x):=\int_{\partial
D}\mathbf{G}^{\alpha}(x,y)\phi(y)\;ds(y)\>\>,x\in Y.$ (2.19)

It follows from [1], for any $\phi\in H^{-1/2}(\partial D)^{3}$

$\displaystyle\mathcal{L}\mathcal{S}^{\alpha}_{D}\phi$
$\displaystyle=0\;\text{in}\;D\;\text{and}\;Y\setminus D,$ (2.20)
$\displaystyle\mathcal{S}^{\alpha}_{D}\phi|_{\partial D}^{-}$
$\displaystyle=\mathcal{S}^{\alpha}_{D}\phi|_{\partial D}^{+},$
$\displaystyle\frac{\partial}{\partial_{\nu}}\mathcal{S}^{\alpha}_{D}\phi|_{\partial
D}^{\pm}$
$\displaystyle=(\pm\frac{1}{2}I+\mathcal{(\tilde{K}^{-\alpha})^{*}})[\phi],$

where $\nu$ is the outward unit normal to $\partial D$ and
$(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}$ is the Neumann Poincaré operator
defined by

$(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}[\phi](x)=p.v\int_{\partial
D}\partial_{\nu_{x}}\mathbf{G}^{\alpha}(x-y)\phi(y)\;ds(y),\;\;x\in\partial D$
(2.21)

where $\tilde{\mathcal{K}}^{\alpha}_{D}$ is the Neumann Poincaré operator

$\tilde{\mathcal{K}}^{\alpha}_{D}[\phi](x)=p.v\int_{\partial
D}\partial_{\nu_{y}}\mathbf{G}^{\alpha}(x-y)\phi(y)\;ds(y),\;\;x\in\partial
D.$ (2.22)

Define $\mathcal{S}^{\alpha}_{\partial
D}\phi=\mathcal{S}^{\alpha}_{D}\phi|_{\partial D}$ for all $\phi\in
H^{-1/2}(\partial D)^{3}$, then we have the following Lemma.

######  Lemma 2.2.

$\mathcal{S}^{\alpha}_{\partial D}:H^{-1/2}(\partial D)^{3}\mapsto
H^{1/2}(\partial D)^{3}$ is invertible.

###### Proof.

We show that $Ker\\{\mathcal{S}^{\alpha}_{\partial D}\\}=\\{0\\}$. Let
$\phi\in H^{-1/2}(\partial D)^{3}$ and suppose $\mathcal{S}^{\alpha}_{\partial
D}\phi=0$. Set $u=\mathcal{S}^{\alpha}_{D}\phi$. Then $u\in W^{\alpha}_{3}$,
and satisfy $\mathcal{L}u=0,\;\text{in}\;D$ with $u|_{\partial D^{-}}=0$.
Hence we have $u=0$ in $D$. Likewise since
$\mathcal{L}u=0,\;\text{in}\;Y\setminus D$ and $u|_{\partial D^{+}}=0$ with
quasi-periodic boundary conditions on $\partial Y$, we conclude $u=0$ in
$Y\setminus D$. Then since $\phi=\partial_{n}u|_{\partial
D^{+}}-\partial_{n}u|_{\partial D^{-}}$, it yields $\phi=0$. To show the
surjectivity of $\mathcal{S}^{\alpha}_{\partial D}$, let $g\in
H^{1/2}(\partial D)^{3}$. Since the trace map $G:W^{\alpha}_{3}\mapsto
H^{1/2}(\partial D)^{3}$ is onto, there exists $u\in W^{\alpha}_{3}$ such that
$Gu=g$. Define $\phi_{u}:=\partial_{n}u|_{\partial
D^{+}}-\partial_{n}u|_{\partial D^{-}}$ and
$w(x):=\mathcal{S}^{\alpha}_{D}[\phi_{u}](x)=\int_{\partial
D}\mathbf{G}^{\alpha}(x,y)\phi_{u}(y)\;ds(y)$. Thus $u,w\in W^{\alpha}_{3}$
and $l:=u-w$ satisfy

$\displaystyle\mathcal{L}l$
$\displaystyle=0\;\text{in}\;D\;\text{and}\;Y\setminus D,$ $\displaystyle
l|_{\partial D^{-}}$ $\displaystyle=l|_{\partial D^{+}},$ $\displaystyle
n\cdot\mathbf{C}^{1}\mathcal{E}(l)|_{\partial D^{-}}$
$\displaystyle=n\cdot\mathbf{C}^{1}\mathcal{E}(l)|_{\partial D^{+}}.$

Then we see that

$\int_{Y}\mathbf{C}^{1}\mathcal{E}(l):\mathcal{E}(l)\;dx=0.$

This implies that $l$ is a rigid body motion and therefore we conclude that
$l=0$. Hence $u=w$. ∎

Let $\mathbf{\Gamma}=(\Gamma_{j,k})_{j,k=1}^{3}$ denote the Kelvin matrix
associated with the fundamental solution of the Lamé operator and has the
following representation

$\Gamma_{j,k}(x)=-\frac{b_{1}\delta_{jk}}{4\pi|x|}-\frac{b_{2}}{4\pi}\frac{x_{j}x_{k}}{|x|^{3}}$
(2.23)

with

$b_{1}:=\frac{1}{2}\Big{(}\frac{1}{\mu_{1}}+\frac{1}{2\mu_{1}+\lambda_{1}}\Big{)}\;\text{and}\;\;b_{2}:=\frac{1}{2}\Big{(}\frac{1}{\mu_{1}}-\frac{1}{2\mu_{1}+\lambda_{1}}\Big{)}.$
(2.24)

We define $\mathbf{F}(x,y):=\mathbf{G}^{\alpha}(x,y)-\mathbf{\Gamma}(x,y)$.
One has the identities

$\mathcal{L}\mathbf{G}^{\alpha}(x,y)=\sum_{n\in\mathbb{Z}^{3}}\delta(x-y-n)I\;,\alpha\neq
0,$ (2.25)
$\mathcal{L}\mathbf{G}^{0}(x,y)=\sum_{n\in\mathbb{Z}^{3}}\delta(x-y-n)I-I.$
(2.26)

We also know that

$-\mathcal{L}\mathbf{\Gamma}(x,y)=\delta(x-y)I.$ (2.27)

Thus on $Y$ for $\alpha\neq 0$, we have

$\begin{cases}-\mathcal{L}\mathbf{F}(x,y)=0\;\text{for}\;x\;\text{in}\;Y\\\
\mathbf{F}(x,y)|_{\partial Y}\text{is continuous.}\end{cases}$ (2.28)

This shows that $\mathbf{F}$ satisfies the homogeneous Lamé equation on $Y$
and we form

$\partial_{\nu_{x}}\mathbf{G}^{\alpha}(x,y)=\partial_{\nu_{x}}\mathbf{\Gamma}(x,y)+\partial_{\nu_{x}}\mathbf{F}(x,y).$
(2.29)

So the Neumann Poincaré operator has the equivalent representation

$\displaystyle(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}[\phi](x)=p.v\int_{\partial
D}\partial_{\nu_{x}}\mathbf{G}^{\alpha}(x,y)\phi(y)\;ds(y)$
$\displaystyle=p.v\\{\int_{\partial
D}\partial_{\nu_{x}}\mathbf{\Gamma}{(x,y)}\phi(y)\;ds(y)$ (2.30)
$\displaystyle+\int_{\partial
D}\partial_{\nu_{x}}\mathbf{F}(x,y)\phi(y)\;ds(y)\\}.$

Recall the identity

$\partial_{\nu_{x}}\mathbf{\Gamma(x,y)}=k_{0}\mathbf{K}_{1}(x,y)+\mathbf{K}_{2}(x,y)$
(2.31)

where

$k_{0}=\frac{-\mu_{1}}{2(2\mu_{1}+\lambda_{1})},$ (2.32)

$\displaystyle\mathbf{K_{1}}(x,y)$
$\displaystyle=\frac{n_{x}(x-y)^{T}-(x-y)n_{x}^{T}}{2\pi|x-y|^{3}},$ (2.33)
$\displaystyle\mathbf{K_{2}}(x,y)$
$\displaystyle=\frac{\mu_{1}}{2\mu_{1}+\lambda_{1}}\frac{(x-y)\cdot
n_{y}}{4\pi|x-y|^{3}}I+\frac{2(\mu_{1}+\lambda_{1})}{2\mu_{1}+\lambda_{1}}\frac{(x-y)\cdot
n_{y}}{4\pi|x-y|^{5}}(x-y)(x-y)^{T}.$ (2.34)

Here $I$ is the $3\times 3$ identity matrix. Then by substituting this to
(2.29) yields

$\partial_{\nu_{x}}\mathbf{G}^{\alpha}(x,y)=k_{0}\mathbf{K}_{1}(x,y)+\mathbf{K}_{2}(x,y)+\partial_{\nu_{x}}\mathbf{F}.$
(2.35)

We now argue as in the fundamental paper [4]. Define

$\mathbf{T}[\phi](x)=p.v\int_{\partial
D}\mathbf{K}_{1}(x,y)\phi(y)\;ds(y),\;\;x\in\partial D.$ (2.36)

Since $\mathbf{K}_{2}$ and $\mathbf{F}$ satisfy the weakly singular conditions
$|\mathbf{K}_{2}(x,y)|\leq C|x-y|^{-1}$ and
$|\partial_{n(x)}\mathbf{F}(x,y)|\leq C|x-y|^{-1}$ the integral operators
$\int_{\partial D}\mathbf{K}_{2}(x,y)\phi(y)\;ds(y)$ and $\int_{\partial
D}\partial_{n(x)}\mathbf{F}_{2}(x,y)\phi(y)\;ds(y)$ are compact on
$H^{-1/2}(\partial D)^{3}$. Therefore we write

$(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}[\phi](x)=k_{0}\mathbf{T}[\phi]+\text{compact
operator}.$

so $(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}-k_{0}\mathbf{T}$ is compact on $Y$
as elements of $H^{-1/2}(\partial D)^{3}$ to conclude that

$p_{3}((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}):=((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*})^{3}-k_{0}^{2}(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*},$

is compact.

The spectrum of $(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}$ is denoted by
$\sigma((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*})$. We conclude using the
spectral mapping theorem that
$p_{3}(\sigma((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*})))=\sigma(p_{3}((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}))$.
Since $p_{3}((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*})$ is compact,
$p_{3}(\sigma((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}))$ consists of
eigenvalues (of finite multiplicities) converging to $0$. Let
$\\{r_{n}(\alpha)\\}$ be the set of eigenvalues of
$p_{3}((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*})$ and suppose
$\zeta\in\sigma((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*})$. Then for each $n$,
we have the equation

$\zeta^{3}-k_{0}^{2}\zeta=r_{n}(\alpha).$

Now set
$\gamma_{n}^{\pm}=\Big{[}\frac{1}{2}(-27r_{n}(\alpha)\pm\sqrt{729r^{2}_{n}+108k^{6}_{0}})\Big{]}^{1/3}$.
By solving this cubic polynomial we get three roots and denote them by
$\zeta_{n_{1}},\zeta_{n_{2}},\zeta_{n_{3}}$ where

$\displaystyle\zeta_{n_{1}}$
$\displaystyle=-\frac{1}{3}\gamma_{n}^{+}-\frac{1}{3}\gamma_{n}^{-}$ (2.37)
$\displaystyle\zeta_{n_{2}}$
$\displaystyle=\frac{1-i\sqrt{3}}{6}\gamma_{n}^{+}+\frac{1+i\sqrt{3}}{6}\gamma_{n}^{-}$
$\displaystyle\zeta_{n_{3}}$
$\displaystyle=\frac{1+i\sqrt{3}}{6}\gamma_{n}^{+}+\frac{1-i\sqrt{3}}{6}\gamma_{n}^{-}.$

Thus for each $n$ we get $3$ different eigenvalues and therefore we will have
$3$ different sequences of eigenvalues with accumulation points $0,k_{0}$ and
$-k_{0}$. We emphasize that while not indicated explicitly these eigenvalues
depend on $\alpha\in Y^{\ast}$. We summarize the results below.

######  Lemma 2.3.

The point spectra of $(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}$ is given by the
three sequences of eigenvalues given by (2.37) and the sequences converge to
the three accumulation points $0$, $k_{0}$, and $-k_{0}$.

Let $G:\,W^{\alpha}_{3}\rightarrow H^{-1/2}(\partial D)^{3}$ be the trace
operator which is bounded one to one and onto.

######  Lemma 2.4.

$\mathcal{S}^{\alpha}_{D}:H^{-1/2}(\partial D)^{3}\mapsto W^{\alpha}_{3}$ is
one to one and onto and moreover
$(\mathcal{S}^{\alpha}_{D})^{-1}=(\mathcal{S}^{\alpha})^{-1}_{\partial D}G$.

###### Proof.

Let $\phi\in H^{-1/2}(\partial D)^{3}$. As shown earlier
$\mathcal{S}^{\alpha}_{D}:H^{-1/2}(\partial D)^{3}\mapsto W^{\alpha}_{3}$ has
$Ker(\mathcal{S}^{\alpha}_{D})=0$ so $\mathcal{S}^{\alpha}_{D}$ is one to one
and $f=\mathcal{S}_{D}^{\alpha}\phi\in W^{\alpha}_{3}$ for every $\phi\in
H^{-1/2}(\partial D)^{3}$. Now suppose $u\in W^{\alpha}_{3}$, and consider
$Gu=u|_{\partial D}\in H^{1/2}(\partial D)^{3}$. Define
$w=\mathcal{S}^{\alpha}_{D}(\mathcal{S}^{\alpha})^{-1}_{\partial D}Gu)$. Since
$u,w\in W^{\alpha}_{3}$, we have $u-w\in W^{\alpha}_{3}$. Also $Gu=Gw$ and
hence $G(u-w)=0$, so $w-u\in(W^{\alpha}_{1}\oplus W^{\alpha}_{2})$. But
$W^{\alpha}_{3}=(W^{\alpha}_{1}\oplus W^{\alpha}_{2})^{\perp}$ and therefore
$w=u$. ∎

We define an auxiliary operator $T:W^{\alpha}_{3}\mapsto W^{\alpha}_{3}$ such
that

$\langle Tu,v\rangle=\frac{1}{2}\int_{Y\setminus
D}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(v)}\;dx-\frac{1}{2}\int_{D}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(v)}\;dx.$
(2.38)

If $\xi$ is an eigenvalue of $T$, then there exists $u\in W^{\alpha}_{3}$ such
that

$Tu=\xi u.$ (2.39)

From $\eqref{H27}$, it is clear that,

$\displaystyle-\frac{1}{2}\langle u,u\rangle$
$\displaystyle=-\frac{1}{2}\int_{{Y\setminus
D}}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(u)}\;dx-\frac{1}{2}\int_{D}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(u)}\;dx$
$\displaystyle\leq\langle Tu,u\rangle=\xi\langle u,u\rangle$
$\displaystyle\leq\frac{1}{2}\int_{{Y\setminus
D}}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(u)}\;dx+\frac{1}{2}\int_{D}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(u)}\;dx=\frac{1}{2}\langle
u,u\rangle.$

Thus for any eigenvalue $\xi$ of $T$, we have the following.

$-\frac{1}{2}\leq\xi\leq\frac{1}{2}.$

The upper bound $1/2$ is the eigenvalue associated with the eigenspace
$W_{1}^{\alpha}$.

The next theorem shows the relation between the auxiliary operator $T$
restricted to $W^{\alpha}_{3}$ and the elastic Neumann Poincaré operator.

######  Theorem 2.5.

For $u\in W^{\alpha}_{3}$ the operator $T$ is given by

$T=\mathcal{S}^{\alpha}_{D}(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}(\mathcal{S}_{D}^{\alpha})^{-1}$
(2.40)

and is self-adjoint.

###### Proof.

Let $u,v\in W^{\alpha}_{3}$. Consider

$\langle\mathcal{S}^{\alpha}_{D}(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}(\mathcal{S}_{D}^{\alpha})^{-1}u,v\rangle=\int_{Y}\mathbf{C}^{1}\mathcal{E}[\mathcal{S}^{\alpha}_{D}(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}(\mathcal{S}^{\alpha}_{D})^{-1}u]:\overline{\mathcal{E}(v)}\;dx$
(2.41) $\displaystyle=\int_{Y\setminus
D}\mathbf{C}^{1}\mathcal{E}[\mathcal{S}^{\alpha}_{D}(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}(\mathcal{S}_{D}^{\alpha})^{-1}u]:\overline{\mathcal{E}(v)}\;dx$
$\displaystyle+\int_{D}\mathbf{C}^{1}\mathcal{E}[\mathcal{S}^{\alpha}_{D}(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}(\mathcal{S}_{D}^{\alpha})^{-1}u]:\overline{\mathcal{E}(v)}\;dx.$

Since $\mathcal{L}\mathcal{S}^{\alpha}_{D}\phi=0$ in $D$ and $Y\setminus D$,
for any $\phi\in H^{-1/2}(\partial D)^{3}$, using integration by parts gives

$\langle\mathcal{S}^{\alpha}_{D}(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}(\mathcal{S}_{D})^{\alpha})^{-1}u,v\rangle=\int_{\partial
D}(n\cdot\mathbf{C}^{1}\mathcal{E}[\mathcal{S}^{\alpha}_{D}(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}(\mathcal{S}_{D}^{\alpha})^{-1}u]|_{\partial
D^{-}}-n\cdot\mathbf{C}^{1}\mathcal{E}[\mathcal{S}^{\alpha}_{D}(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}(\mathcal{S}_{D}^{\alpha})^{-1}u]|_{\partial
D^{+}})\overline{v}.$

Applying the jump conditions from $\eqref{H14}$ gives

$\langle\mathcal{S}^{\alpha}_{D}(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}(\mathcal{S}_{D}^{\alpha})^{-1}u,v\rangle=-\int_{\partial
D}(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}(\mathcal{S}_{D}^{\alpha})^{-1}u\overline{v}\;ds.$
(2.42)

The same jump conditions gives

$(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}[\phi]=\frac{1}{2}(\frac{\partial}{\partial_{\nu}}\mathcal{S^{\alpha}}_{D}\phi|_{\partial
D^{-}}+\frac{\partial}{\partial_{\nu}}\mathcal{S}^{\alpha}_{D}\phi|_{\partial
D^{+}})$

where $\phi=\mathcal{(}{S}^{\alpha}_{D})^{-1}u$ . Thus $\eqref{H30}$ yields

$\langle\mathcal{S}^{\alpha}_{D}(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}(\mathcal{S}_{D}^{\alpha})^{-1}u,v\rangle=-\int_{\partial
D}(\frac{1}{2}\frac{\partial}{\partial_{\nu}}u|_{\partial
D^{-}}+\frac{1}{2}\frac{\partial}{\partial_{\nu}}u|_{\partial
D^{+}})\overline{v}\;ds$ (2.43)
$\displaystyle=-\frac{1}{2}\int_{D}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(v)}\;dx-\frac{1}{2}(-\int_{Y\setminus
D}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(v)}\;dx)$
$\displaystyle=\langle Tu,v\rangle.$

∎

######  Lemma 2.6.

When restricted to $W^{\alpha}_{3}$ the point spectrum of $T$ is given by the
point spectrum of $(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}$ and the essential
spectrum of $T$ is given by the accumulation points $0$, $k_{0}$, and $-k_{0}$
of the eigenvalues of operator $(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}$.

###### Proof.

Since $T$ is self-adjoint, it has empty residual spectrum and
$\sigma(T)=\sigma_{p}(T)\cup\sigma_{ess}(T)$, see, e.g., [32]. We prove the
theorem by showing
$\sigma_{p}(T)=\sigma_{p}((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*})$ and
$\sigma_{ess}(T)=\\{-k_{0},0,k_{0}\\}$. To prove the claim on the point
spectrum, suppose $(\xi,u)\in(-1/2,1/2]\times W^{\alpha}_{3}$ satisfies
$Tu=\xi u.$ Then

$\displaystyle\mathcal{S^{\alpha}}_{D}(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}(\mathcal{S}_{D}^{\alpha})^{-1}u$
$\displaystyle=\xi u$
$\displaystyle(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}(\mathcal{S}_{D}^{\alpha})^{-1}u$
$\displaystyle=\xi(\mathcal{S}_{D}^{\alpha})^{-1}u.$

This shows that $(\mathcal{S}^{\alpha}_{D})^{-1}u$ is an eigenfunction for
$(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}$ associated the with eigenvalue
$\xi.$ On the other hand, if $(\xi,w)\in(-1/2,1/2]\times H^{-1/2}(\partial
D)^{3}$ satisfy $(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}w=\xi w$, then since
the trace map is onto, there is a $u\in W^{\alpha}_{3}$ such that
$w=(\mathcal{S}^{\alpha}_{D})^{-1}u$. Thus replacing
$(\mathcal{S}^{\alpha}_{D})^{-1}u$ for $w$ yields
$(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}(\mathcal{S}^{\alpha}_{D})^{-1}u=\xi(\mathcal{S}^{\alpha}_{D})^{-1}u$
and therefore we obtain
$\mathcal{S}^{\alpha}_{D}(\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}(\mathcal{S}_{D}^{\alpha})^{-1}u=\xi
u$. Which shows that $u$ is an eigenfunction of $T$ associated with the
eigenvalue $\xi.$

We now prove $\sigma_{ess}(T)=\\{-k_{0},0,k_{0}\\}$. Suppose
$\xi\in\\{-k_{0},0,k_{0}\\}$. Note first there exists a sequence
$\\{\xi_{n},\rho_{n}\\}\in\mathbb{R}\times H^{-1/2}(\partial D)^{3}$ such that
$\rho_{n}$ is a sequence of eigenvectors, $\|\rho_{n}\|=1$, and
$\xi_{n}\rightarrow\zeta$ for which

$|\xi_{n}-\xi|=\|((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}-\xi I)\rho_{n}\|.$

Now since $(\mathcal{S}_{D}^{\alpha})^{-1}$ is onto, we can find $u_{n}\in
W^{\alpha}_{3}$ such that $\rho_{n}=(\mathcal{S}_{D}^{\alpha})^{-1}u_{n}$.
Therefore

$\displaystyle\|(T-\xi I)u_{n}\|$
$\displaystyle=\|\mathcal{S}^{\alpha}_{D}((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}-\xi
I)(\mathcal{S}_{D}^{\alpha})^{-1}u_{n}\|$
$\displaystyle=\|\mathcal{S^{\alpha}}_{D}((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}-\xi
I)\rho_{n}\|$
$\displaystyle\leq\|\mathcal{S}^{\alpha}_{D}\|\|((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}-\xi
I)\rho_{n}\|$ $\displaystyle\leq M\|\xi_{n}-\xi|.$

Since $T$ is selfadjoint the eigenfunctions $\\{u_{n}\\}$ form an orthogonal
system and $\\{-k_{0},0,k_{0}\\}$ constitute the essential spectrum of $T$. ∎

From Lemma (2.6), we see that the eigenvalues of the elastic NP operator lie
in $(-1/2,1/2]$. The accumulation points of eigenvalues $k_{0},-k_{0},0$ also
lie in $(-1/2,1/2]$. To see that $k_{0},-k_{0}$ lie in $(-1/2,1/2]$, we use
the relation $K_{1}-\frac{2\mu_{1}}{3}=\lambda_{1}$ where $K_{1}>0$ is the
bulk modulus. Since

$\pm k_{0}=\mp\frac{\mu_{1}}{2(2\mu_{1}+\lambda_{1})},$

and we have

$\displaystyle|k_{0}|=\frac{\mu_{1}}{2(2\mu_{1}+\lambda_{1})}$
$\displaystyle=\frac{\mu_{1}}{4\mu_{1}+2\lambda_{1}}$
$\displaystyle=\frac{\mu_{1}}{4\mu_{1}+2(K_{1}-2\mu_{1}/3)}=\frac{\mu_{1}}{8\mu_{1}/3+2K}$
$\displaystyle\leq\frac{\mu_{1}}{8\mu_{1}/3}=\frac{3}{8}<\frac{1}{2}.$

We now derive an appropriate resolution of the identity on the space
$W^{\alpha}_{3}$ and an associated spectral representation formula for $T$. In
what follows we do not explicitly indicate dependence of eigenvalues on
$\alpha\in Y^{\ast}$ for ease of exposition. Let the three sequences of
eigenvalues for $T$ be denoted by $\\{\xi_{i}^{k}\\}_{i=1}^{\infty}$,
$k=1,2,3$, associated with the three accumulation points $\xi^{1}=-k_{0}$,
$\xi^{2}=0$, and $\xi^{3}=k_{0}$. The invariant subspace associated with each
eigenvalue is denoted by $E_{i}^{k}=\\{u\in
W^{\alpha}_{3}\,;Tu=\xi_{i}^{k}u\\}$ and the orthogonal projection onto this
subspace is denoted by $P_{i}^{k}$, here orthogonality is with respect to the
$\langle\cdot,\cdot\rangle$ inner product. We write
$E^{1}=\sum_{i=1}^{\infty}\oplus E_{i}^{1}$, $E^{2}=\sum_{i=1}^{\infty}\oplus
E_{i}^{2}$, $E^{3}=\sum_{i=1}^{\infty}\oplus E_{i}^{3}$,
$E^{4}=ker\\{\tilde{T}\\}$, and the projection operators onto the spaces are
$P^{k}$, $k=1,2,3$. We define

$\displaystyle\tilde{T}=\sum_{k=1}^{3}\sum_{i=1}^{\infty}(\xi_{i}^{k}-\xi^{k})P^{k}_{i},$
(2.44)

and $E^{4}=ker\\{\tilde{T}\\}$, with projection $P^{4}$. This operator is
selfadjoint and compact. Compactness follows as it is the limit of the rank
one operators

$\displaystyle\tilde{T}^{n}=\sum_{k=1}^{3}\sum_{i=1}^{n}(\xi_{i}^{k}-\xi^{k})P^{k}_{i}.$
(2.45)

One has the following decomposition of $T$ restricted to $W^{\alpha}_{3}$:

######  Lemma 2.7.

$\displaystyle{T}=\sum_{k=1}^{3}\sum_{i=1}^{\infty}\xi_{i}^{k}P^{k}_{i}.$
(2.46)

The identity on $W_{3}$ is given by

$\displaystyle I=\sum_{k=1}^{3}\sum_{i=1}^{\infty}P_{i}^{k}+P^{4}.$ (2.47)

###### Proof.

Since $\tilde{T}$ is compact and selfadjoint on $W^{\alpha}_{3}$, it follows
immediately from the theory that
$W^{\alpha}_{3}=(\sum_{k=1}^{3}\sum_{i=1}^{\infty}\oplus E_{i}^{k})\oplus
ker\\{\tilde{T}\\}$ and (2.47) follows. Writing any element of $u\in
W^{\alpha}_{3}$ as $u=\sum_{k=1}^{3}\sum_{i=1}^{\infty}P_{i}^{k}u+P^{4}u$
shows

$\|T-(-k_{0}P^{1}+k_{0}P^{3})-\tilde{T}^{n}\|\rightarrow 0,$ (2.48)

in the operator norm. Thus given $\epsilon>0$ we can find $N$ such that

$\|T-(-k_{0}P^{1}+k_{0}P^{3})-\tilde{T}\|\leq\|T-(-k_{0}P^{1}+k_{0}P^{3})-\tilde{T}^{n}\|+\|\tilde{T}^{n}-\tilde{T}\|\leq\epsilon,$
(2.49)

for all $n>N$ hence

$T-(-k_{0}P^{1}+k_{0}P^{3})=\tilde{T}.$ (2.50)

and the Lemma follows. ∎

To simplify the exposition we collect all eigenvalues of $T$ restricted to
$W^{\alpha}_{3}$ and denote them as the sequence
$\\{\tau_{n}(\alpha)\\}_{n=1}^{\infty}\in(-1/2,1/2)$ and we have the
definition

######  Definition 2.8.

The structural spectra of the crystal is defined as $\cup_{\alpha\in
Y^{\ast}}\\{\tau_{i}(\alpha)\\}_{i=1}^{\infty}$.

This spectra is independent of contrast encodes the geometry of the crystal.

For fixed $\alpha$ the projections onto their eigenspaces in $W_{3}^{\alpha}$
are denoted as $P_{\tau_{n}(\alpha)}$. We also write $\tau_{0}=0$ and the
projection onto $ker\\{T\\}$ as $P_{\tau_{0}}$. On writing the projections on
$W^{\alpha}_{1}$ and $W^{\alpha}_{2}$ as $P^{\alpha}_{1}$, $P^{\alpha}_{2}$
respectively, we arrive at the desired partition of unity for $u$, $v$ in
$H^{1}_{\\#}(\alpha,Y)^{3}=W^{\alpha}_{1}\oplus W^{\alpha}_{2}\oplus
W^{\alpha}_{3}$ given by

$\langle u,v\rangle=\langle
P^{\alpha}_{1}u+P^{\alpha}_{2}u+(\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}P_{\tau_{i}(\alpha)})u,v\rangle.$
(2.51)

The spectral decomposition for $T_{k}^{\alpha}$ associated with the
sesquilinear form is given by

######  Theorem 2.9.

The linear operator $T_{k}^{\alpha}:H^{1}_{\\#}(\alpha,Y)^{3}\mapsto
H^{1}_{\\#}(\alpha,Y)^{3}$ associated with the

sesquilinear form $B_{k}$ is given by

$\langle T_{k}^{\alpha}u,v\rangle=\langle
kP^{\alpha}_{1}u+P^{\alpha}_{2}u+\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}[k(1/2+\tau_{i}(\alpha))+(1/2-\tau_{i}(\alpha))]P_{\tau_{i}(\alpha)}u,v\rangle$

for all $u,v\in H^{1}_{\\#}(\alpha,Y)^{3}$.

###### Proof.

Recall that for all $u,v\in H^{1}_{\\#}(\alpha,Y)^{3}$

$\displaystyle B_{k}(u,v)$
$\displaystyle=\int_{Y}{\mathbf{C}(x)\mathcal{E}(u):\overline{\mathcal{E}(v)}\;dx}$
$\displaystyle=\int_{Y\setminus
D}\mathbf{C}^{2}\mathcal{E}(u):\overline{\mathcal{E}(v)}\;dx+\int_{D}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(v)}\;dx$
$\displaystyle=k\int_{Y\setminus
D}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(v)}\;dx+\int_{D}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(v)}\;dx.$

Let $u,v\in W_{3}$. Since $P_{\tau_{i}(\alpha)}u$ is the eigenvector
corresponding to eigenvalue $\tau_{i}(\alpha)$ of $T$, we have $\langle
T(P_{\tau_{i}(\alpha)}u),v\rangle=\tau_{i}(\alpha)\langle
P_{\tau_{i}(\alpha)}u,v\rangle$ or equivalently after manipulation

$\displaystyle\int_{Y\setminus
D}\mathbf{C}^{1}\mathcal{E}(P_{\tau_{i}(\alpha)}u):\overline{\mathcal{E}(v)}\;dx$
$\displaystyle=\frac{(\frac{1}{2}+\tau_{i}(\alpha))}{(\frac{1}{2}-\tau_{i}(\alpha))}\int_{D}\mathbf{C}^{1}\mathcal{E}(P_{\tau_{i}(\alpha)}u):\overline{\mathcal{E}(v)}\;dx.$

So we find that

$\displaystyle B_{k}(P_{\tau_{i}(\alpha)}u,v)$
$\displaystyle=[k\frac{(\frac{1}{2}+\tau_{i}(\alpha))}{(\frac{1}{2}-\tau_{i}(\alpha))}+1]\int_{D}\mathbf{C}^{1}\mathcal{E}(P_{\tau_{i}(\alpha)}u):\overline{\mathcal{E}(v)}\;dx.$

We also have

$\int_{D}\mathbf{C}^{1}\mathcal{E}(P_{\tau_{i}(\alpha)}u):\overline{\mathcal{E}(v)}\;dx=(1/2-\tau_{i}(\alpha))\int_{Y}\mathbf{C}^{1}\mathcal{E}(P_{\tau_{i}(\alpha)}u):\overline{\mathcal{E}(v)}\;dx.$

Therefore

$B_{k}(P_{\tau_{i}(\alpha)}u,v)=[k(1/2+\tau_{i}(\alpha))+(1/2-\tau_{i}(\alpha))]\int_{Y}\mathbf{C}^{1}\mathcal{E}(P_{\tau_{i}(\alpha)}u):\overline{\mathcal{E}(v)}\;dx.$

It is easily seen that

$\displaystyle B_{k}(P^{\alpha}_{1}u,v)$ $\displaystyle=k\int_{Y\setminus
D}\mathbf{C}^{1}\mathcal{E}(P^{\alpha}_{1}u):\overline{\mathcal{E}(v)}\;dx$
$\displaystyle B_{k}(P^{\alpha}_{2}u,v)$
$\displaystyle=k\int_{D}\mathbf{C}^{1}\mathcal{E}(P^{\alpha}_{2}u):\overline{\mathcal{E}(v)}\;dx,$

so

$\langle T_{k}^{\alpha}u,v\rangle=\langle
kP^{\alpha}_{1}u+P^{\alpha}_{2}u+\sum_{-\frac{1}{2}<\tau_{i}<\frac{1}{2}}[k(1/2+\tau_{i}(\alpha))+(1/2-\tau_{i}(\alpha))]P_{\tau_{i}(\alpha)}u,v\rangle.$

∎

It is clear that $T_{k}^{\alpha}:H^{1}_{\\#}(\alpha,Y)^{3}\mapsto
H^{1}_{\\#}(\alpha,Y)^{3}$ is invertible when

$k\in\mathbb{C}\setminus Z^{\alpha}\;\text{
where}\;Z^{\alpha}=\\{\frac{\tau_{i}(\alpha)-1/2}{\tau_{i}(\alpha)+1/2}\\}_{\\{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}\\}}.$
(2.52)

So for $z=k^{-1}$, we have

$(T_{k}^{\alpha})^{-1}=zP^{\alpha}_{1}u+P^{\alpha}_{2}u+\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}z[(1/2+\tau_{i}(\alpha))+z(1/2-\tau_{i}(\alpha))]P_{\tau_{i}(\alpha)}.$
(2.53)

For future reference we also introduce the set $S^{\alpha}$ of
$z\in\mathbb{C}$ for which $T_{k}^{\alpha}$ is not invertible given by

$S^{\alpha}=\\{\frac{\tau_{i}(\alpha)+1/2}{\tau_{i}(\alpha)-1/2}\\}_{\\{-1/2\leq\tau_{i}(\alpha)\leq
1/2\\}}$ (2.54)

which also lies on the negative real axis. Collecting results, the spectral
representation of the operator
$-\nabla\cdot(\mathbf{C}^{1}\chi_{D}(x)+\mathbf{C}^{2}\chi_{Y\setminus
D}(x))\mathcal{E}$ on $H^{1}_{\\#}(\alpha,Y)^{3}$ is given by

$-\nabla\cdot(\mathbf{C}^{1}\chi_{D}(x)+\mathbf{C}^{2}\chi_{Y\setminus
D}(x))\mathcal{E}=-\mathcal{L}_{\alpha}T^{\alpha}_{k},$ (2.55)

in the sense of linear functionals over the space $H^{1}_{\\#}(\alpha,Y)^{3}$
and recall that $-\mathcal{L}_{\alpha}$ is the Lamé operator associated with
the bilinear form $\langle\cdot,\cdot\rangle$ defined on
$H^{1}_{\\#}(\alpha,Y)^{3}.$ This formulation is useful since it separates the
effect of the contrast $k$ from the underlying geometry of the crystal. We
note for future use that $(-\mathcal{L}_{\alpha})^{-1}$ is given by

$(-\mathcal{L}_{\alpha})^{-1}u(x)=-\int_{Y}\mathbf{G}^{\alpha}(x,y)u(y)\;dy.$
(2.56)

##  3 Band Structure for Complex Coupling Constant

We set $\omega^{2}=\xi$ in $\eqref{I1}$. The operator representation is
applied to write the Bloch eigenvalue problem as

$\displaystyle-\nabla\cdot(\mathbf{C}^{1}\chi_{D}(x)+\mathbf{C}^{2}\chi_{Y\setminus
D}(x))\mathcal{E}(u)$
$\displaystyle=-\mathcal{L}_{\alpha}T^{\alpha}_{k}u=\xi\rho u$ (3.1)
$\displaystyle(T^{\alpha}_{k})^{-1}(-\mathcal{L}_{\alpha})^{-1}\rho u$
$\displaystyle=\frac{1}{\xi}u.$

We characterize the Bloch spectra by analysing the operator

$B^{\alpha}(k)=(T^{\alpha}_{k})^{-1}(-\mathcal{L}_{\alpha})^{-1},$ (3.2)

and the operator given by the product $B^{\alpha}(k)\rho$.

It is shown in Theorem 13.5 that the operator
$B^{\alpha}(k):L^{2}_{\\#}(\alpha,Y)^{3}\mapsto H^{1}_{\\#}(\alpha,Y)^{3}$ is
bounded for $k\notin Z^{\alpha}$. Thus the product
$B^{\alpha}(k)\rho:L^{2}_{\\#}(\alpha,Y)^{3}\mapsto H^{1}_{\\#}(\alpha,Y)^{3}$
is also bounded. It follows from the compact embedding of
$H^{1}_{\\#}(\alpha,Y)^{3}$ into $L^{2}_{\\#}(\alpha,Y)^{3}$ that
$B^{\alpha}(k)\rho$ is compact on $L^{2}_{\\#}(\alpha,Y)^{3}$ and therefore
has a discrete spectrum $\\{\gamma_{i}(k,\alpha)\\}_{i\in\mathbb{N}}$ with a
possible accumulation point at $0$. The corresponding eigenspaces are finite
dimensional and the eigenfunctions $p_{i}\in L^{2}_{\\#}(\alpha,Y)^{3}$
satisfy

$[B^{\alpha}(k)\rho]p_{i}(x)=\gamma_{i}(k,\alpha)p_{i}(x)\;\text{for}\;x\;\text{in}\;Y$
(3.3)

and also belong to $H^{1}_{\\#}(\alpha,Y)^{3}$. Note further for
$\gamma_{i}\neq 0$ that (3.3) holds if and only if (3.1) holds with
$\xi_{i}(k,\alpha)=\gamma_{i}^{-1}(k,\alpha)$, and
$-\mathcal{L}_{\alpha}T^{\alpha}_{k}p_{i}=\rho\xi_{i}(k,\alpha)p_{i}$.
Collecting results we have the following theorem

######  Theorem 3.1.

Let $Z^{\alpha}$ denote the set of points on the negative real axis defined by
(2.52). Then the Bloch eigenvalue problem (1.1) for the operator
$-\nabla\cdot(\mathbf{C}(x)\mathcal{E}h(x))$ associated with the sesquilinear
form (2.6) can be extended for values of the coupling constant $k$ off the
positive real axis into $\mathbb{C}\setminus Z^{\alpha}$, i.e., for each
$\alpha\in Y^{*}$ the Block eigenvalues are of finite multiplicity and denoted
by $\xi_{j}(k,\alpha)=\gamma_{j}^{-1}(k,\alpha),j\in\mathbb{N}$, and the band
structure

$\xi_{j}(k,\alpha)=\omega^{2},\;j\in\mathbb{N}$ (3.4)

extends to complex coupling constants $k\in\mathbb{C}\setminus Z^{\alpha}$.

##  4 Power series representation of Bloch eigenvalues for high contrast
periodic media

We set $\gamma=\xi^{-1}(k,\alpha)$ and analyse the spectral problem

$[B^{\alpha}(k)\rho]u=\gamma(k,\alpha)u.$ (4.1)

We analyse the high contrast limit by developing a power series in
$z=\frac{1}{k}$ about $z=0$ for the spectrum of the family of operators
associated with (4.1).

$\displaystyle B^{\alpha}(k)$
$\displaystyle:=(T_{k}^{\alpha})^{-1}(-\mathcal{L}_{\alpha})^{-1}$
$\displaystyle=(zP^{\alpha}_{1}u+P^{\alpha}_{2}u+\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}z[(1/2+\tau_{i}(\alpha))+z(1/2-\tau_{i}(\alpha))]P_{\tau_{i}(\alpha)})(-\mathcal{L}_{\alpha})^{-1}$
$\displaystyle=A^{\alpha}(z).$

We define the operator $A^{\alpha}(z)$ such that
$A^{\alpha}(1/k)=B^{\alpha}(k)$ and the associated eigenvalues
$\beta(1/k,\alpha)=\gamma(k,\alpha)$ and the spectral problem is
$[A^{\alpha}(z)\rho]u=\beta(z,\alpha)u$ for $u\in L^{2}_{\\#}(\alpha,Y)^{3}$.
From the above representation, it is easily seen that $A^{\alpha}(z)$ is self-
adjoint for $k\in\mathbb{R}$ and is a family of bounded operators taking
$L^{2}_{\\#}(\alpha,Y)^{3}$ into itself.

######  Lemma 4.1.

$A^{\alpha}(z)$ is holomorphic on $\Omega_{0}:=\mathbb{C}\setminus
S^{\alpha}$, where $S^{\alpha}=\cup_{i\in\mathbb{N}}z_{i}$ is the collection
of points $z_{i}=(1/2+\tau_{i}(\alpha))/(\tau_{i}(\alpha)-1/2)$ on the
negative real axis associated with the eigenvalues
$\\{\tau_{i}(\alpha)\\}_{i\in\mathbb{N}}$. The set $S^{\alpha}$ consists of
poles of $A^{\alpha}(z)$ with accumulation points
$z\in\\{-\frac{3\mu_{1}+\lambda_{1}}{\mu_{1}+\lambda_{1}},-1,-\frac{\mu_{1}+\lambda_{1}}{3\mu_{1}+\lambda_{1}}\\}$.

In section 8 we develop explicit $\alpha$ independent lower bounds
$-1/2<\tau^{-}\leq\tau^{-}(\alpha)=\min_{i}\\{\tau_{i}(\alpha)\\}$, that hold
for generic classes of inclusion domains $D$ and for every $\alpha\in Y^{*}$.
The corresponding upper bound $z^{+}$ on $S^{\alpha}$ is written

$\max_{i}\\{z_{i}\\}=\frac{\tau^{-}(\alpha)+1/2}{\tau^{-}(\alpha)-1/2}=z^{*}\leq
z^{+}<0.$ (4.2)

Let $\beta_{0}^{\alpha}\in\sigma(A^{\alpha}(0)\,\rho)$ with spectral
projection $P(0)$, and let $\Gamma$ be a closed contour in $\mathbb{C}$
enclosing $\beta_{0}^{\alpha}$ but no other element in
$\sigma(A^{\alpha}(0)\,\rho).$ The spectral projection associated with
$\beta^{\alpha}(z)\in\sigma(A^{\alpha}(z)\,\rho)$ for $\beta^{\alpha}(z)\in
int(\Gamma)$ is denoted by $P(z).$ We write
$M(z)=P(z)L^{2}_{\\#}(\alpha,Y)^{3}$ and suppose for the moment that $\Gamma$
lies in the resolvent of $A^{\alpha}(z)\,\rho$ and $\dim(M(0)=\dim(M(z))=m.$
Now define
$\hat{\beta}^{\alpha}(z)=\frac{1}{m}\operatorname{tr}(A^{\alpha}(z)\,\rho\,P(z)),$
the weighted mean of the eigenvalue group
$\\{\beta^{\alpha}_{1}(z),\dots\beta^{\alpha}_{m}(z)\\}$ corresponding to
$\beta^{\alpha}_{0}=\beta^{\alpha}_{1}(0)=\dots=\beta^{\alpha}_{m}(0).$ We
write the weighted mean as

$\hat{\beta}^{\alpha}(z)=\beta^{\alpha}_{0}+\frac{1}{m}\operatorname{tr}[(A^{\alpha}(z)\,\rho-\beta^{\alpha}_{0})P(z)].$
(4.3)

Since $A^{\alpha}(z)$ is analytic in a neighborhood of the origin we write

$A^{\alpha}(z)=A^{\alpha}(0)+\sum_{n=1}^{\infty}z^{n}A^{\alpha}_{n}.$ (4.4)

The explicit form of the sequence $\\{A^{\alpha}_{n}\\}_{n\in\mathbb{N}}$ will
be given later. Define the resolvent of the operator $A^{\alpha}(z)\,\rho$ by

$R(\zeta,z)=(A^{\alpha}(z)\,\rho-\zeta)^{-1},$

and expanding successively in Neumann series and power series as in [23] we
obtain the resolvent as power series in $z$ with coefficents depending on
$\zeta$ and $\rho$

$\displaystyle R(\zeta,z)$
$\displaystyle=R(\zeta,0)[I+(A^{\alpha}(z)\,\rho-A^{\alpha}(0)\rho)R(\zeta,0)]^{-1}$
(4.5)
$\displaystyle=R(\zeta,0)\sum_{p=0}^{\infty}[-(A^{\alpha}(z)\,\rho-A^{\alpha}(0)\,\rho)R(\zeta,0)]^{p}$
$\displaystyle=R(\zeta,0)+\sum_{n=1}^{\infty}z^{n}R_{n}(\zeta,\rho),$

where

$R_{n}(\zeta,\rho)=\sum_{\stackrel{{\scriptstyle
k_{1}+\dots+k_{p}=n}}{{k_{j}\geq
1}}}(-1)^{p}R(\zeta,0)\,A^{\alpha}_{k_{1}}\,\rho\,R(\zeta,0)A^{\alpha}_{k_{2}}\,\rho\,R(\zeta,0)\cdots
R(\zeta,0)A^{\alpha}_{k_{p}}\,\rho\,R(\zeta,0),$

where the sum is taken for all combinations of positive integers $p$ and
$\\{k_{1},\ldots,k_{p}\\}$ such that $1\leq p\leq n$, $k_{1}+\cdots+k_{p}=n$.
Application of the contour integral formula for spectral projections delivers
the spectral projection

$\displaystyle P(z)$ $\displaystyle=-\frac{1}{2\pi
i}\oint_{\Gamma}R(\zeta,z)\;d\zeta$ (4.6)
$\displaystyle=P(0)+\sum_{n=1}^{\infty}z^{n}P_{n}$

where $P_{n}=-\frac{1}{2\pi i}\oint_{\Gamma}R_{n}(\zeta,\rho)\;d\zeta$. Now we
develop the series for the weighted mean of the eigenvalue group associated
with an eigenvalue $\beta^{\alpha}_{0}$ of geometric multiplicity $m$. start
with

$(A^{\alpha}(z)\,\rho-\beta^{\alpha}_{0})R(\zeta,z)=I+(\zeta-\beta^{\alpha}_{0})R(\zeta,z)$
(4.7)

and we have

$(A^{\alpha}(z)\,\rho-\beta^{\alpha}_{0})P(z)=-\frac{1}{2\pi
i}\oint_{\Gamma}(\zeta-\beta^{\alpha}_{0})R(\zeta,z)\;d\zeta,$ (4.8)

so from (4.3)

$\hat{\beta}(z)-\beta^{\alpha}_{0}=-\frac{1}{2m\pi
i}\operatorname{tr}\oint_{\Gamma}(\zeta-\beta^{\alpha}_{0})R(\zeta,z)\;d\zeta.$
(4.9)

Manipulation and integration by parts as in [23], Chap. 2, Sec. 2.2 yields

$\hat{\beta}(z)=\beta^{\alpha}_{0}+\sum_{n=1}^{\infty}z^{n}\beta^{\alpha}_{n},$
(4.10)

where

$\beta^{\alpha}_{n}=-\frac{1}{2m\pi
i}\operatorname{tr}\sum_{\stackrel{{\scriptstyle
k_{1}+\dots+k_{p}=n}}{{k_{i}\geq
1}}}\frac{(-1)^{p}}{p}\oint_{\Gamma}A^{\alpha}_{k_{1}}\,\rho\,R(\zeta,0)A^{\alpha}_{k_{2}}\,\rho\,R(\zeta,0)\cdots
A^{\alpha}_{k_{p}}\,\rho\,R(\zeta,0)\;d\zeta,$ (4.11)

as before the sum is taken for all combinations of positive integers $p$ and
$\\{k_{1},\ldots,k_{p}\\}$ such that $1\leq p\leq n$, $k_{1}+\cdots+k_{p}=n$.

##  5 Spectrum in the high contrast limit: Quasi-periodic case

We now identify the limiting operator $A^{\alpha}(0)\,\rho$ when $\alpha\neq
0.$ Using the representation

$A^{\alpha}(z)=(zP^{\alpha}_{1}+P^{\alpha}_{2}+\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}z[(1/2+\tau_{i}(\alpha))+z(1/2-\tau_{i}(\alpha))]P_{\tau_{i}(\alpha)})(-\mathcal{L}_{\alpha})^{-1},$
(5.1)

we see that

$A^{\alpha}(0)\,\rho=P^{\alpha}_{2}(-\mathcal{L}_{\alpha})^{-1}\,\rho.$ (5.2)

Denote the spectrum of $A^{\alpha}(0)\rho$ by $\sigma(A^{\alpha}(0)\rho).$ The
following theorem provides the explicit characterization of
$\sigma(A^{\alpha}(0)\rho)$. Let $-\mathcal{L}_{D}$ be the Lamé operator
associated with the bilinear form $\langle\cdot,\cdot\rangle$ defined on
$H^{1}_{0}(D)^{3}$. Recall the density is piece wise constant taking the value
$\rho^{1}$ in $D$ and $\rho^{2}$ outside. Consider the Dirichlet eigenvalue
problem

$\mathcal{L}_{D}u=\rho^{1}\eta u\hbox{ for $\eta>0$ and $u\in
H^{1}_{0}(D)^{3}$}.$ (5.3)

The operator $-\mathcal{L}_{D}$ is invertable and the Dirichlet eigenvalues
are given by the reciprocals of the discrete spectrum of
$(-\mathcal{L}_{D})^{-1}\rho^{1}$.

######  Theorem 5.1.

$\sigma\left(A^{\alpha}(0)\rho\right)=\sigma\left((-\mathcal{L}_{D})^{-1}\rho^{1}\right).$

###### Proof.

First we show that the eigenvalue problem

$P^{\alpha}_{2}(-\mathcal{L}_{\alpha})^{-1}\rho^{1}u=\eta u$

with $\eta\in\sigma(A^{\alpha}(0)\rho)$ and eigenfunction $u\in
L^{2}_{\\#}(\alpha,Y)^{3}$ is equivalent to finding $\eta$ and $u\in
W^{\alpha}_{2}$ for which

$(\rho u,v)=\eta\langle u,v\rangle,\;\text{for all}\;v\in W^{\alpha}_{2}.$
(5.4)

To see (5.4), note that we have $u=P^{\alpha}_{2}u$ and for $v\in
H^{1}_{\\#}(\alpha,Y)^{3}$,

$\langle P^{\alpha}_{2}(-\mathcal{L}_{\alpha})^{-1}\rho u,v\rangle=\eta\langle
u,v\rangle=\eta\langle P^{\alpha}_{2}u,v\rangle$ (5.5)

hence

$\langle(-\mathcal{L}_{\alpha})^{-1}\rho u,P^{\alpha}_{2}v\rangle=\eta\langle
u,P^{\alpha}_{2}v\rangle.$ (5.6)

Since $\langle(-\mathcal{L}_{\alpha})^{-1}\rho u,v\rangle=\int_{Y}\rho
u\cdot\overline{v}\;dx=(\rho u,v)$ for any $u\in L^{2}_{\\#}(\alpha,Y)^{3})$
and $v\in H^{1}_{\\#}(\alpha,Y)^{3}$, equation (5.6) becomes

$(\rho u,P^{\alpha}_{2}v)=\eta\langle u,P^{\alpha}_{2}v\rangle,$ (5.7)

since $P^{\alpha}_{2}$ is the projection of $H^{1}_{\\#}(\alpha,Y)^{3}$ onto
$W^{\alpha}_{2}$, the equivalence follows.
We conclude by showing the set of eigenvalues for (5.4) is given by
$\sigma((-\mathcal{L}_{D})^{-1}\rho^{1})$. Let $\mathcal{R}$ be the space of
rigid motions on $Y\setminus D$ and note that the kernel of the symmetric
gradient on $Y\setminus D$ is $\mathcal{R}$. Define $\tilde{H}^{1}_{0}(D)^{3}$
to be the subspace of functions $H^{1}_{0}(D)^{3}$ extended by zero into
$Y\setminus D$. Since $W^{\alpha}_{2}\cap\mathcal{R}=0$ we see that
$W^{\alpha}_{2}=\tilde{H}^{1}_{0}(D)^{3}$. Since $P^{\alpha}_{2}v$ is
supported in $D$ (5.4) is

$\eta^{-1}\int_{D}\rho^{1}u\cdot\overline{P^{\alpha}_{2}v}=\int_{D}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(P^{\alpha}_{2}v)}\;dx.$
(5.8)

Now since $P^{\alpha}_{2}:H^{1}_{\\#}(\alpha,Y)^{3}\mapsto
W^{\alpha}_{2}=\tilde{H}^{1}_{0}(D)^{3}$ is onto, it follows that $\eta^{-1}$
is a eigenvalue of (5.3). ∎

##  6 Spectrum in the high contrast limit: Periodic case

For the periodic case, $P^{0}_{2}$ is the projection onto $W^{0}_{2}$. The
limiting operator is written

$A^{0}(0)\,\rho=P^{0}_{2}(-\mathcal{L}_{0})^{-1}\,\rho.$ (6.1)

Here the operator $(-\mathcal{L}_{0})^{-1}$ is compact and self-adjoint on
$L^{2}_{\\#}(0,Y)^{3}$ and given by

$(-\mathcal{L}_{0})^{-1}u(x)=-\int_{Y}\mathbf{G}^{0}(x,y)u(y)\;dy.$ (6.2)

Denote the spectrum of $A^{0}(0)\rho$ by $\sigma(A^{0}(0)\rho)$. To
characterize this spectrum we introduce the effective mass tensor

$M(\nu)=I\int_{Y}\rho(x)\;dx-\nu\sum_{j\in\mathbb{N}}\frac{\int_{D}\rho^{1}\overline{\psi}_{j}\;dx\otimes\int_{D}\rho^{1}\psi_{j}\;dx}{\nu-\delta^{*}_{j}},$
(6.3)

where $I$ is the $3\times 3$ identity and
$\\{\delta^{*}_{j}\\}_{j\in\mathbb{N}}$ are the Dirichlet eigenvalues of
$(-\mathcal{L}_{D})u=\rho^{1}\eta u$, $u\in H_{0}^{1}(D)^{3}$ associated with
eigenfunctions $\psi_{j}$ for which $\int_{D}\rho^{1}\psi_{j}\;dx\neq 0$.

######  Remark 6.1.

The effective mass tensor $M(\nu)$ is precisely the effective mass tensor of
the high contrast elastic metamaterial [40] and [10] associated with a sub-
wavelength periodic lattice. Matrices of a similar type corresponding to the
effective magnetic permeability tensor for photonic metamaterials with
artificial magnetism are identified in [8], [9], and also appear in the
homogenization theory of high contrast porous media [41].

Next we introduce the sequence of numbers $\\{\nu_{j}\\}_{j\in\mathbb{N}}$
given by the positive roots $\nu$ of the determinant of the effective mass
matrix

$det\left\\{M(\nu)\right\\}=0$ (6.4)

The following theorem provides the explicit characterization of
$\sigma(A^{0}(0)\rho)$.

######  Theorem 6.2.

Let $\\{(\delta^{\prime-1}_{j}\\}_{j\in\mathbb{N}}$ denote the collection of
eigenvalues for $(-\mathcal{L}_{D})^{-1}\rho^{1}$ associated with
eigenfunctions $\psi_{j}$ of (5.2) for which $\int_{D}\rho^{1}\psi_{j}=0$.
Then
$\sigma(A^{0}(0)\rho)=\\{\delta^{\prime-1}_{j}\\}_{j\in\mathbb{N}}\cup\\{\nu^{-1}_{j}\\}_{j\in\mathbb{N}}$.

###### Proof.

We argue as in the previous section to find that the eigenvalue problem

$P^{0}_{2}(-\mathcal{L}_{0})^{-1}\rho=\eta u$

with $\eta\in\sigma(A^{0}(0)\rho)$ and eigenfunction $u\in
L^{2}_{\\#}(0,Y)^{3}$ is equivalent to finding $\eta$ and $u\in W^{0}_{2}$ for
which

$(\rho u,v)=\eta\langle u,v\rangle,\;\text{for all}\;v\in W^{0}_{2}.$ (6.5)

We show that the eigenvalues $\eta_{j}$ for (6.5) are given by the
alternative:

$\eta_{j}=(\delta^{\prime}_{j})^{-1},\hbox{ or }\eta_{j}=(\nu_{j})^{-1}.$
(6.6)

From (2.14) we have the dichotomy: $\int_{D}\tilde{u}\;dx=0$ and
$u=\tilde{u}\in\tilde{H}^{1}_{0}(D)^{3}$ or $\int_{D}\tilde{u}\;dx\neq 0$ and
$u=\tilde{u}-\gamma 1_{Y}$ with
$\gamma=\langle\rho\rangle^{-1}\int_{D}\rho^{1}\tilde{u}\;dx$. For the first
case that the eigenfunction belongs to $\tilde{H}^{1}_{0}(D)^{3}$ and for
$v\in W^{0}_{2}$ given by

$v=\tilde{v}-\langle\rho\rangle^{-1}\big{(}\int_{D}\rho^{1}\tilde{v}\;dx\big{)}1_{Y}\;\text{for}\;\tilde{v}\in\tilde{H}^{1}_{0}(D)^{3}$
(6.7)

the problem (6.5) becomes

$\int_{D}\rho^{1}u\cdot\overline{\tilde{v}}=\eta\int_{D}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(\tilde{v})}\;dx,\;\text{for
all}\;\tilde{v}\in\tilde{H}^{1}_{0}(D)^{3},$ (6.8)

and we conclude that $\tilde{u}$ is a Dirichlet eigenfunction with zero
average over $D$ so $\eta\in\\{\delta^{\prime-1}_{j}\\}_{j\in\mathbb{N}}$. For
the second case, we have $u\in W^{0}_{2}$ and again

$\int_{D}\rho^{1}u\cdot\overline{\tilde{v}}=\eta\int_{D}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(\tilde{v})}\;dx,\;\text{for
all}\;\tilde{v}\in\tilde{H}^{1}_{0}(D)^{3}.$ (6.9)

Writing $u=\tilde{u}-\gamma 1_{Y}$ and integration by parts in (6.9) shows
that $\tilde{u}\in\tilde{H}^{1}_{0}(D)^{3}$ is the solution of

$\mathcal{L}_{D}\tilde{u}+\nu\rho^{1}\tilde{u}=\nu\rho^{1}\gamma\;\text{for}\;x\in
D.$ (6.10)

Since $\tilde{u}\in\tilde{H}^{1}_{0}(D)^{3}$ we can write

$\tilde{u}=\sum_{j=1}^{\infty}c_{j}\psi_{j}$ (6.11)

where, $\psi_{j}$ are the Dirichlet eigenfunctions of (5.3) associated with
eigenvalue $\delta_{j}$ extended by zero to $Y$. Substitution of (6.11) into
(6.10) yields

$\sum_{j=1}^{\infty}(-\delta_{j}\rho^{1}+\nu\rho^{1})c_{j}\psi_{j}=\nu\rho^{1}\gamma.$
(6.12)

Multiplying both sides of (6.12) by $\overline{\psi}_{k}$ over $D$ and
$\int_{D}\rho^{1}\psi_{i}\cdot\overline{\psi}_{j}dx=\delta_{ij}$ shows that
$\tilde{u}$ is given by

$\tilde{u}=\nu\sum_{k\in\mathbb{N}}\frac{\gamma\cdot\,\int_{D}\rho^{1}\overline{\psi}_{k}}{\nu-\delta^{*}_{k}}\psi_{k},$
(6.13)

where $\delta^{*}_{k}$ correspond to Dirichlet eigenvalues associated with
eigenfunctions for which $\int_{D}\rho^{1}\psi_{k}\;dx\neq 0$. Hence

${u}=\nu\sum_{k\in\mathbb{N}}\frac{\gamma\cdot\,\int_{D}\rho^{1}\overline{\psi}_{k}}{\nu-\delta^{*}_{k}}\psi_{k}-\gamma.$
(6.14)

To find $\nu$, we multiply both sides of (6.14) by $\rho(x)$ and integrate
both sides over $Y$ to recover the identity

$\operatorname{det}\left\\{M(\nu)\right\\}=0.$ (6.15)

Hence we conclude that $\eta\in\\{\nu^{-1}_{i}\\}$ and the proof is complete.
∎

We conclude with a varational characterization of the eigenvalues. Let
$\\{\delta_{j}\\}_{j\in\mathbb{N}}$ denote all the Dirichlet eigenvalues of
$(-\mathcal{L}_{D})u=\rho^{1}\delta u$, $u\in H^{1}_{0}(D)^{3}$, i.e.,
$\\{\delta_{j}\\}_{j\in\mathbb{N}}=\\{\delta^{{}^{\prime}}_{j}\\}_{j\in\mathbb{N}}\cup\\{\delta_{j}^{\ast}\\}_{j\in\mathbb{N}}$.
Then one readily obtains the min-max characterizations of $\eta^{-1}_{j}$ and
$\delta_{j}$ given by

######  Lemma 6.3.

$\delta_{j}=\min_{S^{j}\subseteq H^{1}_{0}(Y)^{3}}\Big{\\{}\max_{0\neq u\in
S^{j}}\frac{\int_{Y}\mathbf{C}^{1}\mathcal{E}(u):\mathcal{E}(\overline{u})dx}{(\rho
u,u)}\Big{\\}}$ (6.16) ${\eta^{-1}_{j}}=\min_{S^{j}\subseteq
H_{0}^{1}(D)^{3}}\Big{\\{}\max_{0\neq u\in
S^{j}}\frac{\int_{Y}\mathbf{C}^{1}\mathcal{E}(u):\mathcal{E}(\overline{u})dx}{(\rho
u,u)-\langle\rho\rangle^{-1}|\int_{Y}\rho u|^{2}}\Big{\\}}$ (6.17)

###### Proof.

The min-max formulation for Dirichlet eigenvalues (6.16) is standard [23]. The
second identity follows from the standard min-max formulation on noting that
for $u\in W_{2}^{0}$ that
$u=\tilde{u}-{\langle\rho\rangle^{-1}}\int_{Y}\rho\tilde{u}\,dx$ for
$\tilde{u}\in H^{1}_{0}(D)^{3}$ and $0\leq(\rho
u,u)=(\rho\tilde{u},\tilde{u})-\langle\rho\rangle^{-1}|\int_{Y}\rho\tilde{u}|^{2}$,
when $\int_{Y}\rho\tilde{u}\,dx\not=0$. ∎

##  7 Radius of convergence and convergence rates

Fix an inclusion geometry specified by the domain $D$. Suppose that $\alpha\in
Y^{*}$ and $\alpha\neq 0$. Recall from Theorem 5.1 that the spectrum of
$A^{\alpha}(0)\rho$ is $\sigma((-\mathcal{L}_{D})^{-1}\rho^{1})$. Take
$\Gamma$ to be a closed contour in $\mathbb{C}$ containing an eigenvalue
$\beta^{\alpha}_{j}(0)$ in $\sigma((-\mathcal{L}_{D})^{-1}\rho^{1})$ but no
other element of $\sigma((-\mathcal{L}_{D})^{-1}\rho^{1})$.

$d$$d$$\hat{\beta}^{\alpha}_{j-1}(0)$$\beta^{\alpha}_{j}(0)$$\check{\beta}^{\alpha}_{j+1}(0)$$\Gamma$
Figure 2: $\Gamma$

Define $d$ to be the distance between $\Gamma$ and
$\sigma((-\mathcal{L}_{D})^{-1})\rho^{1})$, i.e.,

$d=\operatorname{dist}(\Gamma,\sigma((-\mathcal{L}_{D})^{-1}\rho^{1})=\inf_{\xi\in\Gamma}\\{\operatorname{dist}(\Gamma,\sigma((-\mathcal{L}_{D})^{-1}\rho^{1}).\\}$
(7.1)

The component of the spectrum of $A^{\alpha}(0)\rho$ inside $\Gamma$ is
precisely $\beta^{\alpha}_{j}(0)$ and we denote this by $\Sigma^{\prime}(0)$.
The part of the spectrum of $A^{\alpha}(0)\rho$ in the domain exterior to
$\Gamma$ is denoted by $\Sigma^{\prime\prime}(0)$ and
$\Sigma^{\prime\prime}(0)=\sigma((-\mathcal{L}_{D}^{-1})\rho^{1})\setminus{\beta^{\alpha}_{j}(0)}$.
The invariant subspace of $A^{\alpha}(0)\rho$ associated with
$\Sigma^{\prime}(0)$ is denoted by $M^{\prime}(0)$ with
$M^{\prime}(0)=P(0)L^{2}_{\\#}(\alpha,Y)^{3}$.

Suppose the lowest quasi-periodic resonance eigenvalue for the domain $D$ lies
inside $-1/2<\tau^{-}(\alpha)$. It is noted that in the sequel a large and
generic class of domains are identified for which there exists $\tau^{-}$,
independent of $\alpha\in Y^{\ast}$ such that
$-1/2<\tau^{-}\leq\tau^{-}(\alpha)$. The corresponding upper bound on the set
$z\in S^{\alpha}$ for which $A^{\alpha}(z)\rho$ is not invertible is given by

$z^{*}=\frac{\tau^{-}(\alpha)+1/2}{\tau^{-}(\alpha)-1/2}<0,$ (7.2)

see 4.2. Now set

$r^{*}=\frac{\mu_{1}|\alpha|^{2}d|z^{*}|}{\frac{\|\rho\|_{L^{\infty}(Y)^{3}}}{1/2-\tau^{-}(\alpha)}+\mu_{1}|\alpha|^{2}d}.$
(7.3)

######  Theorem 7.1.

Separation of spectra and radius of convergence for $\alpha\in
Y^{*},\alpha\neq 0$. The following properties hold for inclusions with domains
$D$ that satisfy (7.2):

  1. 1.

If $|z|<r^{*}$ then $\Gamma$ lies in the resolvent of both $A^{\alpha}(0)\rho$
and $A^{\alpha}(z)\rho$ and thus separates the spectrum of $A^{\alpha}(z)\rho$
into two parts given by the components of spectrum of $A^{\alpha}(z)\rho$
inside $\Gamma$ denoted by $\Sigma^{\prime}(z)$ and components exterior to
$\Gamma$ denoted by $\Sigma^{\prime\prime}(z)$. The invariant subspace of
subspace of $A^{\alpha}(z)\rho$ associated with $\Sigma^{\prime}(z)$ is
denoted by $M^{\prime}(z)$ with $M^{\prime}(z)=P(z)L^{2}_{\\#}(\alpha,Y)^{3}$.

  2. 2.

The projection $P(z)$ is holomorhic for $|z|<r^{*}$ and $P(z)$ is given by

$P(z)=-\frac{1}{2\pi i}\oint_{\Gamma}R(\zeta,z)\;d\zeta.$ (7.4)

  3. 3.

The spaces $M^{\prime}(z)$ and $M^{\prime}(0)$ are isomorphic for $|z|<r^{*}$.

  4. 4.

The power series (4.10) converges uniformly for $z\in\mathbb{C}$ inside
$|z|<r^{*}$.

Suppose now $\alpha=0$. Recall from Theorem 6.2 that the limit spectrum for
$A^{\alpha}(0)\rho$ is
$\sigma(A^{\alpha}(0)\rho)=\\{\delta^{\prime-1}_{j}\\}_{j\in\mathbb{N}}\cup\\{\nu^{-1}_{j}\\}_{j\in\mathbb{N}}$.
For this case take $\Gamma$ to be the closed contour in $\mathbb{C}$
containing an eigenvalue $\beta^{0}_{j}(0)$ in $\sigma(A^{0}(0)\rho)$ but no
other element of $\sigma(A^{0}(0)\rho)$ and define

$d=\inf_{\zeta\in\Gamma}\\{\operatorname{dist}(\zeta,\sigma(A^{0}(0)\rho)\\}.$
(7.5)

Suppose the lowest quasi-periodic resonance eigenvalue for the domain $D$ lies
inside $-1/2<\tau^{-}(0)<0$ and the corresponding upper bound on $S^{0}$ is
given by

$z^{*}=\frac{\tau^{-}(0)+1/2}{\tau^{-}(0)-1/2}<0.$ (7.6)

Set

$r^{*}=\frac{4\pi^{2}\mu_{1}d|z^{*}|}{\frac{\|\rho\|_{L^{\infty}(Y)^{3}}}{1/2-\tau^{-}(0)}+4\pi^{2}\mu_{1}d}.$
(7.7)

######  Theorem 7.2.

Separation of spectra and radius of convergence for $\alpha=0$.
The following properties hold for inclusions with domains $D$ that satisfy
(7.6):

  1. 1.

If $|z|<r^{*}$ then $\Gamma$ lies in the resolvent of both $A^{0}(0)\rho$ and
$A^{0}(z)\rho$ and thus separates the spectrum of $A^{0}(z)\rho$ into two
parts given by the components of spectrum of $A^{0}(z)\rho$ inside $\Gamma$
denoted by $\Sigma^{\prime}(z)$ and components exterior to $\Gamma$ denoted by
$\Sigma^{\prime\prime}(z)$. The invariant subspace of subspace of
$A^{0}(z)\rho$ associated with $\Sigma^{\prime}(z)$ is denoted by
$M^{\prime}(z)$ with $M^{\prime}(z)=P(z)L^{2}_{\\#}(0,Y)^{3})$.

  2. 2.

The projection $P(z)$ is holomorhic for $|z|<r^{*}$ and $P(z)$ is given by

$P(z)=-\frac{1}{2\pi i}\oint_{\Gamma}R(\zeta,z)\;d\zeta.$ (7.8)

  3. 3.

The spaces $M^{\prime}(z)$ and $M^{\prime}(0)$ are isomorphic for $|z|<r^{*}$.

  4. 4.

The power series (4.10) converges uniformly for $z\in\mathbb{C}$ inside
$|z|<r^{*}$.

Next we provide an explicit representation of the integral operators appearing
in the series expansion for the eigenvalue group.

######  Theorem 7.3.

Representation of integral operators in the series expansion of eigenvalues.
Let $P^{\alpha}_{3}$ be the projection onto the orthogonal complement of
$W^{\alpha}_{1}\oplus W^{\alpha}_{2}$ and let $\tilde{I}$ denote the identity
on $L^{2}(\partial D)^{3}$, then the explicit representation for the operators
$A_{n}^{\alpha}$ in the expansion (4.10),(4.11) is given by

$\displaystyle A^{\alpha}_{1}$
$\displaystyle=[\mathcal{S}^{\alpha}_{D}(\mathcal{K}^{-\alpha}_{D})^{*}+\frac{1}{2}\tilde{I})(\mathcal{S}^{\alpha}_{D})^{-1})^{-1}P^{\alpha}_{3}+P^{\alpha}_{1}](-\mathcal{L}_{\alpha})^{-1}\;\text{and}$
(7.9) $\displaystyle A^{\alpha}_{n}$
$\displaystyle=\mathcal{S}^{\alpha}_{D}((\mathcal{K}^{-\alpha}_{D})^{*}+\frac{1}{2}\tilde{I})^{-1}(\mathcal{S}^{\alpha}_{D})^{-1})^{-1}[\mathcal{S}^{\alpha}_{D}((\mathcal{K}^{-\alpha}_{D})^{*}-\frac{1}{2}\tilde{I})\mathcal{S}^{\alpha}_{D}((\mathcal{K}^{-\alpha}_{D})^{*}+\frac{1}{2}\tilde{I})^{-1}(\mathcal{S}^{\alpha}_{D})^{-1}]^{n-1}P^{\alpha}_{3}(-\mathcal{L}_{\alpha})^{-1}.$

######  Theorem 7.4.

Error estimates for the eigenvalue expansion.

  1. 1.

Let $\alpha\neq 0$, and suppose $D,z^{*},$ and $r^{*}$ are as in Theorem 7.1.
Then the following error estimate for the series (4.10) holds for $|z|<r^{*}$:

$\Big{|}\hat{\beta}^{\alpha}(z)-\sum_{n=0}^{p}z^{n}\beta^{\alpha}_{n}\Big{|}\leq\frac{d|z|^{p+1}}{(r^{*})^{p}(r^{*}-|z|)}.$
(7.10)

  2. 2.

Let $\alpha=0$, and suppose $D,z^{*},$ and $r^{*}$ are as in Theorem 7.2. Then
the following error estimate for the series (4.10) holds for $|z|<r^{*}$:

$\Big{|}\hat{\beta}^{0}(z)-\sum_{n=0}^{p}z^{n}\beta^{0}_{n}\Big{|}\leq\frac{d|z|^{p+1}}{(r^{*})^{p}(r^{*}-|z|)}.$
(7.11)

We summarize results in the following theorem.

######  Theorem 7.5.

The Bloch eigenvalue problem (1.1) is defined for the coupling constant $k$
extended into the complex plane and the operator
$u\mapsto\nabla\cdot(\mathbf{C}^{1}\chi_{D}+\mathbf{C}^{2}\chi_{Y\setminus
D})\mathcal{E}(u)$ with domain $H_{\\#}^{1}(\alpha,Y)^{3}$ is holomorphic for
$k\in\mathbb{C}\setminus Z^{\alpha}$. The associated Bloch spectra is given by
the eigenvalues $\xi_{j}(k,\alpha)=(\beta_{j}^{\alpha}(1/k))^{-1}$, for
$j\in\mathbb{N}$. For $\alpha\in Y^{*}$ fixed, the eigenvalues are of finite
multiplicity. Moreover for each $j$ and $\alpha\in Y^{*}$ the eigenvalue group
is analytic within a neighborhood of infinity containing the disk
$|k|>(r^{*})^{-1}$ where $r^{*}$ is given by (7.3) for $\alpha\neq 0$ and by
(7.7) for $\alpha=0$. When $\beta_{j}^{\alpha}(0)$ is simple these conditions
are sufficient for the separation of spectral branches of the dispersion
relation for fixed quasi-momentum within a neighborhood of infinity containing
the disk $|k|>(r^{*})^{-1}$.

The proofs of Theorems 7.1, 7.2 and 7.4 are given in section 13. The proofs of
Theorem 7.3 is given in section 11.

##  8 Bounds on Quasi-static Resonance Spectra for Periodic Scatters of
General Shape

In this section we identify an explicit condition on the inclusion geometry
that guarantees a lower bound $\tau^{-}$ on the quasi-periodic spectra that
holds uniformly for $\alpha\in Y^{*}$.

$-\frac{1}{2}<\tau^{-}\leq\tau^{-}(\alpha)=\min_{i}\\{\tau_{i}\\}\leq\frac{1}{2},$

This provides a lower bound on the structural spectra $\cup_{\alpha\in
Y^{\ast}}\\{\tau_{i}(\alpha)\\}_{i=1}^{\infty}$ that is strictly greater than
$-1/2$.

To begin, note if $(\tau,w)$ is an eigenpair of $T|_{W_{3}}$ and $v\in
H_{\\#}^{1}(\alpha,Y)^{3}$ then,

$\frac{1}{2}\int_{Y\setminus
D}\mathbf{C}^{1}\mathcal{E}(w):\overline{\mathcal{E}(v)}\;dx-\frac{1}{2}\int_{D}\mathbf{C}^{1}\mathcal{E}(w):\overline{\mathcal{E}(v)}\;dx=\tau\int_{Y}\mathbf{C}^{1}\mathcal{E}(w):\overline{\mathcal{E}(v)}\;dx.$
(8.1)

Adding
$\frac{1}{2}\int_{Y}\mathbf{C}^{1}\mathcal{E}(w):\overline{\mathcal{E}(v)}\;dx$
to both sides yields

$\int_{Y\setminus
D}\mathbf{C}^{1}\mathcal{E}(w):\overline{\mathcal{E}(v)}\;dx=(\tau+\frac{1}{2})\int_{Y}\mathbf{C}^{1}\mathcal{E}(w):\overline{\mathcal{E}(v)}\;dx.$
(8.2)

The lower bound is obtained by showing that there exists a $p>0$ such that
$\tau_{i}+\frac{1}{2}\geq p$ independent of $i\in\mathbb{N}$ and $\alpha\in
Y^{*}$.

######  Theorem 8.1.

Let $\tau^{-}(\alpha)$ be the lowest eigenvalue of $T$ in
$W_{3}^{\alpha}\subset H_{\\#}^{1}(\alpha,Y)^{3}$. Suppose there is a
$\theta>0$ independent of $\alpha\in Y^{\ast}$ such that for all $v\in
W^{\alpha}_{3}$ we have

$\int_{Y\setminus
D}\mathbf{C}^{1}\mathcal{E}(v):\overline{\mathcal{E}(v)}\;dx\geq\theta\int_{D}\mathbf{C}^{1}\mathcal{E}(v):\overline{\mathcal{E}(v)}\;dx.$
(8.3)

Let $p=\min\\{\frac{1}{2},\frac{\theta}{2}\\}.$ Then
$\tau^{-}(\alpha)+\frac{1}{2}\geq p$ for all $\alpha\in Y^{*}.$

###### Proof.

We prove by contradiction, so in addition to (8.3), we suppose that
$\tau^{-}(\alpha)+\frac{1}{2}<p$ for some $\alpha\in Y^{*}$. Then
$\tau^{-}(\alpha)+\frac{1}{2}<\frac{1}{2}$ and
$\tau^{-}(\alpha)+\frac{1}{2}<\frac{\theta}{2}.$ Let $u^{-}$ be the
eigenvector of $T$ with eigenvalue $\tau^{-}(\alpha),$ normalized so that
$\|u^{-}\|=1$. Then we obtain

$\int_{Y\setminus
D}\mathbf{C}^{1}\mathcal{E}(u^{-}):\overline{\mathcal{E}(u^{-})}\;dx<\frac{1}{2}$
(8.4)

and

$\frac{\theta}{2}>\int_{Y\setminus
D}\mathbf{C}^{1}\mathcal{E}(u^{-}):\overline{\mathcal{E}(u^{-})}\;dx\geq\theta\int_{D}\mathbf{C}^{1}\mathcal{E}(u^{-}):\overline{\mathcal{E}(u^{-})}\;dx.$
(8.5)

This gives

$\int_{D}\mathbf{C}^{1}\mathcal{E}(u^{-}):\overline{\mathcal{E}(u^{-})}\;dx<\frac{1}{2}.$
(8.6)

Inequalities (8.4) and (8.6) yield

$\int_{Y}\mathbf{C}^{1}\mathcal{E}(u^{-}):\overline{\mathcal{E}(u^{-})}\;dx=\|u^{-}\|^{2}<1.$

This is a contradiction since $\|u^{-}\|=1$. ∎

Clearly the parameter $\theta$ is a geometric descriptor for $D$. We define
the class of inclusion configurations for which the structural spectra is
bounded strictly above $-1/2$.

######  Definition 8.2.

The class of periodic distributions of inclusions for which Theorem 8.1 holds
for a fixed positive value of $\theta$ is denoted by $P_{\theta}$. The
structural spectra $\cup_{\alpha\in
Y^{\ast}}\\{\tau_{i}(\alpha)\\}_{i=1}^{\infty}$ for this class is bounded
above $-1/2$ so $z^{\ast}<0$ is uniformly bounded away from zero for
$\alpha\in Y^{*}$.

With this definition we have the corollary given by:

######  Corollary 8.3.

For every inclusion domain $D$ belonging to $P_{\theta}$ Theorems 7.1 through
7.5 hold with $z^{*}$ replaced with $z^{+}_{\theta}$ given by

$z^{+}_{\theta}=\frac{\tau^{-}+1/2}{\tau^{-}-1/2}<0,$ (8.7)

where $\tau^{-}=\min\\{\frac{1}{2},\frac{\theta}{2}\\}-\frac{1}{2}$.

Now we introduce a wide class of inclusion shapes belonging to $P_{\theta}$
for a given $\theta>0$. Consider an inclusion domain $D$ with smooth
$C^{\infty}$ boundary. Suppose we can surround the inclusion with a security
layer $R$ of given thickness such that their union ${D}\cup R=D^{\prime}$ is
contained inside the interior of $Y$. We show next that there is a $\theta>0$
that will depend on $R$ and $D$ but be independent of $\alpha\in Y^{\ast}$.
Given $f^{\prime}\in H^{1}(R)^{3}$ there is a bounded linear extension
operator $E:H^{1}(R)^{3}\mapsto H^{1}(D^{\prime})^{3}$ such that
$f=E(f^{\prime})$ satisfies $f(x)=f^{\prime}(x)$ for $x\in R$ see [31]. Hence
there is a positive constant $C_{e}$ depending only on $D$ and $R$ such that

$\displaystyle\|E(f^{\prime})\|_{H^{1}(D)^{3}}^{2}$ $\displaystyle\leq
C_{e}\|f^{\prime}\|^{2}_{H^{1}(R)^{3}}.$ (8.8)

The space of rigid body motions on $R$ is written
$\mathcal{R}=\\{u(x)=Qx+c\;;\,\,x\in R,\,\,Q\in SO^{3},c\in\mathbb{R}^{3}\\}$
and its projection with respect to the $L^{2}(R)^{3}$ norm is written
$\mathbb{P}_{\mathcal{R}}$. Now choose $u\in W_{3}^{\alpha}$ and consider
$u-\mathbb{P}_{\mathcal{R}}u$ restricted to $R$, and we have the inequalities:

$\displaystyle\frac{1}{\beta}\int_{D}\,\mathbf{C}^{1}\mathcal{E}E(u-\mathbb{P}_{\mathcal{R}}u):\overline{\mathcal{E}E(u-\mathbb{P}_{\mathcal{R}}u)}\,dy$
$\displaystyle\leq\|E(u-\mathbb{P}_{\mathcal{R}}u)\|_{H^{1}(D)^{3}}^{2}\leq
C_{e}\|u-\mathbb{P}_{\mathcal{R}}u\|^{2}_{H^{1}(R)^{3}}.$ (8.9)

To obtain the first inequality we use $\gamma<\mathbf{C}^{1}<\beta$ in the
sense of quadratic forms, see (1.3). On applying Korn and Poincare
inequalities to the right hand side (see, e.g., [14] pg. 117) delivers
positive constants $K$ and $C$ independent of $\alpha\in Y^{\ast}$ such that

$\displaystyle\|u-\mathbb{P}_{\mathcal{R}}u\|_{H^{1}(R)^{3}}^{2}$
$\displaystyle\leq
K\left(\int_{R}|u-\mathbb{P}_{\mathcal{R}}u|^{2}\,dw+\int_{R}\,\mathcal{E}(u):\overline{\mathcal{E}(u)}\,dx\right),$
(8.10)

and

$\displaystyle\|u-\mathbb{P}_{\mathcal{R}}u\|_{L^{2}(R)^{3}}^{2}$
$\displaystyle\leq C\int_{R}\,\mathcal{E}(u):\overline{\mathcal{E}(u)}\,dx,$
(8.11)

Noting that $u\in W_{3}^{\alpha}$ satisfies $\mathcal{L}u=0$ in $D$ we see
that it is a minimizer of the elastic energy on $D$ for boundary data
$u-\mathbb{P}_{\mathcal{R}}u$ on $\partial D$ hence

$\displaystyle\frac{1}{\beta}\int_{D}\,\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(u)}\,dy$
$\displaystyle\leq C_{e}\|u-\mathbb{P}_{\mathcal{R}}u\|^{2}_{H^{1}(R)^{3}}.$
(8.12)

Application of (8.10), (8.11), (8.12), and (1.3) gives

$\displaystyle\theta\int_{D}\,\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(u)}\,dy$
$\displaystyle\leq\int_{R}\,\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(u)}\,dy,$
(8.13)

with $\theta$ independent of $\alpha\in Y^{\ast}$ and
$\theta=\frac{\gamma}{\beta KC_{e}(1+C)}$ and it follows that

$\displaystyle\theta\int_{D}\,\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(u)}\,dy$
$\displaystyle\leq\int_{Y\setminus
D}\,\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(u)}\,dy,$ (8.14)

##  9 Radius of Convergence and rates of convergence for Dispersions of
Spherical inclusions

We now provide an example where the radius of convergence and separation of
spectra given by Theorems 7.1 and 7.2 is determined explicitly by the radii of
each inclusion, the minimum distance seperating each inclusion and the
Dirichlet spectra of the inclusions. We display this for crystals in
$\mathbb{R}^{3}$ with period cell containing a of spherical inclusion $D$ of
radius $a$ surrounded by a buffer shell $R$ of inner radius $a$ and outer
radius $b$. The spherical inclusion can be located any where inside the unit
cell provided that the buffer shell is also interior to the unit cell. This is
a specific example of the $P_{\theta}$ type geometry, see Figure 3 introduced
in the work of [6].

Figure 3: The shaded region is the inclusion of radius $a$ surrounded by a
shell of thickness $a$ and outer radius $2a$.

Let $q$ denote the ratio between inner and outer radius and to illustrate the
ideas we choose $q=0.5$. Table 2 of [6] delivers $\theta=1/2$ consequently
$\tau^{-}=-0.25$ and together with (8.7) gives
$|z^{*}|\leq\frac{\tau^{-}+1/2}{1/2-\tau^{-}}=1/3$. From Section 5 we have
that the limit spectrum $\sigma(A^{\alpha}(0)\rho)$ is given by the reciprocal
of the Dirichlet eigenvalues of

$\mathcal{L}_{D}u=\rho^{1}\hat{\eta}u\hbox{ for $\hat{\eta}>0$ and $u\in
H^{1}_{0}(D)^{3}$}.$ (9.1)

We set $\eta_{j}=\hat{\eta}^{-1}_{j}$, and write this as
$\sigma(A^{\alpha}(0)\rho)=\\{\eta_{j}\\}_{j=1}^{\infty}$. Let
$\eta_{j}\in\sigma(A^{\alpha}(0)\rho)$ and let $\tilde{\eta}$ be the minimizer
of $\min_{j,j^{\prime}\in\mathbb{N}}|\eta_{j}-\eta_{j^{\prime}}|$. Then
$d=\frac{1}{2}|\eta_{j}-\tilde{\eta}|$. Substitution of $d$ and $z^{*}$ into
(7.3) shows that the radius of convergence $r^{*}$ is defined explicitly in
terms of the physical geometry of the inclusions and the Dirichlet spectra of
the inclusions and is given by

$r^{*}=\frac{\mu_{1}|\alpha|^{2}|\eta_{j}-\tilde{\eta}|}{8\rho^{{}^{\prime}}+3\mu_{1}|\alpha|^{2}|\eta_{j}-\tilde{\eta}|}\;\text{for}\;\alpha\neq
0.$ (9.2)

From Section 6, theorem 6.2 the spectra is discrete and
$\sigma(A^{0}(0)\rho)=\\{\delta^{\prime-1}_{j}\\}_{j\in\mathbb{N}}\cup\\{\nu^{-1}_{j}\\}_{j\in\mathbb{N}}$.
We denote generic elements of this discrete spectra as $\gamma_{j}$ and for a
fixed element $\gamma_{j}$ we let $\tilde{\gamma}$ be the minimizer of
$\min_{j,j^{\prime}\in\mathbb{N}}|\gamma_{j}-\gamma_{j^{\prime}}|$. Then
$d=\frac{1}{2}|\gamma_{j}-\tilde{\gamma}|$. Substitution of $d$ and $z^{*}$
into (7.7) shows that the radius of convergence $r^{*}$ is defined explicitly
in terms of the physical geometry of the inclusions and the Dirichlet spectra
of the inclusions and is given by

$r^{*}=\frac{\pi^{2}\mu_{1}|\gamma_{j}-\tilde{\gamma}|}{2\rho^{{}^{\prime}}+3\pi^{2}\mu_{1}|\gamma_{j}-\tilde{\gamma}|}\;\text{for}\;\alpha=0.$
(9.3)

The convergence rates are given by Theorem 7.4 using.the values of $r^{*}$
given by (9.2) and (9.3)

##  10 Emergence of Bandgaps in the high contrast limit

In this section we identify conditions which are sufficient for the emergence
of band gaps for sufficiently large contrast. Here the contrast $k$ is real
and taken to be in the interval $1\leq k<\infty$. We order the Dirichlet
spectrum of the inclusion $D$ by minmax and
$0<\delta_{1}\leq\delta_{2}\leq\cdots\delta_{j}\leq\delta_{j+1}\cdots$,
$\delta_{j}\rightarrow\infty$ as $j\rightarrow\infty$. Let
$\\{\delta^{\prime}_{j}\\}_{j=1}^{\infty}$ be the Dirichlet eigenvalues
associated with zero average eigenfunctions and let
$\\{\delta^{*}_{j}\\}_{j=1}^{\infty}$ be the Dirichlet eigenvalues for which
there are non-zero average eigenfunctions. Recall the limit spectra defined in
Theorem 6.2

$\sigma(A^{0}(0)\rho)=\\{\delta^{\prime-1}_{j}\\}_{j=1}^{\infty}\cup\\{\nu^{-1}_{j}\\}_{j=1}^{\infty},$
(10.1)

and $\\{\nu_{j}\\}_{j\in\mathbb{N}}$ are the positive roots of (6.4). The
eigenspaces of elements of $\sigma(A^{0}(0)\rho)$ are orthogonal to the $3$
dimensional space of rigid translations. Let
$\omega_{j}=\lim_{k\rightarrow\infty}\xi_{j+3}(k,0)$ and from Theorem 6.2 and
(6.6) one has the alternative $\omega_{j}=\delta^{\prime}_{j}$ or
$\omega_{j}=\nu_{j}$. In what follows we assume that the inclusions are in the
class $P_{\theta}$. We have the interlacing theorem.

######  Theorem 10.1.

$\delta_{j}\leq{\omega}_{j}\leq\delta_{j+3}.$ (10.2)

One can characterize pass bands in the high contrast limit. Set

$a_{j}(k)=min_{\alpha\in Y^{*}}\xi_{j}(k,\alpha),\qquad
b_{j}(k)=max_{\alpha\in Y^{*}}\xi_{j}(k,\alpha),$ (10.3)

The pass band for $k<\infty$ is given by $[a_{j}(k),b_{j}(k)]$ and

######  Theorem 10.2.

$\lim_{k\rightarrow\infty}[a_{j}(k),b_{j}(k)]=[\omega_{j},\delta_{j+3}],$
(10.4)

here $\lim_{k\rightarrow\infty}[a_{j}(k),b_{j}(k)]$ can reduce to the single
point $\omega_{j}=\delta_{j+3}$, and the band structure in the high contrast
limit $k\rightarrow\infty$ is given by

$[0,\delta_{1}]\cup[0,\delta_{2}]\cup[0,\delta_{3}]\bigcup_{j\geq
1}[\omega_{j},\delta_{j+3}].$ (10.5)

A criterion for band gap opening now follows.

######  Theorem 10.3 (Criterion for band gap opening).

A band gap exists if

$\delta_{j+2}<{\omega}_{j}.$ (10.6)

Lemma 6.3 together with Theorem 10.3 shows that the band gap increases with
decreasing $\rho_{2}$, this is consistent with the two dimensional result of
[1] and the experimental findings of [17]. Theorem 10.1, Theorem 10.2 and the
band gap criterion are established in [1] and [2] for the two dimensional
case. The variational arguments of [1] together with Lemma 6.3 can be readily
used to prove Theorem 10.1 in the three dimensional case. We now provide the
proof of Theorem 10.2 in three dimensions using the Lipschitz continuity of
the eigenvalues $\xi_{j}(k,\alpha)$ about $\alpha=0$ for fixed $k$. Using the
minmax formulation of eigenvalues one deduces as in [12] that

$|\xi_{j+3}(k,0)-\xi_{j+3}(k,\alpha)|<Ck|\alpha|,$ (10.7)

where $C$ is independent of $\alpha$ and $k$. Equation (7.11) of Theorem 7.4
gives the convergence

$|\omega_{j}-\xi_{j+3}(k,0)|<C\frac{d}{k(r^{*}-(1/k))}.$ (10.8)

So given any $\epsilon>0$ we can find a pair
$(k^{\prime},\alpha^{\prime})\in\mathbb{R}^{+}\times Y^{\ast}$ such that

$|\omega_{j}-\xi_{j+3}(k^{\prime},\alpha^{\prime})|<\epsilon.$ (10.9)

On the other hand from the minmax formulation $\xi_{j+3}(k,\alpha)$ is
monotone increasing with $k$ and for $\alpha\not=0$, we have from Theorem 7.4
that $\xi_{j+3}(k,\alpha)\rightarrow\delta_{j+3}$ as $k\rightarrow\infty$. So
Theorem 10.2 follows immediately from (10.9) and these observations.

In closing we show that the symmetry of inclusion domains $D$ provides a new
condition on the interlacing of $\\{\nu_{j}\\}_{j=1}^{\infty}$ and
$\\{\delta_{j}^{*}\\}_{j=1}^{\infty}$.

######  Theorem 10.4.

Suppose $D$ is invariant under the cubic group of rotations then we have the
interlacing

$\nu_{j-1}<\delta^{\ast}_{j}<\nu_{j}.$ (10.10)

###### Proof.

Since $D$ is invariant under the cubic group of rotations
$M(\nu)=\lambda(\nu)I$, where $I$ is the $3\times 3$ identity. Here
$\lambda(\nu)$ is a real valued function of $\nu$ and
$\det\left\\{M(\nu)\right\\}=\lambda^{3}(\nu)$ so $\nu_{j}$ are the roots of
the equation $\lambda(\nu)=0$. For any constant vector $\vec{v}$ in
$\mathbb{R}^{3}$ we have

$\lambda({\nu})=\frac{M(\nu)\vec{v}\cdot\vec{v}}{|\vec{v}|^{2}}=\int_{Y}\rho(x)\;dx-\nu\sum_{j\in\mathbb{N}}\frac{a^{2}_{j}}{\nu-\delta^{*}_{j}},$
(10.11)

where
$a^{2}_{j}={|\int_{D}\rho^{1}{\psi}_{j}\;dx\cdot\vec{v}|^{2}}/{|\vec{v}|^{2}}>0$
and $\delta_{j}^{\ast}$ are only associated with nonzero mean eigenfunctions.
For $\delta^{\ast}_{j-1}<\nu<\delta^{\ast}_{j}$, calculation shows
$-\infty<\lambda(\nu)<\infty$, with $\lambda^{\prime}(\nu)>0$. From this we
conclude $\delta^{\ast}_{j}<\nu_{j}<\delta^{\ast}_{j+1}$ and we have the
interlacing $\nu_{j-1}<\delta^{\ast}_{j}<\nu_{j}$. Thus a Dirichlet eigenvalue
associated with non zero mean eigenfunctions always lies strictly between
successive roots $\nu_{j-1}$ and $\nu_{j}$. ∎

##  11 Layer potential representation of operators in power series

In this section we identify explicit formulas for the operators
$A^{\alpha}_{n}$ appearing in the power series (4.11). It is shown that
$A^{\alpha}_{n},n\neq 0$ can be expressed in terms of integral operators
associated with layer potentials and we establish Theorem 7.3.
Recall that $A^{\alpha}(z)-A^{\alpha}(0)$ is given by

$(zP^{\alpha}_{1}+\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}z[(1/2+\tau_{i}(\alpha))+z(1/2-\tau_{i}(\alpha))]^{-1}P_{\tau_{i}(\alpha)})(-\mathcal{L}_{\alpha})^{-1}.$
(11.1)

Notice that

$[(1/2+\tau_{i}(\alpha))+z(1/2-\tau_{i}(\alpha))]^{-1}=(\tau_{i}(\alpha)+1/2)^{-1}\sum_{n=0}^{\infty}z^{n}\Big{(}\frac{\tau_{i}(\alpha)-1/2}{\tau_{i}(\alpha)+1/2}\Big{)}^{n},$
(11.2)

therefore

$A^{\alpha}(z)-A^{\alpha}(0)=(zP^{\alpha}_{1}+\sum_{n=1}^{\infty}z^{n}\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}(\tau_{i}(\alpha)+1/2)^{-1}\Big{(}\frac{\tau_{i}(\alpha)-1/2}{\tau_{i}(\alpha)+1/2}\Big{)}^{n-1}P_{\tau_{i}(\alpha)}P^{\alpha}_{3})(-\mathcal{L}_{\alpha})^{-1}$
(11.3)

It follows that

$A^{\alpha}_{1}=[P^{\alpha}_{1}+\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}(\tau_{i}(\alpha)+1/2)^{-1}P_{\tau_{i}(\alpha)}P^{\alpha}_{3}](-\mathcal{L}_{\alpha})^{-1}.$
(11.4)

and

$A^{\alpha}_{n}=[\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}(\tau_{i}(\alpha)+1/2)^{-1}\Big{(}\frac{\tau_{i}(\alpha)-1/2}{\tau_{i}(\alpha)+1/2}\Big{)}^{n-1}P_{\tau_{i}(\alpha)}P^{\alpha}_{3}](-\mathcal{L}_{\alpha})^{-1}.$
(11.5)

recall also that we have the resolution of the identity

$I_{H_{\\#}^{1}(\alpha,Y)^{3}}=P^{\alpha}_{1}+P^{\alpha}_{2}+P^{\alpha}_{3}\;\text{with}\;P^{\alpha}_{3}=\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}P_{\tau_{i}(\alpha)},$
(11.6)

and the spectral representation

$\displaystyle\langle Tu,v\rangle$
$\displaystyle=\langle(\mathcal{S}^{\alpha}_{D}(\mathcal{K}^{-\alpha}_{D})^{*}(\mathcal{S}^{\alpha}_{D})^{-1})P^{\alpha}_{3}u+\frac{1}{2}P^{\alpha}_{1}u-\frac{1}{2}P^{\alpha}_{2}u,v\rangle$
(11.7)
$\displaystyle=\langle\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}\tau_{i}(\alpha)P_{\tau_{i}(\alpha)}u+\frac{1}{2}P^{\alpha}_{1}u-\frac{1}{2}P^{\alpha}_{2}u,v\rangle.$

Adding $\frac{1}{2}I$ to both sides of the above equation, we obtain

$\displaystyle\langle(Tu+\frac{1}{2}I)u,v\rangle$
$\displaystyle=\langle(\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}(\tau_{i}(\alpha)+\frac{1}{2})P_{\tau_{i}(\alpha)}+P^{\alpha}_{1})u,v\rangle$
(11.8)
$\displaystyle=\langle((\mathcal{S}^{\alpha}_{D}(\mathcal{K}^{-\alpha}_{D})^{*}(\mathcal{S}^{\alpha}_{D})^{-1}+\frac{1}{2}P^{\alpha}_{3})P^{\alpha}_{3}+P^{\alpha}_{1})u,v\rangle$
$\displaystyle=\langle((\mathcal{S}^{\alpha}_{D}(\mathcal{K}^{-\alpha}_{D})^{*}+\frac{1}{2}\tilde{I})(\mathcal{S}^{\alpha}_{D})^{-1})P^{\alpha}_{3}+P^{\alpha}_{1})u,v\rangle,$

where $\tilde{I}$ is the identity on $H^{-1/2}(\partial D)^{3}$. Now from
(11.8) we see that

$\displaystyle\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}(\tau_{i}(\alpha)+\frac{1}{2})^{-1}P_{\tau_{i}(\alpha)}P^{\alpha}_{3}$
$\displaystyle=(\mathcal{S}^{\alpha}_{D}(\mathcal{K}^{-\alpha}_{D})^{*}(\mathcal{S}^{\alpha}_{D})^{-1}+\frac{1}{2}P^{\alpha}_{3})^{-1}P^{\alpha}_{3}$
(11.9)
$\displaystyle=((\mathcal{S}^{\alpha}_{D}(\mathcal{K}^{-\alpha}_{D})^{*}+\frac{1}{2}\tilde{I})(\mathcal{S}^{\alpha}_{D})^{-1})^{-1}P^{\alpha}_{3}.$

Using the first line of (11.4) and (11.9), we obtain

$A^{\alpha}_{1}=[\mathcal{S}^{\alpha}_{D}(\mathcal{K}^{-\alpha}_{D})^{*}+\frac{1}{2}\tilde{I})(\mathcal{S}^{\alpha}_{D})^{-1})^{-1}P^{\alpha}_{3}+P^{\alpha}_{1}](-\mathcal{L}_{\alpha})^{-1}.$
(11.10)

For higher order terms, by the mutual orthogonality of the projections
$P_{\tau_{i}(\alpha)}$, we have that

$\displaystyle\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}(\tau_{i}(\alpha)+1/2)^{-1}\Big{(}\frac{\tau_{i}(\alpha)-1/2}{\tau_{i}(\alpha)+1/2}\Big{)}^{n-1}P_{\tau_{i}(\alpha)}$
(11.11)
$\displaystyle=\Big{(}\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}(\tau_{i}(\alpha)+1/2)^{-1}P_{\tau_{i}(\alpha)}\Big{)}\Big{(}\sum_{-\frac{1}{2}<\mu_{i}<\frac{1}{2}}\Big{(}\frac{\tau_{i}(\alpha)-1/2}{\tau_{i}(\alpha)+1/2}\Big{)}P_{\tau_{i}(\alpha)}\Big{)}^{n-1}$
$\displaystyle=\Big{(}\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}(\tau_{i}(\alpha)+1/2)^{-1}P_{\tau_{i}(\alpha)}\Big{)}\Big{(}\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}(\tau_{i}(\alpha)-1/2)P_{\tau_{i}(\alpha)}\Big{)}^{n-1}\Big{(}\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}(\tau_{i}(\alpha)+1/2)P_{\tau_{i}(\alpha)}\Big{)}^{1-n}.$

As above, we have that

$\displaystyle\Big{(}\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}(\tau_{i}(\alpha)+1/2)^{-1}P_{\tau_{i}(\alpha)}\Big{)}$
$\displaystyle=\mathcal{S}^{\alpha}_{D}((\mathcal{K}^{-\alpha}_{D})^{*}+\frac{1}{2}\tilde{I})^{-1}(\mathcal{S}^{\alpha}_{D})^{-1})^{-1}P^{\alpha}_{3},$
(11.12)
$\displaystyle\Big{(}\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}(\tau_{i}(\alpha)+1/2)P_{\tau_{i}(\alpha)}\Big{)}$
$\displaystyle=\mathcal{S}^{\alpha}_{D}((\mathcal{K}^{-\alpha}_{D})^{*}+\frac{1}{2}\tilde{I})(\mathcal{S}^{\alpha}_{D})^{-1}P^{\alpha}_{3},$
$\displaystyle\Big{(}\sum_{-\frac{1}{2}<\tau_{i}(\alpha)<\frac{1}{2}}(\tau_{i}(\alpha)-1/2)P_{\tau_{i}(\alpha)}\Big{)}$
$\displaystyle=\mathcal{S}^{\alpha}_{D}((\mathcal{K}^{-\alpha}_{D})^{*}-\frac{1}{2}\tilde{I})(\mathcal{S}^{\alpha}_{D})^{-1}P^{\alpha}_{3}.$

Combining (11.12), (11.11), and (11.4) we obtain the layer-potential
representation for $A^{\alpha}_{n}$,

$A^{\alpha}_{n}=\mathcal{S}^{\alpha}_{D}((\mathcal{K}^{-\alpha}_{D})^{*}+\frac{1}{2}\tilde{I})^{-1}(\mathcal{S}^{\alpha}_{D})^{-1})^{-1}[\mathcal{S}^{\alpha}_{D}((\mathcal{K}^{-\alpha}_{D})^{*}-\frac{1}{2}\tilde{I})\mathcal{S}^{\alpha}_{D}((\mathcal{K}^{-\alpha}_{D})^{*}+\frac{1}{2}\tilde{I})^{-1}(\mathcal{S}^{\alpha}_{D})^{-1}]^{n-1}P^{\alpha}_{3}(-\mathcal{L}_{\alpha})^{-1}.$
(11.13)

##  12 Explicit first order correction to the Bloch band structure in the high
contrast limit

In this section we develop explicit formulas for the second term in the power
series

$\beta^{\alpha}_{j}(z)=\beta^{\alpha}_{j}(0)+z\beta^{\alpha}_{j,1}+z^{2}\beta^{\alpha}_{j,2}+\cdots$
(12.1)

for simple eigenvalues. We use analytic representation of $A^{\alpha}(z)$ and
the Cauchy Integral Formula to represent $\beta^{\alpha}_{j,1}$

$\displaystyle\beta^{\alpha}_{j,1}$ $\displaystyle=\frac{1}{2\pi
im}\operatorname{tr}\oint_{\Gamma}A^{\alpha}_{1}\rho R(0,\zeta)\;d\zeta$
(12.2) $\displaystyle=\frac{1}{2\pi
im}\operatorname{tr}(A^{\alpha}_{1}\rho\oint_{\Gamma}R(0,\zeta)\;d\zeta)$
$\displaystyle=\frac{1}{m}\operatorname{tr}(A^{\alpha}_{1}\rho
P(0))=\frac{1}{m}\sum_{k=1}^{m}\langle\varphi_{k},A^{\alpha}_{1}\rho
P(0)\varphi_{k}\rangle_{L^{2}_{\\#}(\alpha,Y)^{3}}$

Here $P(0)$ is the $L^{2}_{\\#}(\alpha,Y)^{3}$ projection onto the eigenspace
corresponding to the Dirichelt eigenvalue $(\beta^{\alpha}_{j}(0))^{-1}$ of
$-\mathcal{L}_{D}$. For simple eigenvalue consider the normalized eigenvector
$P(0)\varphi=\varphi$ and

$\beta^{\alpha}_{j,1}=\langle\varphi,A^{\alpha}_{1}\rho
P(0)\rangle_{L^{2}_{\\#}(\alpha,Y)^{3}}$ (12.3)

We apply the integral operator representation of $A^{\alpha}_{1}$ to deliver
an explicit formula for the first order term $\beta^{\alpha}_{j,1}$ in the
series for $\beta^{\alpha}_{j}(z)$. The explicit formula is given by the
following theorem.

######  Theorem 12.1.

Let $\beta^{\alpha}_{j}(z)$ be an eigenvalue of $A^{\alpha}(z)\rho$. Then for
$|z|<r^{*}$ there is a $\beta_{j}(0)\in\sigma(-\mathcal{L}^{-1}_{D})$ with
corresponding eigenfunction $\varphi_{j}$ such that

$\beta^{\alpha}_{j}(z)=\beta^{\alpha}_{j}(0)+z(\frac{\beta_{j}(0)}{\rho^{1}})^{2}\int_{Y\setminus
D}\mathbf{C}^{1}\mathcal{E}(v):\overline{\mathcal{E}(v)}dx+z^{2}\beta^{\alpha}_{j,2}+\cdots$
(12.4)

where $v$ takes $\alpha-$quasi periodic boundary conditions on $\partial Y$,
and $\mathcal{L}v=0$ in $Y\setminus D$, and takes the Neumann boundary
conditions on $\partial D$ given by

$n\cdot\mathbf{C}^{1}\mathcal{E}(v)|_{\partial
D^{+}}=n\cdot\mathbf{C}^{1}\mathcal{E}(\varphi)|_{\partial D^{-}},$

######  Remark 12.2.

From Theorem 7.5, we have eigenvalues
$\xi_{j}^{\alpha}(k)=(\beta^{\alpha}_{j}(1/k))^{-1}$, for $j\in\mathbb{N}$.
The high coupling limit expansion for $\xi_{j}^{\alpha}(k)$ is written in
terms of the expansion
$\beta^{\alpha}_{j}(z)=\beta_{j}(0)+z\beta^{\alpha}_{j,1}+\cdots$ as

$\displaystyle\xi_{j}^{\alpha}(k)$
$\displaystyle=(\beta_{j}(0))^{-1}-\frac{1}{k}(\beta_{j}(0))^{-2}\beta^{\alpha}_{j,1}+\cdots$
(12.5) $\displaystyle=\xi_{j}(0)-\frac{1}{k\rho^{2}_{1}}\int_{Y\setminus
D}\mathbf{C}^{1}\mathcal{E}(v):\overline{\mathcal{E}(v)}+\cdots,$

where $\xi_{j}(0)=(\beta_{j}(0))^{-1}$ is the $j^{th}$ Dirichlet eigenvalue
for the homogeneous Lamé operator in $D$. This is consistent with the formula
for the leading order terms presented in [2] for 2 dimensional elasticity.

###### Proof.

Recall from the previous section that

$\displaystyle A^{\alpha}_{1}$
$\displaystyle=[\mathcal{S}^{\alpha}_{D}((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}+\frac{1}{2}\tilde{I})^{-1}(\mathcal{S}^{\alpha}_{D})^{-1}P^{\alpha}_{3}+P^{\alpha}_{1}](-\mathcal{L}_{\alpha})^{-1}$
(12.6) $\displaystyle=K^{\alpha}_{1}(-\mathcal{L}_{\alpha})^{-1},$

where
$K^{\alpha}_{1}:=\mathcal{S}^{\alpha}_{D}((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}+\frac{1}{2}\tilde{I})^{-1}(\mathcal{S}^{\alpha}_{D})^{-1}P^{\alpha}_{3}+P^{\alpha}_{1}$.
Moreover,

$(-\mathcal{L}_{\alpha})^{-1}f=-\int_{Y}\mathbf{G}^{\alpha}(x,y)f(y)dy.$
(12.7)

Since $\varphi$ is a Dirichlet eigenvector of the Lamé operator defined on $D$
with eigenvalue $(\beta_{j}(0))^{-1}$ and $\varphi=0$ in $Y\setminus D$, we
have

$\varphi=-\frac{\beta_{j}(0)}{\rho^{1}}\chi_{D}(-\mathcal{L}\varphi).$ (12.8)

Now from (12.8)

$\displaystyle(-\mathcal{L}_{\alpha})^{-1}\varphi$
$\displaystyle=-\frac{\beta_{j}(0)}{\rho^{1}}\int_{Y}\mathbf{G}^{\alpha}(x,y)\chi_{D}(\mathcal{L}\varphi)dy$
(12.9)
$\displaystyle=-\frac{\beta_{j}(0)}{\rho^{1}}\int_{D}\mathbf{G}^{\alpha}(x,y)(\mathcal{L}_{y}\varphi)dy$

Using integration by parts and adopting index notation where repeated indices
indicate summation we get for each component

$\displaystyle{[(-\mathcal{L}_{\alpha})^{-1}\varphi]}_{i}$
$\displaystyle=-\frac{\beta_{j}(0)}{\rho^{1}}(\int_{D}\partial_{j}(\mathbf{G}^{\alpha}(x,y)_{ik}\mathbf{C^{1}}\mathcal{E}(\varphi)_{kj})dy-\int_{D}\mathbf{C^{1}}\mathcal{E}(\varphi)_{kj}:(\mathcal{E}(\mathbf{G}^{\alpha}(x,y)_{i}))_{kj}dy)$
$\displaystyle=-\frac{\beta_{j}(0)}{\rho^{1}}(\mathcal{S}^{\alpha}_{D}[\partial_{n}\varphi|_{\partial
D^{-}}]_{i}(x)-R(x)_{i})$

where the last equality follows from the divergence theorem and definition of
the single layer potential $\mathcal{S}^{\alpha}_{D}$ and

$R(x)_{i}=\int_{D}\mathbf{C^{1}}\mathcal{E}(\varphi)_{kj}:(\mathcal{E}(\mathbf{G}^{\alpha}(x,y)_{i}))_{kj}dy).$
(12.10)

Hence

$[A^{\alpha}_{1}\varphi]_{i}=\frac{\beta_{j}(0)}{\rho^{1}}K^{\alpha}_{1}(\mathcal{S}^{\alpha}_{D}[\partial_{n}\varphi|_{\partial
D^{-}}]_{i}(x)-R(x)_{i}).$ (12.11)

Now we apply the definition of $K^{\alpha}_{1}$ and compute
$P^{\alpha}_{1}R(x)$ and $P^{\alpha}_{3}R(x)$. Integrating by parts, we find

$\displaystyle R(x)_{i}$
$\displaystyle=\int_{D}\mathbf{C^{1}}\mathcal{E}(\varphi)_{kj}:(\mathcal{E}(\mathbf{G}^{\alpha}(x,y)_{i}))_{kj}dy$
(12.12)
$\displaystyle=\int_{D}(\mathcal{E}(\mathbf{G}^{\alpha}(x,y)_{i}))_{kj}:\mathbf{C^{1}}\mathcal{E}(\varphi)_{kj}dy$
$\displaystyle=\int_{D}\partial_{j}(\mathbf{C^{1}}\mathcal{E}(\mathbf{G}^{\alpha}(x,y)_{i})_{kj})\varphi_{k}))dy-\int_{D}-\partial_{j}(\mathbf{C}^{1}\mathcal{E}(\mathbf{G}^{\alpha}(x,y)_{i})_{kj}\varphi_{k}dy$
$\displaystyle=\varphi(x)_{i}.$

Thus $P^{\alpha}_{1}R(x)=P^{\alpha}_{3}R(x)=0$ since $\varphi\in
W^{\alpha}_{2}$. Then we obtain

$\displaystyle\beta^{\alpha}_{j,1}=\operatorname{tr}(A^{\alpha}_{1}\rho P(0))$
$\displaystyle=\langle\varphi,A^{\alpha}_{1}\rho
P(0)\rangle_{L^{2}_{\\#}(\alpha,Y)^{3}}$ (12.13)
$\displaystyle=\langle\varphi,-\frac{\beta_{j}(0)}{\rho^{1}}\mathcal{S}^{\alpha}_{D}((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}+\frac{1}{2}\tilde{I})^{-1}[\partial_{n}\varphi|_{\partial
D^{-}}]\rangle_{L^{2}_{\\#}(\alpha,Y)^{3}}$

Let $v\in H_{\\#}^{1}(\alpha,Y)^{3}$ be defined

$v:=\mathcal{S}^{\alpha}_{D}((\tilde{\mathcal{K}}^{-\alpha}_{D})^{*}+\frac{1}{2}\tilde{I})^{-1}[\partial_{n}\varphi|_{\partial
D^{-}}].$ (12.14)

Then $\mathcal{L}v=0$ in $D$ and $Y\setminus D$, and

$n\cdot\mathbf{C}^{1}\mathcal{E}(v)|_{\partial
D^{+}}=n\cdot\mathbf{C}^{1}\mathcal{E}(\varphi)|_{\partial D^{-}}.$ (12.15)

Then we have

$\displaystyle\beta^{\alpha}_{j,1}$
$\displaystyle=-\frac{\beta_{j}(0)}{\rho^{1}}\langle\varphi,v\rangle=-(\frac{\beta_{j}(0)}{\rho^{1}})^{2}\int_{D}v(\mathcal{L}\overline{\varphi})dy$
(12.16) $\displaystyle=-(\frac{\beta_{j}(0)}{\rho^{1}})^{2}(\int_{\partial
D}n\cdot\mathbf{C}^{1}\mathcal{E}(\varphi)|_{\partial
D^{-}}\overline{v}d\sigma-\int_{D}\mathbf{C}^{1}\mathcal{E}(\varphi):\overline{\mathcal{E}(v)}dx)$
$\displaystyle=-(\frac{\beta_{j}(0)}{\rho^{1}})^{2}(\int_{\partial
D}n\cdot\mathbf{C}^{1}\mathcal{E}(v)|_{\partial
D^{+}}\overline{v}d\sigma-\int_{D}\mathbf{C}^{1}\mathcal{E}(\varphi):\overline{\mathcal{E}(v)}dx)$

Last an integration by parts yields

$\displaystyle\int_{D}\mathbf{C}^{1}\mathcal{E}(\varphi):\overline{\mathcal{E}(v)}dx$
$\displaystyle=\int_{D}\mathbf{C}^{1}\overline{\mathcal{E}(v)}:\mathcal{E}(\varphi)dx$
(12.17)
$\displaystyle=\int_{D}\nabla\cdot(\mathbf{C}^{1}\overline{\mathcal{E}(v)}\varphi)-\mathcal{L}\overline{v}\varphi$
$\displaystyle=n\cdot\mathbf{C}^{1}\overline{\mathcal{E}(v)}|_{\partial
D^{-}}\varphi d\sigma=0.$

Combining this result with the last line of (12.16) and integrating by parts a
one time provides a representation of the second term in (12.1)

$\beta^{\alpha}_{j,1}=(\frac{\beta_{j}(0)}{\rho^{1}})^{2}\int_{Y\setminus
D}\mathbf{C}^{1}\mathcal{E}(v):\overline{\mathcal{E}(v)}dx,$ (12.18)

and the theorem follows. ∎

##  13 Derivation of the convergence radius and the separation of spectra

Here we prove Theorems 7.1 and 7.2. To begin, we suppose $\alpha\neq 0$ and
recall the Neumann series (4.5) and consequently (4.6) and (4.10) converge
provided that

$\|(A^{\alpha}(z)\rho-A^{\alpha}(0)\rho)R(\zeta,0)\|_{L[L^{2}_{\\#}(\alpha,Y)^{3}:L^{2}_{\\#}(\alpha,Y)^{3}]}<1.$
(13.1)

Then we will compute an explicit upper bound $B(\alpha,z)$ and identify a
neighborhood of the origin on the complex plane for which

$\|(A^{\alpha}(z)\rho-A^{\alpha}(0)\rho)R(\zeta,0)\|_{L[L^{2}_{\\#}(\alpha,Y)^{3}:L^{2}_{\\#}(\alpha,Y)^{3}]}<B(\alpha,z)<1,$
(13.2)

holds for $\zeta\in\Gamma$. The inequality $B(\alpha,z)<1$ will be used first
to derive a lower bound on the radius of convergence of the power series
expansion of the eigenvalue group about $z=0$. It will then be used to provide
a lower bound on the neighborhood of $z=0$ where properties 1 through 3 of
Theorem 7.1 hold.
We have the basic statement given by

$\displaystyle\|(A^{\alpha}(z)\rho-A^{\alpha}(0)\rho)R(\zeta,0)\|_{L[L^{2}_{\\#}(\alpha,Y)^{3}:L^{2}_{\\#}(\alpha,Y)^{3}]}\leq$
(13.3)
$\displaystyle\|(A^{\alpha}(z)-A^{\alpha}(0))\|_{L[L^{2}_{\\#}(\alpha,Y)^{3}:L^{2}_{\\#}(\alpha,Y)^{3}]}\|\rho\|_{L^{\infty}(Y)}\|R(\zeta,0)\|_{L[L^{2}_{\\#}(\alpha,Y)^{3}:L^{2}_{\\#}(\alpha,Y)^{3}].}$

Here $\zeta\in\Gamma$ as defined in Theorem 7.1 and elementary arguments
deliver the estimate

$\|R(\zeta,0)\|_{L[L^{2}_{\\#}(\alpha,Y)^{3}:L^{2}_{\\#}(\alpha,Y)^{3}]}\leq
d^{-1},$ (13.4)

where $d$ is given by (7.1).
Next we estimate
$\|(A^{\alpha}(z)-A^{\alpha}(0))\|_{L[L^{2}_{\\#}(\alpha,Y)^{3}:L^{2}_{\\#}(\alpha,Y)^{3}]}$.
Denote the energy seminorm of $u$ by

$\|u\|=\int_{Y}\mathbf{C}^{1}\mathcal{E}(u):\overline{\mathcal{E}(u)}\;dx.$
(13.5)

First we derive the Poincaré inequality between the spaces
$L_{\\#}^{2}(\alpha,Y)^{3}$ and $H_{\\#}^{1}(\alpha,Y)^{3}$ for $\alpha\neq
0$:

######  Lemma 13.1.

$\|u\|_{L^{2}(Y)^{3}}\leq\frac{\alpha^{-1}}{\sqrt{\mu}_{1}}\|u\|.$ (13.6)

###### Proof.

Let $u\in L_{\\#}^{2}(\alpha,Y)^{3}$. Set $\xi=2\pi n+\alpha$ and let

$\displaystyle-l^{-1}(\xi)v\cdot\overline{v}$
$\displaystyle=-(-\frac{1}{\mu_{1}|\xi|^{2}}(I-\frac{\lambda_{1}+\mu_{1}}{\lambda_{1}+2\mu_{1}}\frac{(\xi\otimes\xi)}{|\xi|^{2}}))v\cdot\overline{v}$
$\displaystyle=\frac{1}{\mu_{1}|\xi|^{2}}(I-\frac{\lambda_{1}+\mu_{1}}{\lambda_{1}+2\mu_{1}}\frac{(\xi\otimes\xi)}{|\xi|^{2}}))v\cdot\overline{v},$

then

$\displaystyle((-\mathcal{L}_{\alpha})^{-1}u,u)$
$\displaystyle=\int_{Y}((-\mathcal{L}_{\alpha})^{-1}u(x)\cdot\overline{u(x)}\;dx$
$\displaystyle=\int_{Y}(-\int_{Y}\mathbf{G}^{\alpha}(x,y)u(y)\;dy)\cdot\overline{u(x)}\;dx$
$\displaystyle=\int_{Y}(-\int_{Y}(\sum_{n\in\mathbb{Z}^{3}}l^{-1}(\xi)e^{i\xi\cdot(x-y)})u(y)\;dy)\cdot\overline{u(x)}\;dx$
$\displaystyle=(\sum_{n\in\mathbb{Z}^{3}}-l^{-1}(\xi)\widehat{u(\xi)}\cdot{\overline{\widehat{u(\xi)}}}.$

We obtain an upper bound on the quadratic form
$-l^{-1}(\xi)\widehat{u(\xi)}\cdot{\overline{\widehat{u(\xi)}}}$. Without loss
of generality choose the basis
$a_{1}:=\frac{\xi}{|\xi|}=(1,0,0),a_{2}:=(0,1,0),a_{3}:=(0,0,1)$ and set
$c:=\frac{\lambda_{1}+\mu_{1}}{\lambda_{1}+2\mu_{1}}$. Then

$\displaystyle\frac{1}{\mu_{1}|\xi|^{2}}(I-\frac{\lambda_{1}+\mu_{1}}{\lambda_{1}+2\mu_{1}}\frac{(\xi\otimes\xi)}{|\xi|^{2}}))$
$\displaystyle=\frac{1}{\mu_{1}|\xi|^{2}}\left((1-c)a_{1}\otimes
a_{1}+a_{2}\otimes a_{2}+a_{3}\otimes a_{3}\right)$

Since $0\leq c\leq 1$, we have $0\leq 1-c\leq 1$. Thus

$\displaystyle\frac{1}{\mu_{1}|\xi|^{2}}(I-\frac{\lambda_{1}+\mu_{1}}{\lambda_{1}+2\mu_{1}}\frac{(\xi\otimes\xi)}{|\xi|^{2}}))v\cdot\overline{v}\leq\frac{1}{\mu_{1}|\xi|^{2}}|v|^{2}.$

Hence we obtain the upper bound

$((-\mathcal{L}_{\alpha})^{-1}u,u)=(\sum_{n\in\mathbb{Z}^{3}}-l^{-1}(\xi)\widehat{u(\xi)}\cdot{\overline{\widehat{u(\xi)}}}\leq\sum_{n\in\mathbb{Z}^{3}}\frac{1}{\mu_{1}|\xi|^{2}}|\widehat{u(\xi)}|^{2}$
(13.7)
$\displaystyle\leq\frac{1}{\mu_{1}|\alpha|^{2}}\sum_{n\in\mathbb{Z}^{3}}|\widehat{u(\xi)}|^{2}$
$\displaystyle=\frac{1}{\mu_{1}|\alpha|^{2}}\|u\|^{2}_{L^{2}(Y)^{3}}.$

Let $v\in L^{2}_{\\#}(\alpha,Y)^{3}$. Then notice that

$\displaystyle\langle(-\mathcal{L}_{\alpha})^{-1}v,v\rangle$
$\displaystyle=\int_{Y}\mathbf{C}^{1}\mathcal{E}((-\mathcal{L}_{\alpha})^{-1}v):\overline{\mathcal{E}(v)}\;dx$
$\displaystyle=\int_{Y}-\nabla\cdot\mathbf{C}^{1}\mathcal{E}((-\mathcal{L}_{\alpha})^{-1}v)\cdot\overline{v}\;dx$
$\displaystyle=\int_{Y}-\mathcal{L}_{\alpha}((-\mathcal{L}_{\alpha})^{-1}v)\cdot\overline{v}\;dx$
$\displaystyle=\int_{Y}v\cdot\overline{v}\;dx=\|v\|_{L^{2}(Y)^{3}}.$

Now, from the Cauchy inequality we have

$\|v\|^{2}_{L^{2}(Y)^{3}}=\langle(-\mathcal{L}_{\alpha})^{-1}v,v\rangle\leq\|(-\mathcal{L}_{\alpha})^{-1}v\|\|v\|.$
(13.8)

Applying (13.7) we get

$\|(-\mathcal{L}_{\alpha})^{-1}v\|=\langle(-\mathcal{L}_{\alpha})^{-1}v,(-\mathcal{L}_{\alpha})^{-1}v\rangle^{1/2}=((-\mathcal{L}_{\alpha})^{-1}v,v)^{1/2}\leq\frac{\alpha^{-1}}{\sqrt{\mu}_{1}}\|v\|_{L^{2}(Y)^{3}}$
(13.9)

and the Poincare inequality follows from (13.8) and (13.9). ∎

To obtain the Poincare estimate for $\alpha=0$, we replace $\xi=2\pi n$ in the
above proof. Then following will be the corresponding inequality for
$\eqref{D7}$.

$((-\mathcal{L}_{0})^{-1}u,u)\leq\sum_{n\in\mathbb{Z}^{3}\setminus\\{0\\}}\frac{1}{\mu_{1}|\xi|^{2}}|\widehat{u(\xi)}|^{2}\leq\sum_{n\in\mathbb{Z}^{3}\setminus\\{0\\}}\frac{1}{\mu_{1}|2\pi
n|^{2}}|\widehat{u(\xi)}|^{2}$
$\displaystyle\leq\frac{1}{4\pi^{2}\mu_{1}}\sum_{n\in\mathbb{Z}^{3}\setminus\\{0\\}}|\widehat{u(\xi)}|^{2}$
$\displaystyle=\frac{1}{4\pi^{2}\mu_{1}}\|u\|^{2}_{L^{2}(Y)^{3}}.$

Hence for $\alpha=0$, the Poincare inequality becomes

$\|v\|_{L^{2}(Y)^{3}}\leq\frac{1}{2\pi\sqrt{\mu}_{1}}\|v\|.$

For any $v\in L^{2}_{\\#}(\alpha,Y)^{3}$, we apply (13.6) to find

$\displaystyle\|(A^{\alpha}(z)-A^{\alpha}(0))v\|_{L^{2}(Y)^{3}}$ (13.10)
$\displaystyle\leq\frac{|\alpha|^{-1}}{\sqrt{\mu_{1}}}\|(A^{\alpha}(z)-A^{\alpha}(0))v\|$
$\displaystyle\leq\frac{|\alpha|^{-1}}{\sqrt{\mu_{1}}}\|((T_{k}^{\alpha})^{-1}-P^{\alpha}_{2})(-\mathcal{L}_{\alpha})^{-1}v\|$
$\displaystyle\leq\frac{|\alpha|^{-1}}{\sqrt{\mu_{1}}}\|((T_{k}^{\alpha})^{-1}-P^{\alpha}_{2})\|_{{L[H^{1}_{\\#}(\alpha,Y)^{3}:H^{1}_{\\#}(\alpha,Y)^{3}]}}\|(-\mathcal{L}_{\alpha})^{-1}v\|.$

Applying (13.9) and (13.10) delivers the upper bound:

$\displaystyle\|(A^{\alpha}(z)-A^{\alpha}(0))\|_{L[L^{2}_{\\#}(\alpha,Y)^{3}:L^{2}_{\\#}(\alpha,Y)^{3}]}$
(13.11)
$\displaystyle\leq\frac{|\alpha|^{-2}}{\mu_{1}}\|((T_{k}^{\alpha})^{-1}-P^{\alpha}_{2})\|_{L[H^{1}_{\\#}(\alpha,Y)^{3}:H^{1}_{\\#}(\alpha,Y)^{3}]}.$

The next step is to obtain an upper bound on
$\|((T_{k}^{\alpha})^{-1}-P^{\alpha}_{2})\|_{L[H^{1}_{\\#}(\alpha,Y)^{3}:H^{1}_{\\#}(\alpha,Y)^{3}]}$.
For all $v\in H^{1}_{\\#}(\alpha,Y)^{3}$, we have

$\frac{\|((T_{k}^{\alpha})^{-1}-P^{\alpha}_{2})v\|}{\|v\|}\leq|z|\\{w_{1}+\sum_{i=1}^{n}\tilde{w}_{i}|(1/2+\tau_{i}(\alpha))+z(1/2-\tau_{i}(\alpha))|^{-2}\\}^{1/2}.$
(13.12)

where
$w_{1}:=\frac{\|P^{\alpha}_{1}v\|^{2}}{\|v\|^{2}},\tilde{w}_{i}:=\frac{\|P_{\tau_{i}(\alpha)}v\|^{2}}{\|v\|^{2}}$,
and $w_{1}+\sum_{i=1}^{n}\tilde{w}_{i}\leq 1$. Thus maximizing the right hand
side is equivalent to calculating

$\max_{w_{1}+\sum_{i=1}^{n}\tilde{w}_{i}\leq
1}\\{w_{1}+\sum_{i=1}^{n}\tilde{w}_{i}|(1/2+\tau_{i}(\alpha))+z(1/2-\tau_{i}(\alpha))|^{-2}\\}^{1/2}$
(13.13)
$\displaystyle=\sup\\{1,|(1/2+\tau_{i}(\alpha))+z(1/2-\tau_{i}(\alpha))|^{-2}\\}^{1/2}.$

Hence we maximize the function

$f(x)=|\frac{1}{2}+x+z(\frac{1}{2}-x)|^{-2}$ (13.14)

over $x\in[\tau^{-}(\alpha),\tau^{+}(\alpha)]$ for $z$ in a neighborhood about
the origin. Let $\operatorname{Re}(z)=u$, $\operatorname{Im}(z)=v$ and we
write

$\displaystyle f(x)$
$\displaystyle=|\frac{1}{2}+x+(u+iv)(\frac{1}{2}-x)|^{-2}$ (13.15)
$\displaystyle=((\frac{1}{2}+x+u(\frac{1}{2}-x))^{2}+v^{2}(\frac{1}{2}-x)^{2})^{-1}$
$\displaystyle\leq(\frac{1}{2}+x+u(\frac{1}{2}-x))^{-2}=g(\operatorname{Re}(z),x),$

to get the bound

$\|(T_{k}^{\alpha})^{-1}-P^{\alpha}_{2}\|_{L[H^{1}_{\\#}(\alpha,Y)^{3}:H^{1}_{\\#}(\alpha,Y)^{3}]}\leq|z|\sup\\{1,\sup_{x\in[\tau^{-}(\alpha),\tau^{+}(\alpha)]}g(u,x)\\}^{1/2}\\}.$
(13.16)

We now examine the poles of $g(u,x)$ and the sign of its partial derivative
$\partial_{x}g(u,x)$ when $|u|<1.$ If $\operatorname{Re}(z)=u$ is fixed, then
$g(u,x)=(\frac{1}{2}+x+u(\frac{1}{2}-x))^{-2}$ has a pole when
$(\frac{1}{2}+x)+u(\frac{1}{2}-x)=0.$ For $u$ fixed this occurs when

$\hat{x}=\hat{x}(u)=\frac{1}{2}\big{(}\frac{1+u}{u-1}\big{)}.$ (13.17)

On the other hand if, $x$ is fixed, $g$ has a pole at

$u=\frac{\frac{1}{2}+x}{x-\frac{1}{2}}.$ (13.18)

The sign of $\partial_{x}g$ is determined by the formula

$\partial_{x}g(u,x)=N/D,$ (13.19)

where $N=-2(1-u)^{2}x-(1-u^{2})$ and
$D=((\frac{1}{2}+x+u(\frac{1}{2}-x))^{4}\geq 0.$ Calculation shows that
$\partial_{x}g<0$ for $x>\hat{x}$, i.e. $g$ is decreasing
on$(\hat{x},\infty).$ Similarly, $\partial_{x}g>0$ for $x<\hat{x}$ and $g$ is
increasing on $(-\infty,\hat{x}).$
Now we identify all $u=\operatorname{Re(z)}$ for which $\hat{x}=\hat{x}(u)$
satisfies

$\hat{x}<\tau^{-}(\alpha)<0.$ (13.20)

For such $u$, the function $g(u,x)$ will be decreasing on
$[\tau^{-}(\alpha),\tau^{+}(\alpha)]$, so that $g(u,\tau^{-}(\alpha))\geq
g(u,x)$ for all $x\in[\tau^{-}(\alpha),\tau^{-}]$, providing an upper bound
for $\eqref{D16}$.

######  Lemma 13.2.

The set $U$ of $u\in\mathbb{R}$ for which
$-\frac{1}{2}<\hat{x}(u)<\tau^{-}(\alpha)<0$ is given by

$U:=[z^{*},1]$

where

$-\frac{(\mu_{1}+\lambda_{1})}{(3\mu_{1}+\lambda_{1})}\leq
z^{*}:=\frac{\tau^{-}(\alpha)+\frac{1}{2}}{\tau^{-}(\alpha)-\frac{1}{2}}<0.$

###### Proof.

Noting $\hat{x}=\hat{x}(u)=\frac{1}{2}\big{(}\frac{1+u}{u-1}\big{)}$, we
invert and write

$u=\frac{\frac{1}{2}+\hat{x}}{\hat{x}-\frac{1}{2}}$ (13.21)

We now show that

$z^{*}\leq u\leq 1$ (13.22)

for $\hat{x}\leq\tau^{-}(\alpha).$ Set
$h(\hat{x})=\frac{\frac{1}{2}+\hat{x}}{\hat{x}-\frac{1}{2}}.$ Then

$h^{{}^{\prime}}(\hat{x})=\frac{-1}{(\hat{x}-\frac{1}{2})^{2}},$ (13.23)

and so $h$ is decreasing on $(-\infty,\frac{1}{2}).$ Since
$\tau^{-}(\alpha)<\frac{1}{2},h$ attains a minimum over
$(-\infty,\tau^{-}(\alpha)]$ at $x=\tau^{-}(\alpha).$ Thus
$\hat{x}(u)\leq\tau^{-}(\alpha)$ implies

$z^{*}=\frac{\tau^{-}(\alpha)+\frac{1}{2}}{\tau^{-}(\alpha)-\frac{1}{2}}\leq
u\leq 1$ (13.24)

as desired. ∎

Combining Lemma (13.2) with inequality (13.16), noting that
$-|z|\leq\operatorname{Re}(z)\leq|z|$ and on rearranging terms we obtain the
following corollary.

######  Corollary 13.3.

For $|z|<|z^{*}|$:

$\|(A^{\alpha}(z)-A^{\alpha}(0))\|_{L[L^{2}_{\\#}(\alpha,Y)^{3}:L^{2}_{\\#}(\alpha,Y)^{3}]}\leq\frac{|\alpha|^{-2}}{\mu_{1}}|z|(-|z|-z^{*})^{-1}(\frac{1}{2}-\tau^{-}(\alpha))^{-1}.$
(13.25)

From Corollary 13.3, (13.3), (13.4) we easily seen that

$\displaystyle\|(A^{\alpha}(z)\rho-A^{\alpha}(0)\rho)R(\zeta,0)\|_{L[L^{2}_{\\#}(\alpha,Y)^{3}:L^{2}_{\\#}(\alpha,Y)^{3}]}\leq$
(13.26) $\displaystyle
B(\alpha,z)=\frac{|\alpha|^{-2}}{\mu_{1}}|z|(-|z|-z^{*})^{-1}(\frac{1}{2}-\tau^{-}(\alpha))^{-1}d^{-1}\|\rho\|_{L^{\infty}(Y)^{3}}.$

a straight forward calculation shows that $B(\alpha,z)<1$ for

$|z|<r^{*}:=\frac{\mu_{1}|\alpha|^{2}d|z^{*}|}{\frac{\|\rho\|_{L^{\infty}(Y)^{3}}}{\frac{1}{2}-\tau^{-}(\alpha)}+\mu_{1}|\alpha|^{2}d}$
(13.27)

and property 4 of Theorem 7.1 is established since $r^{*}<|z^{*}|$. Now we
establish properties 1 through 3 of Theorem 7.1. First note that inspection of
(4.5) shows that if (13.1) holds and if $\zeta\in\mathbb{C}$ belong to the
resolvent of $A^{\alpha}(0)\rho$ then it also belongs to the resolvent of
$A^{\alpha}(z)\rho$. Since (13.1) holds for $\zeta\in\Gamma$ and $|z|<r^{*}$,
property 1 of Theorem 7.1 follows. Formula (4.6) shows that $P(z)$ is analytic
in a neighborhood of $z=0$ determined by the condition that (13.1) holds for
$\zeta\in\Gamma$. The set $|z|<r^{*}$ lies inside the neighborhood and
property 2 of Theorem (7.1) is proved. The isomorphimsm expressed in property
3 of Theorem (7.1) follows directly Lemma 4.10 ([23], Chapter I §4) which is
also valid for Banach space.
The proof of Theorem 7.2 proceed along identical lines. To prove Theorem 7.2,
we need the following Poincare inequality between $L^{2}_{\\#}(0,Y)^{3}$ and
$H^{1}_{\\#}(0,Y)^{3}$.

######  Lemma 13.4.

$\|v\|_{L^{2}(Y)^{3}}\leq\frac{1}{2\pi\sqrt{\mu}_{1}}\|v\|.$ (13.28)

This inequality is established using (13.12) and proceeding using the same
steps as in the proof of Lemma 13.1. Using (13.28) in place of (13.6) we argue
as in the proof of Theorem 7.1 to show that

$\|(A^{0}(z)\rho-A^{0}(0)\rho)R(\zeta,0)\|_{L[L^{2}_{\\#}(0,Y)^{3}:L^{2}_{\\#}(0,Y)^{3}]}<1$
(13.29)

holds provided $|z|<r^{*}$, where $r^{*}$ is given by (7.7). This establishes
Theorem 7.2.
The error estimates presented in Theorem 7.4 are easily recovered from the
arguments in([23] Chapter II, §3); for completeness we restate them here. We
begin with the following application of Cauchy inequalities to the
coefficients $\beta_{n}^{\alpha}$ of (4.10) from ([23] Chapter II, §3, pg 88):

$|\beta_{n}^{\alpha}|\leq d(r^{*})^{-n}.$ (13.30)

It follows immediately that, for $|z|<r^{*}$,

$\Big{|}\hat{\beta}^{\alpha}(z)-\sum_{n=0}^{p}z^{n}\beta^{\alpha}_{n}\Big{|}\leq\sum_{n=p+1}^{\infty}|z|^{n}|\beta^{\alpha}_{n}|\leq\frac{d|z|^{p+1}}{(r^{*})^{p}(r^{*}-|z|)}.$
(13.31)

completing the proof.
For completeness we establish the boundedness and compactness of the operator
$B^{\alpha}(k)$.

######  Theorem 13.5.

The operator $B^{\alpha}(k):L^{2}_{\\#}(\alpha,Y)^{3}\mapsto
H^{1}_{\\#}(\alpha,Y)^{3}$ is bounded for $k\notin Z$.

We first prove the result for $\alpha\neq 0$. Let $v\in
L^{2}_{\\#}(\alpha,Y)^{3}$. Then,

$\displaystyle\|B^{\alpha}(k)v\|$
$\displaystyle=\|(T_{k}^{\alpha})^{-1}(-\mathcal{L}_{\alpha})^{-1}v\|$ (13.32)
$\displaystyle\leq\|(T_{k}^{\alpha})^{-1}\|_{L[H^{1}_{\\#}(\alpha,Y)^{3}:H^{1}_{\\#}(\alpha,Y)^{3}]}\|(-\mathcal{L}_{\alpha})^{-1}v\|$
$\displaystyle\leq\frac{\alpha^{-1}}{\sqrt{\mu}_{1}}\|(T_{k}^{\alpha})^{-1}\|_{L[H^{1}_{\\#}(\alpha,Y)^{3}:H^{1}_{\\#}(\alpha,Y)^{3}]}\|v\|_{L^{2}(Y)^{3}},$

where the last inequality follows from (13.9). Now we need to find an upper
estimate for
$\|(T_{k}^{\alpha})^{-1}\|_{L[H^{1}_{\\#}(\alpha,Y)^{3}:H^{1}_{\\#}(\alpha,Y)^{3}]}$.
Observe that

$\frac{\|(T_{k}^{\alpha})^{-1}v\|}{\|v\|}\leq\\{|z|^{2}w_{1}+w_{2}+|z|^{2}\sum_{i=1}^{\infty}|(1/2+\tau_{i}(\alpha))+z(1/2-\tau_{i}(\alpha))|^{-2}\tilde{w}_{i}\\}^{1/2},$
(13.33)

where
$w_{1}:=\frac{\|P_{1}v\|^{2}}{\|v\|^{2}},w_{2}:=\frac{\|P_{2}v\|^{2}}{\|v\|^{2}},\tilde{w}_{i}:=\frac{\|P_{\tau_{i}(\alpha)}v\|^{2}}{\|v\|^{2}}$.
Now Notice that $w_{1}+w_{2}+\sum_{i=1}^{\infty}\tilde{w}_{i}=1$, and
therefore we obtain the upper estimate

$\frac{\|(T_{k}^{\alpha})^{-1}v\|}{\|v\|}\leq M$ (13.34)

where

$M=\max\\{1,|z|^{2},\sup_{i}\\{|(1/2+\tau_{i}(\alpha))+z(1/2-\tau_{i}(\alpha))|^{-1}\\}\\},$
(13.35)

and this completes the proof. For $\alpha=0$ case, the proof is similar.

######  Remark 13.6.

The Poincare inequalities (13.6) and (13.28) together with Theorem 13.5 show
that $B^{\alpha}(k):L^{2}_{\\#}(\alpha,Y)^{3}\mapsto
L^{2}_{\\#}(\alpha,Y)^{3}$ is a bounded linear operator mapping
$L^{2}_{\\#}(\alpha,Y)^{3}$ into itself. The compact embedding of
$H^{1}_{\\#}(\alpha,Y)^{3}$ into $L^{2}_{\\#}(\alpha,Y)^{3}$ shows the
operator is compact on $L^{2}_{\\#}(\alpha,Y)^{3}$.

## Acknowledgements

This research work is supported in part by NSF Grants DMS-1813698 and
DMREF-1921707.

##  Appendix A Appendix: Proof of Lemma 2.1

###### Proof.

Let $x\in Y\setminus D.$ Since $\mathcal{E}(u)=0$ there, then $u$ has to be a
rigid motion in $Y\setminus D.$ However since rigid rotations are not
periodic, $u$ must be a rigid translation. Thus we write $u=c$ for $x\in
Y\setminus D$ where $c$ is a constant vector in $\mathbb{R}^{3}$. From the
continuity of $u$ across the boundary of $D$ we have $u|_{\partial D}=c$ so we
can express $u$ as $u=\tilde{u}+c$ for $x\in Y$ where
$\tilde{u}\in\tilde{H}^{1}_{0}(D)^{3}$. Then the condition $\int_{Y}\rho\
udx=0$ in $Y$ implies that
$c=-\langle\rho\rangle^{-1}\int_{D}\rho^{1}\,\tilde{u}\,dx$, where
$\langle\rho\rangle=\int_{Y}\,\rho\,dx$ and the Lemma follows. ∎

## References

  * [1] H. Ammari, H. Kang, H. Lee. Layer Potential Techniques in Spectral Analysis. American Mathematical Society, Providence, RI, 2009.
  * [2] H. Ammari, H. Kang, H. Lee. Asymptotic analysis of high-contrast phononic crystals and a criterion for the band gap opening.  Arch. Rational Mech.Anal. 193:679–714, 2009.
  * [3] A. Avila, G. Griso, B. Miara, and E. Rohan. Multiscale modeling of elastic waves: Theoretical justification and numerical simulation of band gaps. Multiscale Modeling & Simulation, SIAM, 7:1–21, 2008.
  * [4] K. Ando and H. Kang and Y. Miyanishi. Elastic Neumann-Poincaré operators on three dimensional smooth domains: Polynomal compactness and spectral structure. International Mathematics Research Notices, 12:3883-3900, 2019.
  * [5] K. Ando and Y.G. Ji and H. Kang and K. Kim. Spectral properties of the Neumann-Poincaré operator and cloaking by anomalous localized resonance for the elasto-static system. European J.Appl.Math. 29:189-225, 2018.
  * [6] O.P. Bruno and P.H. Leo. On the Stiffness of materials containing a disordered array of microscopic holes or hard inclusions.  Arch. Rational Mech.Anal. 121:303–338, 1993.
  * [7] O.P. Bruno. The effective conductivity of strongly heterogeneous composites. Proceedings: Mathematical and Physical Sciences, 433:353–381, 1991.
  * [8] G. Bouchitté and D. Felbacq, Homogenization near resonances and artificial magnetism from dielectrics. C. R. Acad. Sci. Paris, Ser. I 339:377–382, 2004.
  * [9] G. Bouchitté, C. Bourel, and D. Felbacq. Homogenization near resonances and artificial magnetism in three dimensional dielectric metamaterials. Archive for Rational Mechanics and Analysis, 225(3):1233–1277, 2017.
  * [10] C. Comi and J.-J. Marigo. Homogenization approach and Bloch-Floquet theory for band-gap prediction in 2D locally resonant metamaterials. J. of Elasticity, 139:61–90 2020.
  * [11] Y. Chen and R. Lipton. Resonance and double negative behavior in metamaterials. Arch. Rational Mech. Anal., 209:835–868, 2013.
  * [12] C. Conca and and M. Vanninathan. Homogenization of periodic structures via Bloch decomposition. SIAM J. Appl. Math., 57(6):1639–1659, 1997.
  * [13] G. Dell’Antonio, R. Figari and E. Orlandi. An approach through orthogonal projections to the study of inhomogeneous or random media with linear response. Annales de l’I.H.P. Physique théorique, 144:1-28, 1986.
  * [14] G. Duvaut and J.L. Lions. Inequalities in Mechanics and Physics. Springer-Verlag, Berlin, Heidelberg, New York, 1976.
  * [15] Y. Deng and H. Li and H. Liu. On spectral properties of Neumann-Poincaré operator and plasmonic resonances in 3D elastostatics. Journal of Sperctral Theory, 9(3):767-789, 2019.
  * [16] E. N. Economou and M. M. Sigalas. Classical wave propagation in periodic structures: Cermet versus network topology, Phys. Rev. B 48, 13434, 1993
  * [17] E. N. Economou and M. M. Sigalas. Stop bands for elastic waves in periodic composite materials. J. Accoust. Soc. Am., 95(4):1734–1740, 1994.
  * [18] R. Hempel and K. Lienau, Spectral properties of periodic media in the large coupling limit, Comm. Part. Diff. Equat. 25:1445–1470, 2000.
  * [19] D. Khavinson, M. Putinar, and H. Shapiro. Poincaré’s variational problem in potential theory. Archive for Rational Mechanics and Analysis, 185:143–184, 2007.
  * [20] Y. Kantor and D. Bergman, Elastostatic resonances - a new approach to the calculation of the effective elastic constants of composites, Journal of Mechanics and Physics of Solids Vol.30:355-376, 1982.
  * [21] T. Kato. On the convergence of the perturbation method, 1. Progr. Theor. Phys., 4:514-523, 1949.
  * [22] T. Kato. On the convergence of the perturbation method, 2. Progr. Theor. Phys., 5:95-101, 1950.
  * [23] T. Kato. Perturbation Theory for Linear Operators. Springer, Berlin, Heidelberg, Germany, 1995.
  * [24] M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani. Acoustic band structure of periodic elastic composites. Physics Review Letters, 71(13):2022–2025, 1993.
  * [25] M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani. Acoustic band structure of periodic elastic composites, Phys. Rev. Lett. 71, 2022, 1993.
  * [26] J. Li, F., Wu, H. Zhong, Y. Yao, and X. Zhang. Acoustic beam splitting in two-dimensional phononic crystals using self-collimation effect. Journal of Applied Physics, 118, 144903, 2015.
  * [27] F. Li and Y. Wang. Elastic wave propagation and localization in band gap materials: a review, Sci. China Phys. Mech. Astron. 55:1734–1746, 2012.
  * [28] R. Lipton and R. Viator. Bloch waves in crystals and periodic high contrast media. ESAIM: Mathematical Modeling and Numerical Analysis, 51:889–918, 2017.
  * [29] R. Lipton and R. Viator. Creating Band Gaps in Periodic Media. SIAM Multiscale Model. Simul., 15(4):1612–1650, 2017.
  * [30] G.W. Milton. The Theory of Composites. Cambridge University Press, Cambridge, 2002.
  * [31] J. Necas. Les Méthodes Directes en Théorie des Equations Elliptiques. Paris. Maison et Cie, 1967.
  * [32] M. Reed and B. Simon. Methods of Modern Mathematical Physics, Vol.1: Analysis, Academic Press, New York, 1980.
  * [33] E. Rohan, B. Miara, and F. Seifrt. Numerical simulation of acoustic band gaps in homogenized elastic composites. International Journal of Engineering Science, 47:573–594, 2009.
  * [34] S. P. Fortes, R. P. Lipton, and S. P. Shipman. Sub-wavelength plasmonic crystals: dispersion relations and effective Properties. Proc. R. Soc. Lond. A Mat., 2010.
  * [35] S. P. Fortes, R. P. Lipton, and S. P. Shipman. Convergent power series for fields in positive or negative high-contrast periodic media. Communications in Partial Differential Equations, Vol. 36 Issue(6): 1016-1043, 2011.
  * [36] M. M. Siglas and N. Garcia. Theoretical study of three dimensional elastic band gaps with the finite-difference time-domain method. Journal of Applied Physics 87, 3122, 2000.