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SPECTRAL PROPERTIES OF CLASSICAL WAVES
IN HIGH-CONTRAST PERIODIC MEDIA∗
A. FIGOTIN†AND P. KUCHMENT‡
SIAM J. APPL.MATH .c
1998 Society for Industrial and Applied Mathematics
Vol. 58, No. 2, pp. 683–702, April 1998 014
Abstract. We introduce and investigate the band gap structure of the frequency spectrum for
classical electromagnetic and acoustic waves in a high-contrast, two-component periodic medium.
The asymptotics with respect to the high-contrast is considered. The limit medium is described
in terms of appropriate self-adjoint operators and the convergence to the limit is proven. These
limit operators give an idea of the spectral structure and suggest new numerical approaches as well.
The results are obtained in arbitrary dimension and for rather general geometry of the medium.
In particular, two-dimensional (2D) photonic band gap structures and their acoustic analogues are
covered.
Key words. propagation of electromagnetic and acoustic waves, band gap structure of the
spectrum, periodic dielectrics, periodic acoustic media, high contrast
AMS subject classifications. 35B27, 73D25, 78A45
PII. S0036139996297249
1. Introduction. Wave propagation in inhomogeneous media is qualitatively
different than in homogeneous ones. The reason for this is the coherent multiple
scattering and interference. In periodic media the multiple scattering can manifest
itself in the rise of the so-called stop bands or gaps in the frequency spectrum. If the
wave frequency falls in a gap, then such a wave cannot propagate in the medium.
In disordered media, coherent multiple scattering and interference can result in ex-
istence of exponentially localized eigenmodes (the phenomenon known as Anderson
localization). The transport of classical electromagnetic, acoustic, and elastic waves
in periodic and disordered media has recently been studied very intensively, both the-
oretically and experimentally (see [12], [13], [20], [1], [11], [16], [21]). In particular,
artificial periodic structures have been studied with the goal of finding conditions
that favor band gaps in the spectrum. Fabrication of materials with prescribed spec-
tral gaps could lead to exciting opportunities in manufacturing entirely new and/or
improved devices (see [1], [13], [21], and references therein).
This paper is a continuation of the series of papers [4], [5], [6], [7], [8], [9], whose
goal is to develop some mathematical tools that can provide qualitative and quantita-
tive understanding of the dependence of spectral properties (such as band gap struc-
ture, eigenmodes, etc.) on the parameters of the periodic medium. High-contrast
media, due to their strong scattering properties, constitute a natural and important
subject of theoretical investigation. Let us describe the definition of the high-contrast
∗Received by the editors January 17, 1996; accepted for publication (in revised form) November
2, 1996. The U.S. Government is authorized to reproduce and distribute reprints for governmental
purposes notwithstanding any copyright notation thereon. The views and conclusions contained
herein are those of the authors and should not be interpreted as necessarily representing the official
policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research,
the National Science Foundation, or the U.S. Government.
http://www.siam.org/journals/siap/58-2/29724.html
†Mathematics Department, University of North Carolina, Charlotte, NC 28223 (figotin@
mosaic.uncc.edu). The research of this author was supported by U.S. Air Force, Air Force Office of
Scientific Research, and Air Force Material Command grant F49620-94-1-0172DEF.
‡Department of Mathematics and Statistics, Wichita State University, Wichita, KS 67260-0033
(kuchment@twsuvn.uc.twsu.edu). The research of this author was partially supported by Division
of Mathematical Sciences of the NSF grant DMS 910211 and by an NSF EPSCOR grant.
683
684 A. FIGOTIN AND P. KUCHMENT
9
5
9
5
FIG.1. The figure on the left shows a periodic pattern when the light component of the medium is
disconnected. The figure on the right shows a periodic pattern when the light component is connected.
media adopted in this paper. Suppose that spacial properties of the medium are de-
scribed by a scalar position–dependent function ε(x) (which is the dielectric constant
for dielectric media, or the compressibility for acoustic media). Suppose also that the
medium consists of two components and correspondingly the function ε(x) takes on
just two values, say 1 and >1. Let us call the component of the medium where
ε(x) = 1 “light” and the second one where ε(x)=“dense.” We define a high-
contrast medium as one having the following properties: (i) the dense component is
a set of thin slabs of the thickness α1 , so its volume fraction is of order α1;
(ii) the total “mass” of the dense component per unit volume α does not approach
zero (in particular, it can be very large).
An important example of high-contrast periodic media is the one where the dense
component forms a connected set, while the light component forms a disconnected one
(see Fig. 1). In this case we can view the medium as a periodic array of “air bubbles”
separated by a thin, dense film of the thickness α. The case of such 2D periodic
dielectric or acoustic media, when the domains are columns of square cross-section
and the periods form a simple cubic lattice, was studied in our papers [6], [7], [8], [9]
under the conditions that
α 1,α
21.(1)
Along with the detailed analysis of the band gap structure (in particular, exis-
tence of absolute gaps), we also discovered two types of eigenmodes that correspond
to two qualitatively different parts of the frequency spectrum. The eigenmodes of the
first type are well confined to the thin, dense component. The corresponding part of
the spectrum consists of small bands of order (α)−1,which alternate with small gaps
of the same order. The structure of those eigenmodes suggests that they presumably
arise due to the phenomenon of total internal reflection, which is well known in optics.
In that case the thin film of the dense material plays the role of a wave guide for these
SPECTRAL PROPERTIES OF CLASSICAL WAVES 685
eigenmodes. There also is another part of the spectrum that corresponds to waves
whose energy resides primarily in the light component. This part of the spectrum
asymptotically (i.e., under conditions in (1)) tends to be almost discrete with large
gaps between narrow spectral bands of order (α)−1(see [6], [7], and [8] for details).
Similar results also should hold for two-component media of rather arbitrary compli-
cated geometries as long as the conditions (1) are satisfied. The natural idea is that
for the high-contrast media (in particular, under the limit conditions (1)) the operator
governing the spectrum asymptotically splits into two operators that are responsible
for two different parts of the spectrum. One can expect from the results of [6], [7],
[8], [9] that the operator governing the waves that are concentrated in the “light”
component is either Dirichlet or Neumann (depending on polarization) Laplacian in
the single cell of the “light” material. It is not immediately clear, however, what
operator is responsible for the waves propagating along the thin, dense component.
It is even less clear how the asymptotic splitting occurs. Another problem is that the
technique used in [6], [7], [8], [9] is based on separation of variables and therefore is
not applicable to geometries more complicated than the cubic one.
In this paper we do the following:
1. Find a high-contrast limit operator (which in some interesting cases can be
represented as a version of the so-called Dirichlet-to-Neumann map) that
is responsible for the spectrum of waves residing in the thin film of dense
component.
2. Give a rigorous proof that this operator is the true limit of the operators with
finite and αand that it gives the correct approximation for the spectrum
in the case of high contrast.
3. Provide an approach that works in arbitrary dimension and for general ge-
ometry.
4. Reduce (1) to much less restrictive asymptotic conditions.
The results of the paper also suggest some new numerical approaches to the
analysis of photonic band gap structures and of their acoustic counterparts. The
numerical implementation, as well as analysis of the complete Maxwell system, will
be discussed elsewhere.
Now we will introduce the mathematical framework of the problem and formulate
the main results.
Let us consider first the case when the dense component is connected and forms
a thin film of the thickness α(see Fig. 1). Consider the Euclidean space Rd. Suppose
that the space is tiled periodically by polyhedra (polygons in the 2D case) Ωpwithout
overlapping: Rd=SpΩp. We also assume that this tiling is invariant with respect to
some lattice Γ (discrete group of translations with a compact fundamental domain)
in Rd. The simplest example is the cubic structure, when Γ is the integer lattice
Γ=Z
d={n=(n
1
, ..., nd)|nj∈Z,j=1, ..., d},
the fundamental polyhedron is
Ω0={x=(x
1
, ..., xd)|0≤xj≤1,j=1, ..., d},
and all other polyhedra are obtained by Γ translations of Ω0:
Ωp=p+Ω
0,p∈Γ.
686 A. FIGOTIN AND P. KUCHMENT
Let ∂Ω denote the boundary of the domain Ω. We introduce the following surface:
Σ=[
p
∂Ω
p
(in the 2D case this is just a periodic graph in the plane). This surface (graph) plays
the central role in our consideration.
Given a small parameter α<1, we construct the α-neighborhood Σαof Σ, namely
Σα=x∈Rd|dist (x, Σ) ≤α.
Then we define smaller domains Ωα
pas
Ωα
p=Ω
p
\Σ
α
,Ω
α=[
p
Ω
α
p,
and
Σα=Rd\Ωα,Rd=Σ
α[Ω
α
.
In fact, this particular construction of the α-neighborhood Σαis not very important.
For instance, if Ωpis convex, then we can choose a point zpin each of the domains
Ωpin such a way that this choice is invariant under the group Γ and introduce Ωα
pas
the result of the dilation with the factor (1 −α/2) and the center zpapplied to Ωp.
The general case, when the dense component of the medium is not connected
(see Fig. 1), can be described by a simple modification of the construction for the
connected case. Namely, instead of Σ we just take some of its periodic polyhedral
subset ˘
Σ⊂Σ. Then we deal with the sets ˘
Σ,˘
Σα, and ˘
Ωα
p=Ω
p
\˘
Σ
αinstead of Σ,Σα,
and Ωα
p, and most of the constructions do not change.
Suppose now that two parts of the space Rd=Σ
αSΩ
αare filled with two different
materials. We assume that ε(x)=1onΩ
α(i.e., Ωαconsists of the “air bubbles” Ωα
p)
and ε(x)=>>1 on the rest of the medium. Hence, the coefficient describing our
medium is defined by
ε(x)=1ifx∈Ω
α
,
if x∈Σα.
Note that, due to the well-known scaling properties of the Maxwell system and of the
equations for acoustic waves, the assumption that = 1 is not actually restrictive.
We will also need the auxiliary parameter
w=(α)−1.(2)
We will refer to the function ε(x) as to the dielectric function, although all considera-
tions are applicable to the acoustic case as well. In both cases of electromagnetic and
acoustic waves, the following spectral problem is important for describing propagation
of monochromatic waves (see, for instance, [9], [17]):
−∆ϕ(x)=λε(x)ϕ(x),x∈Rd.(3)
Here the spectral parameter represents the time frequency of the wave. For instance,
in the electromagnetic case λ=(ω/c)2, where ωis the time frequency and cis the
speed of light. In the case of electromagnetic waves, however, this equation does not
tell the whole story, since one needs here the complete Maxwell system. Furthermore,
for d= 2 this problem describes one possible wave polarization: electromagnetic
SPECTRAL PROPERTIES OF CLASSICAL WAVES 687
waves polarized in such a way that the electric field Eis normal to the plane of
the material. (The second polarization when the electric field is orthogonal to the
plane was considered in [6].) Anyway, in both acoustic and electromagnetic cases it is
important to understand the structure of the spectrum of this problem (in particular,
existence of spectral gaps). We will consider spectral properties of (3) under the
conditions significantly weaker than (1).
The case of the 2D square structure (see above) was thoroughly studied in [7],
[8], [9]. Let us assume now that d=2,Γ=Z
2(the integer lattice), and
Ω0={x=(x
1
,x
2)|0≤x
j≤1,j=1,2}.
The result of [7], [8], [9] was that under the conditions in (1) any finite part of the
spectrum splits into two subspectra with different asymptotic behavior. Namely, the
following statement was proven.
THEOREM 1. The spectrum σof the problem (3) (in the case of square geometry
in R2) splits into two subspectra
σ=σH∪σE
that satisfy the following properties:
(i) For any finite interval [0,N]and if the values α4/3and w=(α)−1are small
enough, the intersection of the subspectrum σHwith the interval [0,N]consists of the
following union of intervals:
[
n∈Z2
+,n6=0 h(πn)2−ρ−
n,(πn)2+ρ+
ni\[0,N].(4)
Here, for the nonnegative integer vector n=(n
1
,n
2)∈Z
2
+we denote n2=n2
1+n2
2.
The quantities ρ±
nsatisfy the estimate
ρ±
n≤cw
for all nsuch that the intersection in (4) is nonempty. All the intervals above contain
nonempty portions of the spectrum.
In other words, any finite portion of the subspectrum σHconcentrates around the
discrete set
[
n∈Z2
+,n6=0 n(πn)2o,
which in fact is the Dirichlet spectrum for the Laplacian in the “air bubble” Ω0.
(ii) Under the same conditions as in (i), the intersection of the subspectrum σE
with the interval [0,N]can be described as follows:
σE\[0,N]=
[
n∈Z
+
[a
n
,b
n]
\[0,N],(5)
where the intervals [an,b
n]are disjoint and their lengths and gaps between them satisfy
the estimates
c1w≤|a
n−b
n
|≤c
2
w,
c1w≤|b
n−1−a
n
|≤c
2
w.
688 A. FIGOTIN AND P. KUCHMENT
(iii) The Floquet–Bloch eigenmodes corresponding to the subspectrum σHare lo-
calized mostly in the “air bubbles” Ωp. The Floquet–Bloch eigenmodes corresponding
to the subspectrum σEmostly reside in the region Σαfilled with the optically dense
component.
The intervals [an,b
n] in (5) can be described as follows: they behave asymptoti-
cally as [ ewθ−
n,ewθ+
n], where
ew=w
1−−1=1
α(−1) ∼w,
and the constant (i.e., not depending on the parameters of the problem) intervals
[θ−
n,θ
+
n] can be well approximated by the set
[0,4] [([
n>0
[2πn, 2πn +π])
(6)
in the sense that
lim
n→∞ θ−
n−2πn=0,lim
n→∞ θ+
n−2πn −π=0.(7)
This result shows that under the conditions (1) any finite part of the spectrum
σconsists of two parts superimposed: σHis localized around the (discrete) Dirichlet
spectrum in the air bubble and is produced by the waves confined mostly to the “air
bubbles”; σE(“the fine spectrum”) has the fine structure of the set Sn≥0[θ−
n,θ
+
n] re-
scaled by the small parameter ew. In particular, one can open up an arbitrary number
of small gaps separated by small spectral bands in an arbitrary part of the spectrum.
One of our main aims here is to understand what operator is responsible for
the spectrum Sn≥0[θ−
n,θ
+
n] and why it is necessary to rescale this spectrum by the
small parameter ew. This problem will be resolved for general geometry in arbitrary
dimension and under weaker asymptotic conditions than (1), namely
α→0 and w=(α)−1→W6=∞.(8)
Let us formulate the main result of the paper. The spectrum we are interested in
can be naturally described in terms of the following operator pencil in L2(Rd):
P(θ)=−∆−θδΣ−θW,(9)
where ∆ is the Laplace operator in Rd,θis a scalar parameter (playing the role of
the spectral parameter), Wis defined in (8), and δΣis the δ-function of the surface Σ
(i.e., the distribution that integrates test functions over Σ). The rigorous definition
of P(θ) as a self-adjoint operator in L2(Rd) requires introduction of the appropriate
quadratic forms and some analysis, which are provided in Section 3. We will call the
set of values θ, for which the operator P(θ) is not invertible, the spectrum of the
pencil P(θ):
σ(P)=θ|P(θ) is not invertible in L2(Rd).
Let us introduce a new spectral parameter θrelated to the primary spectral parameter
λas follows:
θ=w−1λ=(α)−1λ.(10)
The next theorem is the main result of this paper.
SPECTRAL PROPERTIES OF CLASSICAL WAVES 689
THEOREM 2. Let [a, b]be any finite interval of the θ-axis. Then under the condi-
tions (8) the intersection of the spectrum of the problem (3) (in terms of the spectral
parameter θ(10)) with the interval [a, b]tends to the intersection with [a, b]of the
spectrum of the pencil (9). In particular, when W=0,i.e., when α →∞, any
finite part of the spectrum of (3) tends to the corresponding part of the spectrum of
the pencil −∆−θδΣ.
A corollary on convergence of Floquet–Bloch eigenfunctions will also be proven.
Let us compare this statement with the statement of Theorem 1. In that theorem
we assumed that W= 0. If we now look at the relation to (10) we will find that after
rescaling and switching to the parameter θin the limit (1) the spectrum σHlifts up
and disappears at infinity. Hence, the spectrum of the pencil −∆−θδΣis associated
only with the “fine” spectrum σE. In particular, we will show that the spectrum of
this pencil easily leads to the asymptotic relations (7) of the intervals [θ−
n,θ
+
n]. This
implies that the pencil −∆−θδΣis the one that governs propagation of waves confined
to the thin, dense component of the medium. The conclusion is that to understand
the spectrum and behavior of such waves one needs to study spectral properties of
this operator pencil.
Theorem 1 also provides detailed asymptotic representation for the spectrum σH,
which Theorem 2 does not. The latter, on the other hand, covers any dimension and
arbitrary geometry of the lattice of periods and of the “air bubbles” Ωp. Moreover,
the case when W6= 0 is now covered (it could not be included into Theorem 1, since
(1) requires that W= 0). Here the spectrum σHdoes not go to infinity anymore and
also is presented in the limit. This case of W6= 0 was not covered by Theorem 1 at
all. Another advantage of our current result is that it provides some new approaches
to numerics, which will be discussed elsewhere. We would like to mention also that
the proofs of the results of this paper are much simpler than the ones of [7], [8], and
[9] (on the other hand, the results from [9] cover any finite portion of the spectrum,
whereas Theorem 2 describes for W= 0 only a finite number of the first bands).
The spectrum of the pencil −∆−θδΣ−θW can sometimes be described in terms
of the spectrum of an interesting pseudodifferential operator on Σ (a kind of the
popular Dirichlet-to-Neumann map). This operator is defined as follows (the details
will be described in the subsequent sections): let ϕbe a function (from an appropriate
functional class) on the surface Σ. Using ϕas the boundary data, we solve in each of
the polygons Ωpthe Dirichlet boundary value problem for the equation −∆u=λu.
The resulting functions in different polygons match on the common boundaries, but
their normal derivatives do not match. The jump of the normal derivative across Σ
gives another function ψ. This way we determine the Dirichlet-to-Neumann operator
N(λ):ϕ→ψ. Now the spectrum of the pencil (9) can be found from the following
spectral problem for the operator N(λ):
θ∈σ(N(θW)).
Hence, the original spectral problem (3) in the limit (8) reduces to a spectral prob-
lem for the (self-adjoint) operator N. However, this reduction faces some additional
complications. First of all, the spectral parameter θenters in a nonlinear way, except
when W= 0 (in that case one gets the standard spectral problem for the operator
N(0)). Secondly, when λbelongs to the spectrum of the Dirichlet Laplacian in Ωp,
the operator N(λ) cannot be defined. Finally, rigorous definition of Nas of a self-
adjoint operator apparently requires additional conditions on the polygons. Hence,
consideration of the operator pencil (9) is preferable in comparison with using the
Dirichlet-to-Neumann map.
690 A. FIGOTIN AND P. KUCHMENT
We adopt the following notation:
L2(Ω) is the Hilbert space of complex valued square integrable functions on Ω.
H1(Ω) is the Sobolev space of functions uequipped with the norm
kuk2
H1(Ω) =ZΩ|u(x)|2+|∇u(x)|2dx.
H2(Ω) is the Sobolev space of functions uequipped with the norm
kuk2
H2(Ω) =ZΩ
|u(x)|2+|∇u(x)|2+X
i,j ∂2
ij u(x)2
dx.
2. Heuristic arguments. This section is devoted to some heuristic arguments
that show how two spectra, σHand σE, can arise. We make all these considerations
rigorous in the consequent sections of this paper.
Our main object is the problem
−∆ϕ(x)=λε(x)ϕ(x),x∈Rd
(11)
under the conditions in (8). Let us recall that ε(x) is very large closely to the surface
Σ and is equal to 1 outside of some vicinity of Σ. This leads to the natural idea
of singling out of ε(x) some function that in the limit (8) approaches the surface
delta-function δΣwhich is defined as the following distribution:
hδΣ,ϕi=ZΣϕ(x)dx.
If we introduce
ρα(x)=((−1) α)−1(ε(x)−1) = α−1in Σα,
0inΩ
α
,
(12)
then the following limit holds in the distributional sense:
ρα(x)→δΣwhen α→0.
We can now rewrite the original equation (11) in the form
−∆ϕ(x)=λ(−1) αρα(x)ϕ(x)+λϕ(x),(13)
or
−∆ϕ(x)=θρα(x)ϕ(x)+θewϕ(x),
where
θ=λ(−1) α=λew−1,λ=θew.(14)
In these new notations we obtain the equation
−∆ϕ(x)=θρα(x)ϕ(x)+θewϕ(x).(15)
If α→0 and w=(α)−1→W<∞, then wand eware equivalent, i.e.,
lim wew=1.
SPECTRAL PROPERTIES OF CLASSICAL WAVES 691
In the limit (8) our problem becomes
−∆ϕ(x)=θδΣ(x)ϕ(x)+θWϕ(x).(16)
So we can expect the result of Theorem 2 to be true.
Now our goal is to rewrite this problem in a different way (namely, using the
Dirichlet-to-Neumann map). Any distributional solution of this equation in Rdis
smooth in each of the “faces” Ωpand is continuous through the interfaces; however,
the derivatives do not match. Applying in the left-hand side the Laplace operator
(in the distributional sense) to this piecewise-smooth function, one gets the following
reformulation of (16):
−∆ϕ(x)=θWϕ(x)ineachofΩ
p
,
{The jump of ∂νϕ(x) across Σ}=θϕ(x)onΣ.
Here ∂νϕ(x) is the normal to Σ derivative of ϕ(x).
This leads to the definition of the following version of the Dirichlet-to-Neumann
map. Let us assume that λis not in the spectrum of the Dirichlet Laplacian on Ωp
and consider a reasonably well-behaved and decaying function ϕ(x) on Σ. Consider
now any “face” Ωpand solve the following Dirichlet problem:
−∆up(x)=λup(x),x∈Ωp;up(x)=ϕ(x),x∈∂Ω
p
.(17)
Let νpbe the outward normal vector to ∂Ωpand ∂pbe the corresponding normal
derivative. The following formula defines a function on Σ that we denote by Apϕ:
Apϕ(x)=∂
p
u
p
(x) for x∈∂Ωp,
0 for x∈Σ−∂Ωp.
Now we introduce the Dirichlet-to-Neumann map by the formula
NΣ(λ)ϕ(x)=X
p
A
p
ϕ(x),x∈Σ.(18)
Notice that due to the outward directions of the normal vectors, the two normal
derivatives from the opposite sides of Σ in (18) are actually subtracted one from
another. Observe now that the problem (16) can be rewritten as the eigenvalue
problem for our Dirichlet-to-Neumann map:
NΣ(θW)ϕ(x)=θϕ(x),x∈Σ.(19)
In particular, for W= 0 we get
NΣ(0)ϕ(x)=θϕ(x),x∈Σ.
The apparent conclusion is that the operator NΣgoverns propagation of waves con-
fined to the thin, dense component of the medium.
3. Definitions of the operators. In this section we give precise construction
of the operators
S(λ)=−∆−λε(x),(20)
P(θ, λ)=−∆−θδΣ−λ,(21)
and the Dirichlet-to-Neumann operator NΣ(λ). We will use the quadratic form ap-
proach, i.e., the domains of some of these operators will not be described explicitly.
692 A. FIGOTIN AND P. KUCHMENT
Defining the operator −∆−λε(x) is simple. On one hand, one can define this
operator according to its formula with the Sobolev space H2(Rd)⊂L2(Rd)asthe
domain. Using the theory of elliptic operators one can verify its self-adjointness. We
choose, however, an alternative way. Namely, we define the quadratic form of this
operator as
sλ[u]=k∇uk2
L2(Rd)−λZL2(Rd)ε(x)|u(x)|2dx.
The domain of this quadratic form is chosen to be H1(Rd). This form is closed, which
amounts to showing that for some large constant Cthe sum sλ[u]+Ckuk
2
L
2
(R
d
)is
equivalent to kuk2
H1(Rd), which is straightforward due to boundedness of ε(x).One can
also check that these two definitions of the operator S(λ) are equivalent, although we
will not use this fact (only the quadratic form approach will be used).
Let us turn now to the operator P(θ, λ). Since
P(θ, λ)=P(θ, 0) −λI,
we need to define only the operator P(θ, 0) in the space L2(Rd) of square integrable
functions in Rd. We define now a quadratic form p[u] with the domain DPconsisting
of all functions u∈L2(Rd) such that u|Ωp∈H1(Ωp) for any pand such that the
traces of u|Ωpfor different values of pmatch on common boundaries (these traces are
defined, due to the standard embedding theorems (see [3] and [18])). The natural
quadratic form for the operator P(θ)=P(θ, 0) is defined by
p[u]=X
p∈Γk∇uk2
L2(Ωp)−θku|Σk2
L2(Σ) ,u∈D
P.
L
EMMA 3. The form p[u]with the domain DPis closed in L2(Rd).
Proof. It is sufficient to show that for some Cthe expressions
p[u]+Ckuk
2
L
2(R
d)=X
p∈Γk∇uk2
L2(Ωp)−θku|Σk2
L2(Σ) +CX
p∈Γkuk2
L2(Ωp)
and kuk2
H1(Rd)are equivalent. Rewriting p[u]+Ckuk
2
L
2(R
d)we get
p[u]+Ckuk
2
L
2(R
d)=kuk
2
H
1(R
d)+(C−1) kuk2
L2(Rd)−θku|Σk2
L2(Σ) .
Now we have to show that
kuk2
H1(Rd)+Ckuk2
L2(Rd)−θku|Σk2
L2(Σ)
is equivalent to kuk2
H1(Rd)when the constant Cis large enough. First of all, according
to the embedding theorems (see for instance [18] and [3])
θku|Σk2
L2(Σ) ≤C1kuk2
H1(Rd),
which gives
kuk2
H1(Rd)+Ckuk2
L2(Rd)−θku|Σk2
L2(Σ)
≤(1+C1)kuk2
H1(Rd)+C||u||2
L2(Rd)≤C2kuk2
H1(Rd).
SPECTRAL PROPERTIES OF CLASSICAL WAVES 693
Now we have to show that for some constants C1,C
2
kuk2
H1(Rd)≤C
1kuk2
H1(Rd)+C
2kuk2
L
2(Rd)−C
1θku|Σk2
L
2(Σ) ,
or
ku|Σk2
L2(Σ) ≤C1−1
C1θkuk2
H1(Rd)+C2
C1θkuk2
L2(Rd).
In other words, we need to show that for any >0 there exists C>0 such that
||u|Σ||2
L2(Σ) ≤kuk2
H1(Rd)+Ckuk2
L2(Rd).(22)
Let us first consider a cylindric domain Ω = [0,a]×V. Then in the corresponding
coordinates x=(x, y)∈[0,a]×Vthe standard trace theorem says that
ku(0,y)k2
L
2(V)≤Ck∇uk2
L2(Ω) +Ckuk2
L2(Ω) .
Applying this inequality to the function eu(x)=u(C−1x, y) and changing the vari-
ables C−1x=x0afterward we get
ku(0,y)k2
L
2(V)≤k∇uk2
L2(Ω) +Ckuk2
L2(Ω) .
If the domain becomes cylindric after a linear change of variables, the same type of
estimate (with a different constant) holds. Since Ωpcan be covered by a finite number
of such subcylinders with bases in ∂Ωp, one can easily conclude that the estimate (22)
is true. This completes the proof of the lemma.
We will define now the Dirichlet-to-Neumann operator NΣ(λ). Our Hilbert space
will be L2(Σ), i.e., the space of measurable functions on Σ that are square integrable
with respect to the Lebesgue measure on Σ. Let a number λbe given. Assume that
λdoes not belong to the spectrum of the Dirichlet Laplacian in Ωp. We define the
following quadratic form in L2(Σ) :
nλ[ϕ]=ZR
d|∇u|2dx,
where the function u(x)inR
dis defined as follows:
−∆u(x)=λu(x)ineachΩ
p
,
u|
∂Ω
p=ϕ|
∂Ω
pfor each p.
(23)
In other words, the function u(x) solves the Dirichlet boundary value problem for the
operator (−∆−λ) on each “face” Ωpof Σ with the function ϕas the boundary data.
We need to describe the domain of definition of this form. The domain is the space
H1/2(Σ) defined as follows:
H1/2(Σ) = ϕ(x)onΣ|∃u∈H
1
(R
d
) such that u|Σ=ϕ.(24)
The norm of ϕ∈H1/2(Σ) is defined as the minimum of kuk2
H1(Rd)over all uin (24).
LEMMA 4. If λis not in the spectrum of the Dirichlet Laplacian in Ωpand if
Ωpis convex, then the form nλ[ϕ]with the domain H1/2(Σ) is defined and closed in
L2(Σ).
694 A. FIGOTIN AND P. KUCHMENT
Proof. We have already shown that for λnot in the spectrum of the Dirichlet
Laplacian in Ωpthe form is defined. It is sufficient to show now that the expressions
nλ[ϕ]+kϕk
2
L
2(Σ) and kϕk2
H1/2(Σ) are equivalent. Let u(x) be the solution of (23) in
eachofΩ
p
. Then
kuk2
H1(Rd)=kuk2
L2(Rd)+k∇uk2
L2(Rd)=kuk2
L2(Rd)+nλ[ϕ].
Due to standard theorems on elliptic boundary value problems (see for instance [10]
and [3]) applied to each “face” Ωp, the expressions kuk2
H1(Rd)and ku|Σk2
H1/2(Σ) =
kϕk2
H1/2(Σ) are equivalent. Using this and trace theorems we get the inequality
nλ[ϕ]+kϕk
2
L
2(Σ) ≤Ckuk2
H1(Rd)≤Ckϕk2
H1/2(Σ) ,
where Cdenotes different constants.
Now we have to show that
kϕk2
H1/2(Σ) ≤Cnλ[ϕ]+kϕk
2
L
2(Σ).
Due to the equivalence of norms mentioned above, we conclude that
kϕk2
H1/2(Σ) ≤Ckuk2
H1(Rd)=C(nλ[ϕ]+kuk
2
L
2(R
d)).
Hence, it is sufficient to show that
kuk2
L2(Rd)≤Cnλ[ϕ]+kϕk
2
L
2(Σ).
In other words, we have to show that
kuk2
L2(Ωp)≤Ck∇uk2
L2(Ωp)+kuk2
L2(∂Ωp)
for any p. Let
cp=1
Vol (Ωp)ZΩp
u(x )dx.
Then, according to the Poincar´e inequality [3],
ku−cpkH1(Ωp)≤Ck∇ukL2(Ωp).
Nowwehave
kuk
H
1
(Ωp)≤Ck∇ukL2(Ωp)+|cp|.
It remains to get an estimate
|cp|≤Ck∇ukL2(Ωp)+kukL2(∂Ωp),(25)
which can be justified as follows: if we have a cylindric domain Ω = [0,a]×V, then
in the corresponding coordinates x=(x, y)∈[0,a]×Vthe computation is
ZΩu(x)dx=Za
0ZVu(0,y)+Zx
0∂
x
u(τ, y)dτdydx
≤C||u||L2(V)+Za
0ZVZx
0|∂xu(τ,y)|dτ dydx ≤C||u||L2(V)+C||∇u||L2(Ω).
If the domain becomes cylindric after a linear change of variables, the same type
of estimate (with a different constant) holds. Since Ωpcan be covered by a finite
number of such subcylinders with bases in ∂Ωp, the estimate (25) holds. This finishes
the proof of the lemma.
Now all our self-adjoint operators are defined by their quadratic forms.
SPECTRAL PROPERTIES OF CLASSICAL WAVES 695
4. Proof of the main result. In this section we provide the proof of Theorem 2
formulated in the introduction. Namely, we show that under the asymptotic relation
(8) the spectrum of the operator pencil S(θew) defined by (20) and rescaled according
to (10) tends to the spectrum of the pencil (21). Under additional conditions discussed
above, the main statement also can be reformulated in terms of the Dirichlet-to-
Neumann operator N. For instance, when W= 0 we observe that after the rescaling
any part of the spectrum that is not infinitesimally close to zero goes to infinity, so
we are essentially discussing only the behavior of the lower part of the spectrum of
(11) of order w=(α)−1. Since the subspectrum σHis well separated from zero, it
disappears in the limit after rescaling, so we are describing the part σEonly. This
explains the nature of σEdiscovered in [7], [8], [9]. However, our proof now covers
any periodic geometry and arbitrary dimension and the condition W= 0 is relaxed
to arbitrary values of W<∞.
For the sake of simplicity we give the proof for the case of the integer lattice of
periods Γ = Zdonly. The entire proof can be carried out without any substantial
changes for general lattices. Notice that no constraints are imposed on the geometry
of the polyhedral domains Ωp.
THEOREM 5. After introducing a new rescaled spectral parameter θ=λew−1any
finite part of the spectrum of the problem
−∆u(x)=θραu(x)+θewu(x),x∈Rd
(26)
tends (under the conditions (8)) to the spectrum of the problem
−∆u(x)=θδΣu(x)+Wu(x),x∈Rd.(27)
Proof of Theorem 5.The spectrum of the problem (26) will be denoted as σα(S)
(although it actually depends on both and α). The spectrum of (27) will be denoted
as σα(P). Both problems (26) and (27) are periodic with respect to the lattice Zd.
The standard Floquet theory arguments (see, for instance, [14], [15], and [19]) lead to
the following representation for the spectra of these problems:
σ(P)=[
k
σ(P
k
),(28)
where σ(Pk) is the spectrum of the problem
−∆u=θδΣu+Wu,
u(x+γ)=e
ik·γu(x) for any γ∈Zdand x∈Rd.
(29)
The quasimomentum kcan be chosen in the cube (Brillouin zone)
B={k||k
j
|≤π, j =1, ..., d}.
The same assertion is true for the spectrum of (26):
σα=[
k
σα(k),(30)
where σα(k) is the spectrum of the problem
−∆u(x)=θρα(x)u(x)+θewu(x),
u(x+γ)=e
ik·γu(x) for any γ∈Zdand x∈Rd.
(31)
696 A. FIGOTIN AND P. KUCHMENT
After commuting with the exponent eik·x, one can reduce both problems to spectral
problems on the unit torus, i.e., on periodic functions
−∆ku=θδΣu+Wu(32)
and
−∆ku(x)=θρα(x)u(x)+θewu(x)(33)
correspondingly, where the functions u(x) are periodic with respect to the lattice Zd
(and therefore can be treated as functions on the torus Tn=Rn/Zd). The differential
expression −∆kis defined as
−∆k=(i∇−k)
2.
The expansions (30) and (28) enable us to consider the problems (32) and (33) on a
compact manifold instead of the entire space. In particular, we will prove appropriate
convergence of the spectra σα(k)toσ(P
k
).
Distributional solutions of both problems (32) and (33) automatically belong to
the Sobolev space H1(Td) (according to the definition of the operators). So, we will
consider only functions u(x) from this space. The expression (−∆k) determines a
bounded operator from the space H1(Td)intoH
−1
(T
d
). Let us denote this operator
by A(k):
A(k):H
1
(T
d
)→H
−1
(T
d
),A(k)u(x)=−∆
k
u(x),
where −∆kis applied in the distributional sense. A(k) depends continuously (and even
analytically) on the quasimomentum k. This follows immediately from the explicit
formula
A(k)=−∆−2ik ·∇+k
2.
This operator is invertible for all k∈Bexcept k= 0, which follows from the
Fourier expansions
u(x)=X
l∈Z
d
u
l
e
2πil·x,A(k)u(x)= X
l∈Z
d
(2πl +k)2ule2πil·x.
This “bad” point k= 0 will cause only some minor problems later on. Note that the
operator A(0) is a Fredholm operator of index zero with one-dimensional kernel and
cokernel (this follows from the Fourier series representation).
Consider now the following operators acting from H1(Td)intoH
−1
(T
d
):
Bα:H1(Td)→H−1(Td),B
α
u(x)=ρ
α
(x)u(x)+ ewu(x)
and
B:H1(Td)→H−1(Td),Bu(x)=δ
Σ
(x)u(x)+Wu(x).
LEMMA 6. The operators Bαand Bare compact operators from H1(Td)into
H−1(Td).
Proof. The operator u(x)→ρα(x)u(x) factors into the composition of two opera-
tors: the first one multiplies u(x)∈H1(Td)byρ
α
(x). This is obviously a bounded op-
erator from H1(Td)intoL
2
(T
d
). The second operator is just the embedding of L2(Td)
SPECTRAL PROPERTIES OF CLASSICAL WAVES 697
into H−1(Td), and it is compact due to standard embedding theorems. The embed-
ding operator u(x)→ewu(x) from H1(Td)intoH
−1
(T
d
) is also compact. Hence, the
operator Bαis compact.
Consider the operator Bnow. The term Wu(x) is proportional to the embedding
of L2(Td)intoH
−1
(T
d
) and, hence, is compact. The main term to discuss is δΣu.It
also factors in the way similar to the one we applied to Bα. The only difference is
that multiplication by δΣ(x) does not act as a continuous operator from H1(Td)into
L
2
(T
d
). However, it acts between H1(Td) and some H−q(Td) with q<1. Then the
compact embedding of H−q(Td)intoH
−1
(T
d
) completes the proof. So, let us find
the number q. We claim that any q∈(1
2,1) is suitable. Namely, we need to show that
for u∈H1(Td) the distribution δΣubelongs to H−q(Td). In other words, it suffices
to show that the value (δΣu, v) is defined for any u∈H1(Td) and v∈Hq(Td). Note
that
(δΣu, v)=ZΣ/Z
d
u(x)v(x)dx.(34)
Due to standard embedding theorems, the spaces H1(Td) and Hq(Td) for any q∈
(1
2,1) are continuously embedded into the space C(Σ/Zd) of continuous periodic func-
tions on the surface Σ. Hence, the expression (34) defines a continuous bilinear form
on H1(Td)×Hq(Td), which proves that δΣubelongs to H−q(Td) and finishes the
proof of the lemma.
Now our two spectral problems can be rewritten as
A(k)u=θBu, A(k)u=θBαu.
The previous lemma implies the following important statement.
COROLLARY 7. The operators A(k)−θB and A(k)−θBαare Fredholm operators
from H1(Td)into H−1(Td). The operator A(k)−θB (respectively, A(k)−θBα)is
invertible if and only if θ/∈σ(P
k
)(respectively, θ/∈σ
α
(k)).
Proof. Fredholmity follows from the Fredholmity of A(k) and from compactness of
Band Bα. Since A(k) has index zero, the same is true for A(k)−θB and A(k)−θBα.
A Fredholm operator of index zero is noninvertible only if it has a nontrivial kernel,
which finishes the proof of the corollary.
The next assertion is now trivial.
COROLLARY 8. The spectra σ(Pk)and σα(k)are discrete for any kand depend
continuously on k.
We also need the following statement.
LEMMA 9. Under the condition (8) we have the following convergence in the
operator norm from H1(Td)into H−1(Td):
kBα−BkH1→H−1=0.
Proof. Since the norm of the difference operator u→(ew−W)utends to zero
due to the asymptotic (8), we can neglect this part. Doing this, we estimate the norm
kBα−BkH1→H−1by its definition:
kBα−BkH1→H−1= sup
u6=0 k(ρα(x)−δΣ(x))u(x)kH−1
ku(x)kH1
= sup
kukH1=kvkH1=1 ((ρα−δΣ)u, v).
698 A. FIGOTIN AND P. KUCHMENT
So we have to estimate from the above expression ((ρα−δΣ)u, v) to show that it tends
to zero uniformly with respect to kukH1=kvkH1= 1. Polarization of this bilinear
form enables us to set u=vand to consider just
((ρα−δΣ)u, u)= 1
αZΣ
α|u(x)|
2dx −ZΣ|u(x)|2dσ.
Here dσ is the surface measure on Σ and we abuse notations identifying the sets Σ
and Σαwith their images Σ/Zdand Σα/Zdin the torus Td. Now let us assume first
that we are dealing with a cylindric domain Ω = [0,a]×Vand x=(x, y)∈[0,a]×V
are the corresponding coordinates. Then we can write
((ρα−δΣ)u, u)= 1
αZVZα
0(|u(x, y)|2−|u(0,y)|2)dxdy
≤1
αZVZα
0|u(x, y)−u(0,y)||u(x, y)+u(0,y)|dxdy
≤1
αZVZα
0Zα
0|∇u(τ,y)|dτ |u(x, y )+u(0,y)|dxdy
≤α1/2
αZα
0ZVZα
0|∇u(τ,y)|2dxdy1/2
dτ ×ZVZα
0|u(x, y)+u(0,y)|2dxdy1/2
.
The first integral in this product can be estimated from above when kukH1=1by
Cα. The second one under the same conditions is estimated by a constant due to
trace theorems. The whole expression is finally estimated by Cα1/2and therefore
tends to zero. This argument holds also if one can arrive to a cylindric domain after
a linear change of variables. Now, a neighborhood of Σ can be covered by a finite
number of such skew cylinders, which finishes the proof of the lemma.
We are ready now to complete the proof of the theorem. Let [a, b]⊂R\{0}
belong to the complement of σ(P). This means that [a, b]⊂R\σ(Pk) for any kfrom
the Brillouin zone B. Then we get an invertible family of operators
A(k)−θδΣ−θW :H1(Td)→H−1(Td)
continuously depending on parameters (k, θ)∈B×[a, b]. Since B×[a, b] is compact,
we conclude (see, for instance, [22]) that the inverse operator family
(A(k)−θδΣ−θW)−1:H−1(Td)→H1(Td)
is also continuous, and hence
M= sup
B×[a,b]
(A(k)−θδΣ)−1
H−1→H1<∞.
Now, according to the previous lemma we have that when αand ( ew−W) are small
enough, the norm of the difference
k(A(k)−θρα−θew)−(A(k)−θδΣ−θW)kH1→H−1
≤|θ|·(kρ
α−δ
Σk
H
1→H
−1+|ew−W|)
SPECTRAL PROPERTIES OF CLASSICAL WAVES 699
becomes small uniformly with respect to (k, θ)∈B×[a, b]. When this norm becomes
less than M−1, the operator family (A(k)−θρα) becomes invertible and hence the
interval [a, b] does not belong to the union of spectra σα(k) and therefore to the
spectrum σα. On the other hand, let a point θ6= 0 belong to the spectrum σ(P).
Then for any small β>0 there exists a value k06= 0 of the quasi momentum such
that there is a point θ06= 0 in the β-neighborhood of θsuch that θ0∈σ(Pk0).
This means that the point θ−1
0is an eigenvalue of the compact operator A(k0)−1δΣ
in H1(Td). Since the compact operators A(k0)−1ραconverge to A(k0)−1δΣin the
operator norm, we conclude that when αand ( ew−W) are small enough there is a
point θ1∈σα(k0)⊂σαin the 2β-vicinity of θ0. (Let us notice that here we had to
bypass the point k= 0, at which the operator A(k) is not invertible. We also notice
that this operator convergence is uniform with respect to k0in any compact in Bthat
does not contain the point k= 0.) This finishes the proof of Theorem 5.
Theorem 2 is a consequence of Theorem 5.
The end of the proof of Theorem 5 shows that in fact we have not only conver-
gence of spectra, but also convergence of corresponding Floquet–Bloch eigenfunctions.
Namely, the following assertion holds.
COROLLARY 10. Let us assume under the conditions of Theorem 5that θn(k)
is the nth eigenvalue of the problem (29). Suppose that ψn(k)∈H1(Td)are the
corresponding eigenfunctions. Let also Kbe a subcompact of the Brillouin zone B
such that 0/∈K. Then for any ξ>0there exist ˙
β>0and functions b
θn(k)∈R
and b
ψn(k)∈H1(Td)(k∈K) such that b
ψn(k)is an eigenfunction of the problem (31)
corresponding to the eigenvalue b
θn(k),
θn(k)−b
θn(k)<ξ,
and
ψn(k)−b
ψn(k)H1(Td)<ξ
when α<β,
W−(α)−1<β,and k∈K.
It is necessary to prove only the convergence of eigenfunctions. It follows from
the ending remarks of the previous proof about the uniform on Kconvergence of
compact operators A(k)−1(ρα+ew) to the compact operators A(k)−1(δΣ+W), since
convergence of compact operators implies not only convergence of spectra, but also
convergence of eigenfunctions.
5. The case of square 2D structures. Let us consider the case that was earlier
treated in the papers [7], [8], and [9], namely the 2D case of the square structure, when
Ω0is the unit square in the plane and when (1) holds. In this case Σ is the graph
described as the set of all points in the plane R2having at least one integer coordinate.
We will show that the result of this paper leads easily to the asymptotic (7).
Consider our spectral problem for the Dirichlet-to-Neumann map with a fixed
quasi momentum k:
−∆ku(x)=θδΣ(x)u(x),
where x=(x, y)∈[0,1] ×[0,1]. (We remind the reader that under the condition
(1) one has W= 0.) Due to the square structure, this problem allows separation of
variables. Namely, let u(x)=ϕ(x)ψ(y), where u(0) = ϕ(0) = ψ(0) = 1. Then we
700 A. FIGOTIN AND P. KUCHMENT
have
ψ(y)(i∂x−kx)2ϕ(x)+ϕ(x)(i∂y−ky)2ψ(y)=θϕ(x)ψ(y)δ(y)+θψ(y)ϕ(x)δ(x)
or
ψ(y)(i∂x−kx)2ϕ(x)−θψ(y)ϕ(x)δ(x)=−ϕ(x)(i∂y−ky)2ψ(y)−θϕ(x)ψ(y)δ(y),
(i∂x−kx)2ϕ(x)−θδ(x)ϕ(x)ψ(y)=−ϕ(x)(i∂y−ky)2ψ(y)−θδ(y)ψ(y).
Separating variables, we get
(i∂x−kx)2ϕ(x)−θδ(x)ϕ(x)=µ
2
ϕ(x),
(i∂y−ky)2ψ(y)−θδ(y)ψ(y)=−µ
2
ψ(y)
for some µ>0, or vice versa,
(i∂x−kx)2ϕ(x)−θδ(x)ϕ(x)=−µ
2
ϕ(x),
(i∂y−ky)2ψ(y)−θδ(y)ψ(y)=µ
2
ψ(y).
So we have to consider the system
(i∂x−kx)2ϕ(x)−θδ(x)ϕ(x)=µ
2
ϕ(x),
(i∂y−ky)2ψ(y)−θδ(y)ψ(y)=−µ
2
ψ(y)
with 1-periodic functions ϕ(x) and ψ(y). This leads to the following discriminant
equations (see, for instance, [2] and [19] for the general discussion of such equations,
and [7] and [9] for this particular case of δ-potentials):
cos µ−θ
2µsin µ= cos kx,cosh µ−θ
2µsinh µ= cos ky,
or symmetrically
cos µ−θ
2µsin µ= cos ky,cosh µ−θ
2µsinh µ= cos kx.
These two systems lead to the same spectra, so we will use only one of them.
In other words, the spectrum (values of θ) corresponds to solutions of the following
transcendental system:
cos µ−θ
2µsin µ≤1,
cosh µ−θ
2µsinh µ≤1,
(35)
where µ>0. One can check that solutions of this system produce a sequence of
intervals [θ−
n,θ
+
n] for n≥0 that have properties described in the introduction. Our
SPECTRAL PROPERTIES OF CLASSICAL WAVES 701
main theorem shows that these are exactly the intervals found earlier with a different
approach in [7], [8], and [9]. Now a simple analysis of the system (35) enables us to
get the asymptotic of [θ−
n,θ
+
n] when n→∞that was obtained in the papers quoted
above, avoiding the complicated considerations applied there. Namely, we can see
from the second inequality of (35) that when θ→∞the auxiliary parameter µmust
also tend to infinity. So, let now θ, µ →∞. Let us denote
a= cosh µ−θ
2µsinh µ
and assume that |a|≤1. Then
θ
2µ=a−cosh µ
sinh µ→1.
This leads to the conclusion that for µ→∞the set of µ-solutions of the inequality
cos µ−θ
2µsin µ≤1
approaches the set of solutions of the inequality |cos µ−sin µ|≤1, which consists of
intervals
"nπ, (2n+1)π
2#.
Taking into account the asymptotic θ
2µ→1 we conclude that the intervals of values
of θapproach
[2nπ, (2n+1)π],
which is exactly the result of [7] and [8] about the intervals [θ−
n,θ
+
n]. Numerical
experiments show that the convergence is very fast.
Acknowledgment. The second author expresses his gratitude to Dr. I. Pono-
maryov for useful comments.
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