#' lafigotin/papers/figotin kuchment band...key words. propagation of electromagnetic and...

SIAM J. ApPL. MATHVol. 56, No.6, pp. 1561-1620, December 1996

© 1996 Society for Industrial and Applied Mathematics003

BAND-GAP STRUCTURE OF SPECTRA OF PERIODICDIELECTRIC AND ACOUSTIC MEDIA. II. TWO-DIMENSIONAL

PROTONIC CRYSTALS*

A. FIGOTINt AND P. KUCHMENT+

Abstract. We consider the two-component dielectric medium consisting of a periodic array ofparallel air columns of square cross section embedded into a lossless optically dense host materialwith the dielectric constant ( > 1. We show that if ( is large enough and the relative distance 8between the air columns is such that (8 » 1 and (82 « 1, then the corresponding Maxwell operatorhas a series of gaps in the spectrum. We also provide some analytic formulas that enable one todetect location of bands and gaps in the spectrum. In particular, the typical wavelength exhibiting

a photonic band gap is 21rL #' where L is the distance between the axes of adjacent air columns.We also give some estimates on the space distribution of electric field energy for different eigenmodes.

Key words. propagation of electromagnetic and acoustic waves, band-gap structure of thespectrum, periodic dielectrics, periodic acoustic media

AMS subject classifications. 35B27, 73D25, 78A45

1. Introduetion. Periodic dielectrics, often referred to as photonic crystals,have attracted much attention in recent years. These crystals as conducting mediafor electromagnetic waves are expected to exhibit properties similar to those of solidcrystals as conducting media for electronic waves. It is well known from the quantumtheory of solids that the energy spectrum of an electron in a solid consists of bandsseparated by gaps (see, for instance, [AM]). This band-gap structure arises due toperiodicity of the underlying crystal, and it is common for many periodic differentialoperators (see the so-called Floquet-Bloch theory in [E], [K93], [RS]). It has been expected that the Maxwell operators that govern propagation of electromagnetic wavesthrough photonic crystals must exhibit similar properties. This is certainly a soundidea, but proof of existence of gaps for a specific periodic medium, not to mentionmore complete investigation of geometry of the spectrum as a function of parametersof the media, turns out to be a hard mathematical problem. The goal of our series ofpapers (the first part was [FK], and subsequent parts are planned) is to provide somebasic mathematical theory that describes how the band gaps arise in periodic dielectric and acoustic media. We also plan to consider problems of numerical estimates ofthe spectrum in a separate publication.

The idea of designing periodic dielectric materials that exhibit gaps in the spectrum was introduced in papers [V] and [J87]. The basic physical reason for the riseof gaps lies in the coherent multiple scattering and interference of waves (see, forinstance, [J91] and references therein). The tremendous number of applications thatare expected in optics and electronics (including high-efficiency lasers, laser diodes,etc.) warrants thorough investigation of this matter. Thus, one is not surprised by thepersistent attention that this problem has attracted. The most recent theoretical andexperimental results on the photonic band-gap structures are published in the seriesof papers [DE] and in the book [JMW]. The experimental results (see [vAL], [DG],

*Received by the editors April 28, 1995; accepted for publication (in revised form) October 1,1995.

tDepartment of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223.The research of this author was supported by U.S. Air Force grant F49620-94-1-0172DEF.

+Department of Mathematics and Statistics, Wichita State University, Wichita, KS 67260-0033.The research of this author was partially supported by NSF grant DMS 910211 and an NSF EPSCORgrant.

1561

1562 A. FIGOTIN AND P. KUCHMENT

[YG]) for periodic and disordered dielectrics indicate that the photonic gap regimecan be achieved for some nonhomogeneous materials. Fabrication of two-dim~nsional(2D) photonic crystals in the nanoscale was recently reported in [WVG].

Theory of 2D photonic crystals reduces to investigation of two scalar equations associated with the so-called E-polarized and H-polarized electromagnetic fields. Someof these equations have the same form as the equations for nonhomogeneous acousticmedia, which makes the development of theory of 2D photonic crystals even moreimportant. Various theoretical methods for 2D periodic dielectrics were developed in[VP] for sinusoidally and rectangularly modulated dielectric constants and in [PM]'[MM] for a periodic array of parallel dielectric rods of circular cross section, whoseintersections with a perpendicular plane form a triangular or square lattice. Similarstructures were studied theoretically and experimentally in [MBRJ] and [MPD]. Allthese results (see also [EZ], [LL], [ZS], [HCS], [SSE]) indicate the possibility of a gap(or pseudogap) regime for some two-component periodic dielectrics. Our analysis ofthe spectral attributes of 2D photonic crystals is based on a thorough analysis of therelevant one-dimensional equations. These equations are similar to those that arise indiffraction problems associated with dielectric and conducting lamellar gratings andwhich have been proven to be valuable and efficient tools in the analysis of the diffraction patterns [lB], [2B], [M], [PSS], [RMP]. The list of publications on the subject isalready rather lengthy, and we do not pretend to present the complete bibliography.

In this paper we consider the two-component dielectric medium that consists ofa periodic array of parallel air columns with square cross section imbedded into anoptically dense host material (see Fig. 1). We assume that the dielectric constantcontrast (the mass density, or the elasticity contrast in the acoustic case) tends toinfinity, and the distance between the air columns tends to zero. We show that ifthe rates of these two convergencies are properly related to each other, then one canguarantee existence of gaps in some prescribed parts of the spectrum. If there is nosuch coordination between the two rates, then one should expect the rise of pseudogapsrather than of real gaps [F94]. We also show that the spectrum associated with thatmedium splits into two subspectra, which we refer to as the E-subspectrum and theH-subspectrum. Both of them have band-gap structure, but the typical sizes of thebands and gaps for the E-subspectrum are asymptotically much smaller than those for

2D photonic crystal

FIG. 1. The dielectric constant equals 1 in the vertical columns, and it equals c » 1 in therest of the media. The distance between the centers of adjacent columns equals L, and the spacingbetween them equals 8L . We shall assume that L == 1 for simplicity.

TWO-DIMENSIONAL PHOTONIC CRYSTALS 1563

the H-subspectrum. The spatial energy distribution of the corresponding eigenmodesis also different for these subspectra. We have discovered recently that the statementon the spectral structure of the Maxwell operator that we made in our paper [FK94]apparently requires some modifications, since we overlooked there the possibility of asubspectrum analogous to the E-subspectrum for 2D photonic crystals.

This is a long paper, filled with plenty of technical details (mostly related to athorough analysis of some auxiliary one-dimensional problems). Unfortunately, thereis no natural way of splitting this work into smaller pieces. We hope that an outlineof main results and arguments provided below will help the reader to survive a messof approximations, asymptotic formulas, and estimates. Some additional technicaldetails omitted in the paper can be found in the preprint [FK95].

2. Statement of results. Our consideration of dielectric media is based on theMaxwell equations

(1) V7. D = 0,1 aB

D=cE,V7xE=---c at '

(2) V7. B = 0, V7 x H = ~ aD B = J-LH,c at '

where E and D are, respectively, the electric field and electric induction, Hand Bare, respectively, the magnetic field and magnetic induction, and c is the velocityof light. We shall assume that J-L == 1. (This condition holds for many dielectricmaterials of interest.) The dielectric constant c is assumed to be position dependent,i.e., c = c(x) ~ 1.

Let us introduce the standard basis vectors ej, 1 :::; j :::; 3, in the space R 3 andthe domains in R 2:

(3) x = [0, 1] x [0, 1], 08 = [8,1] x [8,1],

where 8 E (0,1) (see Fig. 2). We will also need the domain

(4) X' = [8/2,1+8/2] x [8/2,1+8/2].

Since we are going to consider 2D periodic media, we impose the following conditions on- the dielectric constant c(x), x = (Xl, X2, X3):

(5)

with the periodicity condition

(6)

where Z is the set of integers. For this kind of periodic media we will be interested inthe waves propagating along the plane (el' e2), which leads to the condition that themagnetic and electric fields Hand E depend on coordinates Xl and X2 only. That is,from now on we shall assume that

(7)

Let us describe first the relevant operators without rigorous technical details,which will be provided in §7. We define the Maxwell operator as acting on an appropriate space of pairs E(x), H(x) as follows:

(8) M [E(X)] = ci [c(x)-lV7 x H(X)] .H(x) - V7 x E(x)


x x'

FIG. 2. Domains X and X'. The darker area indicates the location of the optically dense component where the dielectric constant equals ( > 1. The light area corresponds to the air component.

The operator M is self-adjoint. If w, which can be viewed as the frequency of aharmonic eigenmode, is a point in the spectrum a(M), and the pair {Ew , H w } is thecorresponding generalized eigenfunction, then we have

(9)

(10)

These relations imply

c(X)-lV X Hw(x) = -i~Ew(x),c

.wV x Ew(x) = ~-Hw(x).

c

(11)

(12)

MHHw(x) = \1 x (c(x)-l\1 x Hw(x)) = (~) 2Hw(x),

MEEw(x) = c(x)-l\1 x (\1 x Ew(x)) = (~r Ew(x).

The operators MH and ME can be viewed as self-adjoint, nonnegative, unboundedoperators on appropriate subspaces (which incorporate the zero divergence conditionsfrom (1) and (2)) of the Hilbert spaces L2 (R3 , C 3 ) and L2 (R3 , C 3 , cdx), respectively.We denote here by L 2 (R3

, C3) the space of C 3-valued square integrable vector functions with the scalar product Jw* <.P dx, and by L 2 (R3 , C 3 , cdx) the analogous space,where the scalar product is defined as Jw*<.p cdx. We note that the spaces L2 (R3 , C3)and L 2 (R3 , C 3 , cdx) consist of the same functions, and their norms are equivalent.

It will be shown in §7.1 that for the case of 2D photonic crystals when the wavespropagate in the crystal's plane the following decomposition of spectra holds:

(13)

TWO-DIMENSIONAL PROTONIC CRYSTALS

where operators r c and 8 c are defined as

1565

(14)

(15)

f c H 3 == -(81c- 181 + 82c- 182 )H3 ,

8 c E3 == -c-1 (8f + 8~)E3

on appropriate domains in £2 (R3 , C 3 ) and £2 (R3 , C 3 , cdx), respectively. Here (J"(fc)and (J"(8c ) represent the spectra for H-polarized and E-polarized fields. Abusing thenotation we will sometimes mean by (J"(M) the set of values (w/c)2 instead of valuesW E (J"(M).

The relations between two parameters ( and 0 that lead to gaps in the spectrumroughly speaking are as follows:

(16) (03/

2 « 1, (0)> 1,

which, in particular, imply that

(17) (» 1, 0« 1.

In other words, our main results are of asymptotic nature. We shall also need theauxiliary parameters

In particular, the original parameters ( and 0 can be expressed as

The following trivial statement follows readily from the equalities (19).LEMMA 2.1. Suppose that f3 > C (or f31 > C, or f32 > C) for some positive

constant C. Then w ---7 0 implies ( ---7 00 and 0 ---7 O.For instance, the conditions (16) are met if

(20) ( == CO-(l+17 ) , o< TJ < 1/2.

The spectrum (J"(M) that we are interested in naturally splits into two subspectra(J"E(M) and (J"H(M), which we shall call, respectively, the E-subspectrum and theH-subspectrum (see Fig. 3). They can be described as follows. Let wbe defined as

(21)~ w ww--------

- 1 - wO - 1 - (-1 .

(In particular, under the conditions of the Lemma 2.1 we have w rv w.) Then

(22) (J"E(M) == U[wD~((,o),wD;t-((,o)],n~O

where the intervals [D~((,o),D:};((,o)]are disjoint for different values ofn, and theirendpoints can be described as follows:

(23) D;((,0)==27rn(1+x;), n21; Do((,o) ==0, Dt((,0)==4+xci,

(24) D;t- ((,0) - D;; ((,0) == 7r + TJ;t-, D;;+l ((,0) - D;; ((,0) == 27r + TJ;; .


Qualitative picture of2D H- and E- subspectra

o 20 40 60

E-subspectrum

80 100 120 140 160

FIG. 3. The horizontal axis is the spectral axis corresponding to (w / c)2. The black columnsabove the axis display the location of the H -subspectrum, whereas the columns below the axis displaythe location of the E-subspectrum.

The subspectrum (J"H(M) can be represented as

(25) (J"H(M) = U [(7rn)2 + Pn, (7rn)2 + pii]·n~o

Under appropriate conditions on ( and 8 the quantities X~, TJ~, and p~ are small.It turns out that under some conditions on ( and 8 any finite number of the

first bands and gaps in the spectrum can be found very accurately in terms of someabsolute set:

(26) (J"E = U[D~, D;t-] ~ [0,4] U{U [27rn, 27rn + 7r]} ,n~O n>O

where the intervals [D;;:-, D;t] do not intersect and their endpoints D~ do not dependon any parameters of the problem. Properties of these endpoints are described in,Lemmas 4.12 and 4.13.

Our main results are as follows.THEOREM 2.2. For any natural number No > 1 and any positive constant C

there exists a positive constant c such that for any (31 > C and for any w < c thefollowing is true:

(27) 0"(8c;)n I = [ U [WD~((,8),WD~((,8)]]nI, where 1= [0, 27rw(No-l)],O~n~No

TWO-DIMENSIONAL PROTONIC CRYSTALS 1567

and the endpoints D~((,8) can be approximated by the numbers D~ from (26) asfollows:

(29)

Under somewhat stronger conditions we can describe the location of the spectrumin any finite interval of the spectral axis.

THEOREM 2.3. For any positive constant N there exist positive constants C andc such that for any {32 > C and for any w < c the following is true:

(30) wE o-(M)n[O,Nj {=? (~)2 E [o-E(M)Uo-H(M)] n[O,NJ,

where aE(M) and aH(M) satisfy (22)-(25), the quantities X;, TJ~, and p~ satisfy therelations

(31) Ix;1 :::; 100N{3-1 + 6e-O.3n + 10N2{323wn-1,

(32) ITJ;I :::; 5.103 N{3-1 + 2· 103e-o.3n + 3n-1 + 301w,

and for some constant C1

(33) Ip~1 :::; C1w, Inl :::; JN1r-1

.

In particular, the spectrum of values of (w / c)2 has adjacent bands and gaps of approximately same size (of order w) .

Specific estimates for the quantities D~, x;, TJ~, and p~ as well as the structureof eigenmodes are discussed in the subsequent sections.

Summary of the results. 1. The spectrum a(M) of the Maxwell operatorconsists of two subspectra of different structure, which we shall refer to as the Esubspectrum and the H-subspectrum.

2. The E-subspectrum is generated exclusively by E-polarized fields. It consistsof almost equal bands and gaps of the approximate width 1rW. The typical dimensionof the wavelength exhibiting a photonic band gap is 21rLV(S, where L is the distancebetween the centers of adjacent air columns.

3. The H-subspectrum arises due to both polarizations. H-subspectra for bothpolarizations are about the same and consist of bands of width of order w which arelocated approximately at the points (1rn)2. The only difference is that H-subspectrumfor E-polarization starts approximately at the point 1r2 , whereas H-subspectrum forH-polarization starts at O.

4. The E-subspectrum and the H -subspectrum differ also in the way the energy ofthe corresponding eigenmodes is distributed in the space. The eigenmodes associatedwith the E-subspectrum have most of the electric field energy residing in the areaswhere the dielectric constant is large (i.e., in the optically dense component), althoughthe relative area occupied by this component is small. On the contrary, most of theelectric field energy for the eigenmodes associated with the H-subspectrum for Epolarized fields resides in the air columns, where the dielectric constant is 1. Theeigenmodes associated with E-subspectrum are exponentially confined to the dielectricand can be viewed as a manifestation of the total reflection phenomenon.

We finish this section by adopting the following notation:L~(X) == [L2 (X)]3 == L2 (X, dx, C 3 );

L~ c(X) == [L2 (X, c(x) dx)]3 == L2 (X, c(x) dx, C 3 );

l~ '== l2(Z3, C 3 ) is the Hilbert space of C 3-valued functions cI>(m) , m E Z3 suchthat 11cI>11 2

== Em 1cI>(m)1 2 < 00;


hr is the Hilbert space of C 3-valued functions <1>(m) , m E Z3 such that 11<1>11 2 ==I:m(l + m2 )1<1>(m)1 2 < 00;

8m , n is the Kronecker symbol;Z+ is the set of nonnegative integers;Zi is the Cartesian product of d copies of Z+.For a vector n == (n1' ... nd) E R d, n > 0 means that all components of the vector

n are strictly positive, and Inl == In11 + ... + Ind I·3. An outline of the main arguments. The proofs of the main statements

are lengthy. In this section we give an outline of the basic arguments that lead toTheorem 2.3. We believe that this will be a reader's guide to the rest of the text. Theproof of Theorem 2.2 is similar and in fact simpler.

The problem of finding the spectrum a(M) reduces to the two eigenvalue problemsfor a scalar function <p(x), x == (Xl, X2) E R 2

:

r c<p (x) == - [01 c-1 ( X ) 01 + 02c-1 ( x) 02] <p (x) == A<p (x),

8 c<p(x) == -c(x)-l~<p(x) == A<p(X),

where the operators r c and 8 c act in the Hilbert spaces L 2(R2) and L 2(R2,c(x) dx),respectively. (See §7.1 for details.) We have investigated the spectrum of the operatorrein the first part [FK] of the paper. Hence, here we need to investigate the spectrumof the operator 8 c only. It differs significantly from the spectrum of the operator r c'

Since the dielectric constant c(x) is a periodic function, the standard FloquetBloch theory (see, for instance, [E], [RS] , [K82] , [K93]) says that we need to considerthe eigenvalue problem on the fundamental cell X:

8k,c<P(x) == -c(x)-l~<p(x) == A<p(X) ,

where <p satisfies the boundary conditions

xEX,

(34)Ok ok

<p(1,x2) == e~ 1<p(0,X2), <P(X1' 1) == e~ 2<P(X1,0),

\7<p(1,x2) == e~kl\7<p(0,X2)' \7<p(X1' 1) == eik2 \7<p(X1'0)

for k == (k 1 , k2 ), kj E [0, 21r]. The spectrum of the operator 8 c coincides with theunion (for all k) of spectra of these problems.

It turns out that the spectrum of the operator 8 c will remain almost the same ifwe replace the original periodic function c(x) defined in (5) with

where the auxiliary function pC,8(Y), Y E R, is defined as

{( - 1/2 if 0:::; Y < 8,

P(,8(Y) = 1/2 if 8::::: Y < 1, p(y) = p(y + n), nEZ, Y E R.

It is easy to see that the functions c(p, x) and c(x) coincide all over the space R 2 ,

except for small squares with sides of length 8. The advantage of the new eigenvalueproblem is that it has separable variables. This, in turn, will enable us to reduce theproblem to a one-dimensional one. Thus, let us now consider the following eigenvalueproblem:

(35) xEX.


The circle on the top of an operator will indicate that we deal with a separate variablescase. Let us rewrite the problem (35) as

(36) Sk A<p == -~<p - AV(p, X)<p == A<p,, V(p,x) == L (PC,8(XJ ) -1/2).j=I,2

This problem can be viewed as the spectral problem for the Schrodinger operator withthe potential AV (i.e., the potential depends on the spectral parameter). To reducethat to a standard spectral problem we consider the eigenvalues of the followingSchrodinger operator satisfying the cyclic boundary conditions (34):

(37)

where m is an index which counts the eigenvalues ~(m, k, A) of the Schrodinger operator Sk A' This operator involves a new independent parameter A 2: O. Hence theeigenvalu~s of the problem (35) are the solutions for A 2: 0 of the equations

(38) ~(m, k, A) == A,

with the index m counting these eigenvalues.

Equation (38) is our key equation. To understand its solutions we needto analyze the eigenvalues of the Schrodinger operator Sk A as functions of all thevariables (, 8, A, and k. At this point we observe that the potential AV is the sum oftwo one-dimensional ones, and therefore

(40)

where ~m(k, A), m == 0,1, ... , are the eigenvalues of the Floquet-Bloch componentQ K, A of the one-dimensional Schrodinger operator

82

QA == -- - A(pr 8(Y) - 1/2).8y2 ,:-,

Hence, the key equation (38) can be rewritten as follows:

(41)

We note now that pc, 8(Y) - 1/2 is a well potential which equals zero everywhereexcept for an interval of the length 8, where it is equal to ( - 1 » 1 (in view of theconditions (16)). It is important that this potential appears in the operator Q K, A withthe negative amplitude - A.

We will use the following parameterization of the operator Q K, A:

(42) Q(D, 8)~(y) == -~I/(y) - qD, 8(Y)~(Y), yER,

q(y) == q(y + n), nEZ, y E R.(43)

where q is the periodic potential that depends on two parameters D 2: 0 and 8 > 0 as

{D8-1 if 0:::; y < 8,

QD,8(Y)=. 0 if8:S;y<1,

(44)

The idea behind introducing the parameter D is that the potential ql, 8 approximatesthe Dirac's 8-function when 8 is small. Comparing the values of the potentials A(p 1/2) and qD,8 in the area 0 :::; y < 8, we get the relations

A(( -1) == D8- 1, D == A(8( - 8), or

A == Dw (1- (-1)-1 == Dw.


Let us denote by Q ("'" D, 8) the Floquet-Bloch component of the operatorQ (D, 8) that corresponds to the quasi momentum"" (i.e., the operator Q (D, 8) withthe boundary conditions (34)). Then the following simple relation holds: '

When

(45)

iV-I == w- 1 - 8.

w == ((8)-1 -+ 0,

the limit operator

a2

Q(D) = - 8y

2 - D L 8(y - n),nEZ

where 8(y) is the Dirac 8-function, naturally arises. The eigenvalues of the operatorsQ(D,8) and Q(D) are denoted respectively by ~n("'" D, 8), ~n("'" D), n == 0,1, ....These eigenvalues are positive for n ~ 1. It is convenient to introduce the quantities

It is well known from the spectral theory of one-dimensional periodic operators (see,for instance, [E], [RS]) that in order to find the endpoints of bands of the spectrumit is sufficient to consider just the values"" == 0, 1r. We introduce the quantities

(47) Jl;~(D, 8) == max Jl;n("'" D, 8),K=O,1r

Jl;;; (D, 8) == min Jl;n("'" D, 8),K=O,1r

which correspond to the endpoints of bands of the spectrum of the periodicSchrodinger operators Q(D, 8) (see Fig. 4). Both eigenvalues ~o("'" D, 8) and ~o("'" D)

12r~================================:L==Y3 +31t

10

Y2 +21t

Vo

o 10D

15 20 25

FIG. 4. The graphs of the functions J-L~(D) = lim6--+0J-L~(D,8), vj(D) = lim6--+0Vj(D,8),j = 0,1. The limit functions equal J-Lt (D) = 1rn + l'n (D), J-L;; (D) = 1rn, vJ (D) ~ D /2.


are negative for D > 4. They play a significant role in all further analysis. We alsointroduce the real-valued functions

(48)

and the corresponding functions Va (D) and VI (D) for the operator Q (D). The functionVI is defined when ~a(7r, D, 8) < 0 (i.e., for D > 4) (see Fig. 4).

Analysis shows that structure of equations (41) significantly depends on whetherone of the indices mj is zero. Thus, we should investigate separately two seriesof equations depending on this circumstance. These two sets of eq~ations producecorrespondingly two series of bands in the spectrum of the operator eC,8. The firstseries of bands, which we call the E-subspectrum is associated with the equationswhere at least one of the indices mj is zero. The H-subspectrum corresponds to theequations where both mj are positive. The first subset of equations (41), when atleast one mj is zero, after switching to the parameter D == A w- 1 can be rewritten as

(49) ~a(kl,D,8)+~n(k2,D,8)==Dw, w==w(1-w8)-I, n~O.

For a fixed n ~ 0 the set of values Dw for all solutions D to this equation (when bothk j run over the interval [0,1r]) yields the nth band of the E-subspectrum. We denotethis band, which is an interval, by

[wD~((,8), wD~((, 8)].

The second set of equations takes the form

(50)

For a fixed n == (nl' n2) > 0 the set of solutions A to this equation (when both kj runover the interval [0,1r]) yields the nth band of the H-subspectrum. We show that

(51)

(52)

and

Jl;:;; (D, 8) ~ 1rn, Jl;~(D, 8) ~ 1r(n + 1n(D)), n ~ 1, where2 D

1n(D) ~ 1 - - arctan -, D ~ 0, n ~ 1,1r 21rn

(53) va(D, 8) == (1 + 0((3-1 ))D/2, va(D,8) - VI (D, 8) rv e- D / 2.

Using (46), (47), and (48), we can rewrite (49) for the endpoints D~((, 8) of the nthband as

(54) fJ,;;(D,8) = Vv3(D,8) + DiiJ = vo(D,8)(1 + O(w)) for D;;((,8),

(55) fJ,;;(D,8) .= Vvr(D, 8) + DiiJ = VI (D, 8)(1 + O(w)) for D;;((, 8).

Based on these relationships it may be shown that the following approximations to(54) and (55) hold:

(56)

(57)

1rn~(1+0((3-1))D/2 forD~((,8),

1r(n + 1 - ~ arctan~) ~ (1 + 0((3-1))D/2 for D~((, 8).1r 21rn


Solutions to these equations can be found in the form of asymptotic formulas (23)and (24), which give our description of the E -subspectrum. (All these considerationswill be made precise later in the paper.)

As far as equations (50) are concerned, we note that under the condition (45) weobtain for n 2: 1 from (51) and (52) the approximate formula

(58)

This immediately implies that the solution An to (50) is approximately equal to(rrn)2 + O(w). This observation leads directly to the formula (25) for the Hsubspectrum. Now we see that the H-subspectrum has a structure similar to thespectrum of the operator r€, and hence the spectrum of the Maxwell operator Msatisfies the statement of the main theorem, Theorem 2.3.

The rest of the paper contains thorough spectral analysis of the Schrodinger operators Q(D,8), Q(D), analysis of functions f.J;~(D, 8), vo(D,8), VI (D, 8), justificationsof all approximations, precise definitions of the Maxwell operators, and the corresponding version of the Floquet theory.

4. One-dimensional Schrodinger operator with a periodic point potential. We consider in this section some auxiliary one-dimensional problems. Based onthis study we will be able to obtain necessary estimates for location of the spectrumof the relevant 2D and three-dimensional scalar models.

Let us consider the one-dimensional Schrodinger operator Q(D,8) with the wellpotential defined in (42), (43). Let D 2: 0 and 8 ---+ O. Then the potential converges(in the distributional sense) to the function D EnEZ 8(y - n), where 8(y) is the Diracdelta function. The "limit" operator (and we will not make the meaning of the word"limit" precise) should be

(59)d2

Q(D) = - dy

2 - D L 8(y - n).nEZ

Note that this operator can be considered as a periodic version of Kronig-Pennymodel. We have to describe the domain and the action of this operator. Its naturaldomain consists of functions ~(x) E L 2 (R) such that ~ belongs to the Sobolev spaceH 2[n, n + 1] for any integer n and satisfies the conditions

n

(61) ~(n + 0) == ~(n - 0), ~/(n + 0) - ~/(n - 0) == -D~(n), n E Z.

The operator acts on functions from the domain as -d2/ dy2 away from the integer

points. The condition (61) can be explained as follows. Let us have Q(D, 8)~ == f forsome function f in L 2 . Then, solving this ordinary differential equation on the interval[n, n + 8] with the initial data ~(n) and ~'(n) and taking the limit in ~'(n+ 8) when8 ---+ 0, one easily gets (61). It can be shown that the operator Q(D) is self-adjointand can also be defined by the quadratic form QD[~] == J I~' 12 - DEnEZI~(n)12 withthe domain HI (R) .

4.1. Eigenvalues and spectra. Let us consider the standard cyclic (Floquet)boundary value problems related to the operator Q(D):

(62) Q(~, D)~(y) == ~~(y), 0:::; y :::; 1,

TWO-DIMENSIONAL PROTONIC CRYSTALS

where the operator Q("", D) is defined as

1573

on functions ~(x) E L 2 ,loc(R) such that ~ E H 2 [n,n + 1] for any integer n and ~

satisfies the conditions (61) and the Floquet conditions

(63)

Due to (63), the function ~ is completely defined by its restriction to the interval[0,1]. The operator Q("", D) can now be described as follows:

(64) Q("", D)~(y) == _~"(y),

~(1) == eiK~(O),

0:::; y:::; 1,

~/(1) == eiK[~/(O) +D~(O)], -Jr:::; "":::; Jr,

where the last condition for the derivative of ~ combines (63) with (61). Consideringthe equation conjugate to (62), we find that the spectra of the operators Q("", D) andQ( -"", D) are identical and the corresponding eigenfunctions are conjugate. Thus fromnow on we shall consider just the values 0 :::; "" :::; Jr. Let ~o("'" D) :::; ~1("", D) :::; ... bethe set of eigenvalues of problem (62) counted with their multiplicity. Our equation(62) can be rewritten as a system with respect to the vector Up(x) == (~(X),~/(X)).

The corresponding monodromy matrix is

(65)== [cos~ - D~-1/2 sin~ ~-1/2 sin~]

Wp -Dcos~ - ~sin~ cos~ .

For a given k the sequence of eigenvalues ~l, l 2: 0, can be found as solutions of thediscriminant equation (see [E], [RS])

(66)

TrWp == 2 cos "", or

cos If. - DfC sin If. = cos K,.

2v~

If ~(x) is an eigenfunction of problem (62), then WpUp(O) == eiKUp(O). It is not hardto find the corresponding eigenvector Up:

[~(-0) ] [ ~-1/2 sin ~1/2 ]

(67) Up(-O)= 'ljJ'(-O) = -cose/2+eiK:+D~-1/2sine/2

[~-1/2sin~1/2 ]

== (D/2)~-1/2 sin ~1/2 + i sin ""

(where the last equality follows using (66)). It also follows from (61) and (63) that

(68)

(69)

~(+O) == ~(-O),

~(1) == eiK~( -0),

~/(+O) == ~/(-O) - D~(-O),

~' (1) == eiK~' (-0),

[~(+O) ] [ ~-1/2 sin ~1/2 ]

(70) Up(+O)= 'ljJ'(+O) = -(D/2)~-1/2sine/2+isinh; .

The left side ~D(~) == cos~ - (D/2~) sin~ of (66) is an entire function of theparameter ~. In particular, it is well defined for all ~ E R. For positive and negative


(71)

values of ~ (66) can be rewritten respectively as

COSf-l- D sinf-l = COSK, where ~ = f-l2,2M

(72)D.

cosh v - - sinh v == cos "", where ~ == _v2•

2v

The function cos M- (D12M) sin M, which is an entire function with respect to M, willbe denoted by A(D, M) and will be used frequently later.

For D == °the eigenvalues ~l can be easily found, namely,

(73) ~z\D=O == MTID=O' where

MoID=o == "", M2m-1ID=O == -"" + 21fm,

M2mID=o==""+21fm, m==I,2, ....

We will need some properties of the function dD(e) for ~ E R that we will collect inthe next lemma.

LEMMA 4.1. The function dD(e) possesses the following properties:(i) It is monotonic in any interval where IdD(~)1 < 1.

(ii) It has infinitely many oscillations for positive values of e(according to (i),all peaks must happen outside of the horizontal strip Idl < 1.)

(iii) If we denote by eo(k, D) :::; e1 (k, D) :::; ... the sequence of solutions (countedwith multiplicities) of the equation dD(e) == cos k, k E [0,1f], then eo(O, D) < 0, andfor D > 4 also ~o(1f, D) < 0. For all other values of l the solutions el are positive.

(iv) For any D ~ °we have

(v) All functions ~l(k, D) are monotonically increasing functions of k for evenvalues of l and decreasing ones for odd values of l.

(vi) All functions ~l(k, D) are decreasing functions of D.(vii) ~l (k, D) =1= em(k, D) for l =1= m.Proof. A standard proof from the Floquet theory (see, for instance, Theorem

XII.89 in [RS]) can be applied to show (i).Statement (ii) follows from the fact that the graph of dD (~) hits infinitely many

times both lines dD == ±1:

This equality can be checked by direct substitution into the definition of dD(~)'To prove (iii) we introduce as before a new positive parameter v such that ~ == -v2

and the function

AD(v) = cosh v - D sinh v.2v

Considering k == 0, we get the equation AD(v) == 1, which can be rewritten as

Dtanhvv==-------

2 (1 - (cosh v) -1 ) ,

D v D (1 + e-V)

v = "2 coth 2' or v = 2 (1 _ e-v ) ·


The function in the right side of the last equality decreases from infinity to (asymptotically) D/2. This shows existence of a unique solution Va, or ~a(O,D) = -(va,)2.

Now consider the case k == 1r, i.e., the case of the equation V == ~(({;:~:]. Here thefunction on the right side starts with the zero value at v = 0, and increases (beingconvex) to D /2. Then the existence of a negative solution is governed by D: for largevalues of D (namely, for D > 4) the tangent line at the origin to the graph will makethe angle more than 1r/ 4 with the v axis, so there will be a unique intersection of twographs. When D < 4, there is no intersection beyond the origin. This proves (iii).

Let us turn to the proof of (iv) now. First of all, the equalities (75) show that thenumbers [2m1r]2 are among the members of the sequence ~l (0, D) and the numbers[(2m -1)1r]2 are among the members of the sequence ~l(1r, D). The only problem is tocheck the correspondence between the values of m and l, which amounts to checkingthe locations of other solutions of the equations dD (~) = ±1. We already know thesituation for negative values of ~, so we assume now that ~ 2 o. Consider first the caseof k == 0 (or of the equation dD(~) == 1). We introduce here an important auxiliaryfunction of the argument fJ; == ~, which we will use rather often:

(76) <PD(fJ;) == arctan[D/(2fJ;)], fJ; 2 0, 0 ~ <PD(fJ;) ~ 1r/2.

In terms of this function we can express dD(~) as follows:

(77) ~D(~) = COS(fL + CPD(fL)) .cos(<PD (fJ;))

Then the equation dD(~) = 1 reduces to

cos (J,t + <PD (J,t)) == cos(<PD (J,t) ) or

J,t + <PD(J,t) == ±<pD(J,t) + 21rm.

Choosing different signs in the right side, we get two equations: fJ; == 21rm, whichprovides exactly the points ~ == [2m1r]2 that we already know, and

(78) fJ; + 2<pD(fJ;) == 21rm or 21rm - fJ; == 2<pD(fJ;).

The graphs of the functions dD = 21rm - fJ; provide a set of parallel lines in the(dD' fJ;)-plane. Consider the graph of the function 2<pD(fJ;). First of all, 2<pD(fJ;) > 0and 2<pD(0) == 1r. Therefore, it can intersect only the lines for m == 1,2, .... The lowestof these lines starts for fJ; == 0 at the height 21r. Now, differentiation shows that thefunction 2<pD (fJ;) is decreasing and convex downward. Then the first intersection canhappen not earlier than at fJ; == 1r. Direct calculation shows that the absolute valueof the derivative of 2<pD (fJ;) for fJ; > 1r is less than one. This means that the equation(78) has exactly one solution in each interval (21rm, 21r(m + 1)). Hence, the numbers(21rm)2 for m == 1,2, ... provide all roots ~2m(0,D). A similar consideration worksfor the case k == 1r or for the equation dD(~) == -1.

The statement (v) is an obvious consequence of (i)-(iii).The statement (vi) follows from the following inequality between the correspond

ing quadratic forms: QD1 ~ QD2 for D 1 > D 2 . Validity of (vii) follows from thefact that 8J-LA(D,fJ;) (where, as before, A(D,fJ;) == dD(fJ;2)) is not equal to zero at thepoints 1rm (m == 1,2, ... ), and from (iv). This finishes the proof. 0

We conclude that for any positive D

~2m-2(0, D) ~ ~2m-2(K,D) ~ ~2m-2(1r,D)

< ~2m-l(1r,D) ~ ~2m-l(K,D) ~ ~2m-l(0,D)

(79) < ~2m(0, D) :::; ~2m(K, D) :::; ~2m(1r, D), m == 1,2, ....


We also notice from (66) that according to the standard scheme of Floquet theory(which can be easily implemented in our situation), the spectrum a(Q(D)) coincideswith the set of the eigenvalues en(K, D) for all n ~ 0, 0 :::; K :::; 7r. Therefore, thisspectrum can be described as follows.

LEMMA 4.2. For any D > 0 the spectrum a(Q(D)) consists of nonoverlappingintervals Iz(Q (D )) such that

(80) a(Q(D)) = UIn(Q(D)),n2:0

Io(Q(D)) = [-(vO(D))2, eo (D)],

(81)

where

n = 1,2, ... ,

(82)

(83)

(84)

vo(D) = J -eo(O, D), eo(D) = eo('rr, D),

1r(n + I'n(D)) = { J~n(O, D) if n is odd, n = 1,2, ... ,Jen(1r, D) if n is even,

eo(O,D) < 0, eO(1r,D) < 1r2, 0 < I'n(D) < 1.

In particular,

(85) n = 1,2, ....

Proof The statements of the lemma follow readily from Lemma 4.1. 0We shall study dependence of the spectrum a(Q(D)) on D. To do this we rewrite

the basic equations (71) and (72) for the spectra for K = 0 and for K = 1r as

(86)

(87)D tanh v

v = 2 (1 =f (cosh v) -1 ) ,v ~ 0,

respectively. Here "-" and "+" correspond to K = 0 and k = 1r. We also remind thereader that (86) and (87) define the endpoints of bands of the spectrum.

4.2. Negative eigenvalues. Let us consider the lowest band of a(Q(D)),Io(Q(D) = [-(vo(D))2,eo(1r,D)].

LEMMA 4.3. The functions vo(D) and eo(1r, D) possess the following propertiesfor D ~ 0:

(i) vo(D) is the unique solution of the equation

(88)D (1 + e- V

)

v=----2(1- e-v )'

v ~ 0,

and is a strictly increasing function of D.(ii) eo(1r, D) is a continuous strictly decreasing function of D such that

(89) 0:::; eO(1r,D):::; 1r2 forO:::; D:::; 4,

eO(1r, D) < 0 for D > 4.


Besides, for D > 4 we have eO(1r,D) = -(vI(D))2, where vI(D)« vo(D)) is theunique positive solution of the equation

(90)D (1 - e-V

)

v=----2 (1 + e-v ) ,

V> O,D > 4.

(91)

(92)

vI(D) is a strictly increasing function of D.(iii) The functions vj(D), j = 0, 1, satisfy the inequalities

D D D D(1+e-D/2)2" < vo(D) < 2" coth"4 = 2 (1 _ e-D / 2 ) , D > 0,

D { 2e-D/2 } D D D2" 1- 1 + e-D/2 = 2" tanh "4 < vl(D) < 2" for D > 4.

Proof (i) Equation (88) has been discussed already in the proof of Lemma4.1. Monotonicity of Vo follows from the decreasing character of the function(e V + l)/(eV

- 1).(ii) It was mentioned in the proof of the Lemma 4.1 that eo (1r, 4) = 0, eo (1r, D) 2

o for D E [0,4], and eO(1r,D) < 0 for D > 4. Besides, eO(1r,D) < el(1r,D) = 1r2.Equation (90) has also been established earlier. Increasing behavior of VI (D) followsfrom (90) and from convexity and monotonicity of (e V

- l)/(eV + 1).

(iii) Since ~g~:~:l > D/2, (88) implies that vo(D) > D/2. This inequality

and monotonicity of (1 + e- V )/(l - e-V), together with (88), imply that vo(D) <

D (l+e-D

/2

) I ··1 . D(I-e- V

) D/2 t (D) D/2 d"2 (l-e-D/2)· n a SImI ar manner, sInce 2(I+e- v ) < , we ge VI < , anthen from (90) vI(D) > (D/2)tanhD/4. This finishes the proof. 0

LEMMA 4.4. The following relationships hold:

(93)

(94)

8DVj(D) 2 0.3, j = 0,1, for D 2 4,

8Deo(1r, D)I D=4 = -3.

Proof. Differentiating (88) and (90) with respect to D and then using the inequalities (91) and (92), we easily obtain

8 V (D) _ ~ coth(vo(D)/2) > ~ coth[(D/4)coth(D/4)]D 0 - 21+ (D/4) sinh-2 (vo(D)/2) - 21+ (D/4) sinh-2 (D/4) ,

8DV1(D) = ~ tanh (vl(D)/2) > ~ tanh [(D/4)tanh(D/4)] .

21- (D/4) cosh-2 (vI(D)/2) - 21- (D/4) cosh-2 (D/4)

The function in the right side of the first inequality is increasing for D 2 4, and itsvalue at D = 4 is not less than 0.3. The function in the right side of the secondinequality decreases to 1/2. This gives (93). To prove (94) we differentiate (66)with respect to D and then evaluate the result at D = 4, taking into account thate= o. 0

4.3. Positive eigenvalues. Let us turn now to the bands In(Q(D)),n 2 1,lying above 1r2 according to Lemma 4.2. As follows from that lemma, analysis ofthese bands amounts to analysis of the functions ryn(D). Proof of Lemma 4.1 showsthat J-l = 1r(n + ryn(D)) satisfies the equation

(95)


LEMMA 4.5. The functions ryn(D), n ~ 1, have the following properties:(i) ryn(D) are strictly decreasing functions of D > 0 taking values in the interval

(0,1). For each n ~ 1, ryn(D) is the unique solution of the equation

(96)

In addition,

(97)

(98)

In particular,

(99)

o< ryn(D) < ryn+l(D) < 1,2 D

'Yn(D) = 1 - - arctan [ (D)]'7r 27r n + ryn

2 D 2 D1 - - arctan - < 'Yn(D) < 1 - - arctan ( 1) .

7r 27rn 7r 27r n +

(100)

(101)

(ii) For any positive constant C

. 2 Chm ryn(Cn) == 1 - - arctan -2 'n~oo 7r 7r

4n'Yn(D) = D [1 + O(D- 1

)] , D -HXJ.

Proof Equation (96) follows from (95) and from our analysis of ez in Lemma 4.1.Then we notice that the function <PD(J-l) is strictly decreasing with respect to J-l > 0and increasing with respect to D > o. In addition,

(102)

n ~ 1,

This implies

8 1-8 [7rry + 2<pD(7r(n + ry))] ~ 7r - - > 0,

ry n

which means that the left side of the equation (96) is a strictly increasing functionof the argument ry for any n ~ 1. We also note that this left side takes at ry == 0the value 2<pD(7rn) < 7r and at ry == 1 takes the value 7r + 2<pD(7r(n + 1)) > 7r. Thisimplies that (96) has the unique solution ryn(D). The inequalities (97) follow readilyfrom the monotonicity properties of the function <po The identity (98) is a straightconsequence of (96) and (76). The inequalities (99) easily follow from (98) and (97).The relation (100), in turn, follows straightforwardly from (99). Now consider (101).We will show it only for odd values of n. The considerations for the even values aresimilar. Consider (78), which determines e2m-l. From this equation, using the Taylorexpansion of arctan(l/x) at 00, we get

Jt = 7l'(2m - 1) + 47l'(2~ - 1) + O(D-2 ),

which finishes the proof. 0LEMMA 4.6. Let p and a be real numbers such that Ipl < 1 and lal ~ 47r. Then

the following inequality holds for n ~ 1:

(103) Iryn ((1 + p)27rn + a) - 1/21 ~ Ipl + 3/n.


Proof. Using the relations (98), (97) and the elementary inequality Iarctan(tl) arctan(t2) I :::; It I - t2 1, we easily obtain

21 (1+p)21rn+a II1'n ((1 + p)2rrn + a) - 1/2/ = ;: arctan 2rr [n + I'n(D)] - arctan 1

:::; Ipl + (Ial + 21r)/(21rn),

which together with lal :::; 41r leads to (103). 0It will be convenient to introduce the quantities

(104) J-l:;;(D) = min J-ln(K, D) = 1rn,~=o, 7r

J-l~(D) = max J-ln(K, D) = 1r(n + f!n(D)),~=o, 7r

n ? 1.

(105)

Note that (101) implies that for any n ~ 1

4nJt~(D) = rrn + D [1 + O(D-1

)] , D~oo.

4.4. Perturbations of the spectrum of the Schrodinger operator withpoint potential. We plan to investigate later on the location of the spectrum of theSchrodinger operator Q(K, D, 8) as a perturbation of the operator Q(K, D). To do thiswe need to establish some properties of (71) for K = 0, 1r and to study behavior ofcorresponding solutions under some "small" perturbations of the equation.

Let, as before, A(D, J-l) = cos J-l- ~ sinJ-l, D ? O.LEMMA 4.7. Suppose that K = 0 or k = 1r. Let us rewrite (71) in the form

(106) A(D,J-l) = COSK (= ±1).

Let J-ll (K, D) :::; J-l2(K, D) :::; ... be its solutions in the interval [1r, (0), and 8J-LA(D, J-l) isthe derivative of the function A with respect to J-l. Then 8J-LA(D, J-lz) =1= 0 for l = 1,2, ....Besides, for J-l = 1rn, and J-l = 1r(n + f!n(D)), n ? 1, we have

(107)

In particular,

(108)

Proof. The validity of (107) for J-l = 1rn is the subject of simple direct calculation,so let us prove it for J-l = 1r(n + f!n(D)). It was established in the proof of Lemma 4.1that

(109)

Differentiating, we get

A(D ) = cos (J-l + 'PD(J-l)), J-l () ,cos 'PD J-l

J-lER.


(110)

Combining terms with 8j.L'PD(J-l), and performing some simple trigonometric conversion, we get

() A(D, p,) = - sin (p, + <PD(p,)) _ sinp, () <PD(p,).j.L cos 'PD(J-l) cos2 'PD(J-l) j.L

In the first term we multiply and divide by sin( 'PD(J-l)) and then use the equalitytan( 'PD(J-l)) = D j2J-l in both terms. This leads to

() A(D, p,) = D [_ sin (~ + <PD(p,)) _ sin p,] .j.L 2J-l SIn 'PD(J-l) J-l

Since k = 0 or k = 7r, (106) leads to Icos (J-l + 'PD(J-l)) I = Icos 'PD(J-l) I, which in turnimplies Isin (J-l + 'PD(J-l)) I = Isin 'PD(J-l) I· In fact, more precisely

sin (~+ <Ptjp,)) = (_l)n if p, = 7l'(n + 'Yn(D)).SIn 'PD J-l

This gives (107), which implies the inequality (108). 0We are interested now in what happens if we are "almost" at a solution of (106),

Le., if IA(D, J-l) I is close to 1.LEMMA 4.8. Let us assume that for some J-l 2:: 2

(111)

(112)

Then

IIA(D,J-l)1-11 ~ a < 1,

tan 'PD(J-l) = D j(2J-l) 2:: lX.

(113) 1{)/LA(D,p,)1 ~ [(1- a) (1- a(2 + a)(l + a-2)) - ~] (D/2p,).

If lX and a satisfy the condition

(114)

in particular, if

(115)

then

(116)

6J(i ~ lX < 1,

Proof We note first that based on the J.epresentation (109) we can rewrite (111)as

(117)II

cos (J-l + 'PD(J-l)) 1- 11 ~ a.cos 'PD(J-l)

Then formula (110) together with the condition J-l 2:: 2 gives

TWO-DIMENSIONAL PHOTONIC CRYSTALS

Representing

I

sin (J-l + 'PD(J-l)) I = Icos (J-l + 'PD(J-l)) 1.1 tan (J-l + 'PD(J-l)) Isin 'PD(J-l) cos 'PD(J-l) tan 'PD(J-l )

and using (117), we get

Now,

1581

(1 - a) I tan (J-l + 'PD(J-l)) I = (1 - a) (tan2

(J-l + 'PD(J-l)) - 1) + 1tan 'PD(J-l) tan2 'PD(J-l)

2:: (1 - a) (1 -I tan2

(J-l + 'PD(J-l)) - 11)tan2 'PD(J-l)

=(1_a)(1_ICOS2

(J-l+'PD(J-l))_11 1 )cos2 'PD(J-l) sin2 'PD(J-l)

2:: (1 - a) (1 - a(2 + a)(l + a-2)) .

We used here the elementary inequality IvT+U - 11 ~ lui for lui ~ 1 and the estimate

This gives (113). The inequality (116) follows straightforwardly from (113) and(114). 0

Given a positive a we introduce the set

(118) A = {J-l2:: 2: IIA(D,J-l)1-11 ~ a}

and represent it as A = UZ>l Jz, where Jz are some disjoint intervals.LEMMA 4.9. Suppose that numbers a, N, and D satisfy the condition

(119) min(Dj(7N), 1) > 6J(i.

Then there exist only a finite number L of intervals Jz = [1Jz, 1Jt] such thatJz n[O, N] =1= 0, and each of them contains exactly one solution 1Jz of exactly one ofthe equations (106). In addition,

(120) I1JZ - 1JF I ~ aj min(D j (7N), 1) ~ (lj6)J(i.

Proof. We note first of all that finiteness of the set of the intervals Jz such thatJz n[O, N] =1= 0 follows from analyticity of the function A(D, J-l). Due to (119), thereexists a number a E (0,1) such that Dj(7N) > a, and 6J(i < a < 1 (and so (114)is satisfied). It is easy to see that in view of Lemma 4.8 and the conditions (119)the derivative 8p,A(D, J-l), being a continuous function, satisfies exactly one of theinequalities

J-l E Jz.


This implies monotonicity of the function A(D, j-t) on each of the intervals Jz. Therefore, each interval Jz = [1Jz, 1Ji] contains exactly one solution 1Jz to exactly on~ of theequations (106). Besides,

(122) A(D,1JF)=W1,z±W2,za, IWj,zl=l, j=1,2.

The relations (121) and (122) now imply that the solution 1Jz satisfies the inequalities(120). 0

We consider now a perturbation

of the function A(D, j-t), where sJ-L and vJ-L are functions of j-t that will be assumed to beclose in a smooth sense to 1 and 0 correspondingly. (See details in the next lemma.)We need to investigate solutions (for k = 0 or 7r) of the equation

(123) A(D,j-t) = cosk(= ±1).

LEMMA 4.10. Suppose that the functions SJ-L and vJ-L are continuously differentiablewith respect to j-t, and satisfy (together with parameters D and N) the conditions

(124) ISJ-L - 11 :::; v, IVJ-LI :::; v for 2 :::; j-t :::; N + 1,

(125) 18JV < min{D/(7(N + 1)), I},

(126) 18J-LsJ-L1 :::; 1/8, 18J-LvJ-L1 :::; D/(16j-t) for 2 :::; j-t :::; N + 1.

Then there exists a finite system of disjoint intervals Jj that satisfies the followingconditions:

(i) The length of each of these intervals is less than 8v/ min{D/(7N), I}.(ii) Every solution j-t E [3, N] of the equation

(127) A(D,j-t) = cosk(= ±1)

belongs to only one of the intervals.(iii) Each of these intervals contains exactly one solution if of (123).(iv) Every solution v E [2, N] of (123) belongs to some of these intervals.

Proof The estimates can be derived from Lemmas 4.8 and 4.9. 0

The last lemma implies the following statement, which we shall use later.THEOREM 4.11. Let 2 :::; j-t1(D) :::; ... :::; j-tL(D) :::; N be all the solutions of (106)

in the interval [2, N]. Suppose also that 1J1 (D) :::; ... :::; 1JL(D) are the first L solutionsin the interval [2,00] of the equations

If the numbers D and N and the functions SJ-L and VJ-L satisfy the conditions (124),(125), and (126), then

l1Jz(D) - j-tz(D)1 :::; 8v/ min{D/(7(N + 1)), I}, 1 :::; l :::; L.


4.5. Auxiliary spectra. In this section we introduce and investigate propertiesof some auxiliary sets in terms of which the spectrum of the Maxwell operator c,an bedescribed. We use here some previously defined functions. Let us introduce for anynonnegative integer n the set

(128)

Due to continuity of the functions en(k, D), these sets are in fact closed intervals ofthe real line. Consider also the sets

(129) Bn = {D ~ 0 : Bn(D) 30}, n ~ O.

LEMMA 4.12. For any integer n ~ 0 the following statements hold:(i) The interval Bn(D) can be represented as

(130) Bo(D) = [-2(vo(D))2, 2eo(1r, D)],

(131) Bn(D) = [-(vO(D))2 + (1rn)2, eO(1r, D) + [1r(n + I'n(D))]2], n > 0,

and its endpoints are continuous strictly decreasing functions of the argument D.(ii) The set B n coincides with the set of values of D such that for some k1 ,

k2 E [0,1r] the following equality holds:

(132)

(iii) The set B n is an interval, i.e., Bn = [D~, D~], D~ ~ D~.

(iv) The endpoints of the interval B o are

(133) Do =0, D + -4o - ,

and Dt is the unique solution of the equation eO(1r, D) = O.(v) For n ~ 1 D~ and D~ are the unique solutions of the equations

(134) vo(D) = 1rn and vl(D) = 1r(n + I'n(D)).

(vi) The following inequalities are true:

(135)

Proof. (i) The left and right endpoints of the interval Bn(D) coincide correspondingly with the sum of the left and right endpoints of the intervals Io(Q(D)) andIn(Q(D)). This, along with Lemmas 4.2 and 4.3, implies statement (i).

Statement (ii) is just a rephrasing of the definition of the sets.Since the endpoints of the interval Bn(D) are continuous strictly decreasing func

tions of D, we get (iii). The rest of the statements follows from the properties ofthe functions vo(D), vl(D), eO(1r, D), and I'n(D) described in the Lemmas 4.2, 4.3,and 4.5. 0

The interva!s Bn do not depend on any parameters and can be found once andforever. They are closely related to the spectra of interest. The next lemma providesa rather accurate description of the location of the intervals Bn .

LEMMA 4.13. The endpoints D~ and D~ satisfy the estimates

(136)

(137)

Do = 0, Dt = 4,

-t I21rn - 4 t e -7 < D;; < 21rn,1 - e t=7rn-l

n ~ 1,


(139)

(138) D;; > 21r [n + 1 - ~ arctan n+ 1n- 1r-

1

] ,

D:}; < 21r [n + 1 - ~ arctan~] + 27 e-T

_T

I '1r n + 1 + e r=7rn

In particular, when n ~ 00 we get the asymptotic formulas

n ? 1.

(140)

The intervals B n = [D~, D;iJ, n ? 0, are disjoint, i.e., B n nB m = 0 for n =1= m.Proof The statements of the lemma can be derived from Lemmas 4.3, 4.5,

and 4.12. 0The numbers D~ are the absolute constants that are involved in the description

of the set (J"E (see (26)). This set plays an important role in our description ofthe structure of aJ1.Y finite number of bands and gaps in the spectra of the Maxwelloperators M, ME, and MH, as well as of the operator ee'

We also notice that the Lemma 4.13 justifies the approximation of the set (J"E

indicated in (26).

4.6. Eigenfunctions. We shall denote the eigenfunction of the operator Q(K, D)associated with the eigenvalue eZ(K, D) by fZ(K, D; y).

Let us consider the Cauchy problem for the operator Q(K, D):

(141) - P"(y) - D8(y)P(y) = ep(y), 'l/J(-O) = 'l/Jo, 'l/J'(-O) = 'l/Jl, Y? O.

Taking into account (68), we can represent the solution to this problem in the form

(142)

The choice of the parameters 'l/Jo and 'l/Jl in (141) will be convenient when we treatlater the operators Q(K, D) as limits of the operators Q(K, D, 8) with well potentials.Assume now that 'l/J is an eigenfunction of the operator Q(k, D) defined by (62), (63).Then we have the representation (67) for the vector ('l/Jo, 'l/Jl):

'l/Jo = e-1/ 2 sin e1/ 2, 'l/Jl == - cos e1/ 2 + eik + De- 1

/2 sin e1/ 2

.

The equality (142) gives

(143) 'l/J(K, j-l, y) = e-1/ 2 sine1/ 2 cose1/2y + (- cose1/2 + eik )e-1/

2 sine1/ 2y

== j-l-l [ei~ sin j-ly + sin j-l(1 - y)] , where j-l == e1/ 2 .

For k = 0 or k = 1r we get correspondingly

(144) 'l/J(O, j-l, y) = j-l-l [sinj-ly + sinj-l(l - y)] == 2j-l-l sin(j-l/2) cos (j-l(Y - 1/2)),

(145) 'l/J(1r, j-l, y) = j-l-l [- sin j-ly + sin j-l(1 - y)]

~ -2j-l-l cos(j-l/2) sin (j-l(Y - 1/2)) .

Now we calculate the normalization factor for 'l/J(x):

(146) I(K" p) =11

lei/< sin py + sin p(l - y) 12 dy

= l+coSK [j-l-1sinj-l-cosj-l] -j-l-1sinj-lcosj-l


for e2:: O. For instance, for k = 0 and k = 7r we get

(147) I(O,j-l) = 1+j-l-Isinj-l-cosj-l-j-l-Isinj-lcosj-l

= (1 - cos j-l) (1 + j-l-I sin j-l) = 2 sin2(j-l/2) (1 + j-l-I sin j-l) ,

(148) I (7r , j-l) = 1 - (j-l-I sin j-l - cos j-l) - j-l-I sin j-l cos j-l

= (1 + cos j-l)(1 - j-l-I sin j-l) = 2 cos2(j-l/2) (1 - j-l-I sin j-l).

For e= -v2 < 0 we have

(149) I(K, v) = -1 - cos K [v- I sinh v - cosh vJ + v-I sinh v cosh v,

which can be transformed for k = 0, 7r analogously to the previous case.Now the normalized eigenfunctions can be represented as

(150) !Z(K, D; y) = V(K, eZ(K, D), y), 0 :::; y :::; 1, I = 0,1, ... , where

(151) V(K, j-l, y) = I(K, j-l)-1/2 [ei~ sinj-lY + sinj-l(1 - y)J .

In particular, we get from (144), (145), (147), and (148)

1585

(152)

(153)

V(O )= V2cos[j-l(y-(1/2))],Il, Y VI I . ,+ j-l- Slnj-l

V(O, v, y) = V2 cosh [v.(y - (1/2))] ,Vv- I sInh v + 1

V(7r )= V2sin[j-l(y-(1/2))],Il, Y VI I' ,- j-l- slnj-l

V(rr,v,y) = V2sinh[v(y-(1/2))].Vv- I sinhv-I

Remark 4.14. Note that in the formula (67) the vector Up(O) degenerates to 0 ifel / 2 is a multiple of 7r and K = 0 or 7r. Under these conditions I(K,e) is equal to O.The formulas (152) show that after the normalization this degeneration disappears,and we can define

(154) !Z(K, D; x)I~=o 7r = lim !Z(K, D; x)., ~---+o, 7r

In particular, using (152), (71), and (74) we easily obtain

(155) !2m-1 (7r, D; y) = V2 sin(2m - 1)7ry, 0 :::; Y :::; 1,

(156) !2m(0, D; y) = V2 sin 2m7ry, 0 :::; y :::; 1, m = 1,2, ....

5. Relations between Schrodinger operators with point and well potentials. We are now interested in the following eigenvalue problems related to theperiodic Schrodinger operator Q(D,8):

(157) Q(K,D,8)'l/J(y) = -'l/J"(y) - qD,8(y)'l/J(y) = e'l/J(y), 0:::; y:::; 1,'l/J(I) = ei~'l/J(O), 'l/J'(I) = ei~'l/J'(O), -7r:::; K :::; 7r.

The main objective of this section is to show that for 8 ~ 0 the operatorsQ(K, D, 8) and their eigenvalues and eigenfunctions are small perturbations of theoperators Q(K, D) (see (62)) and of their eigenvalues and eigenfunctions correspondingly.


o~ Y ~ 8,(158)

Let eO(K, D, 8) ~ el (K, D, 8) ~ ... be the eigenvalues of the problem (157) countedwith their multiplicity. Consider the vector U(x) = ('l/J(x) , 'l/J' (x)). Solving the equation(157) on the interval [0, 8], we get

U( ) = ['l/J(O) cos PlY + 'l/J' (O)PI-1

sin PlY]Y -'l/J(O)PI sin PlY + 'l/J' (0) cos PlY ,

where PI = VD8-1 + e. Solving the problem on the whole interval [0,1], we find themonodromy matrix

(159)

(160)

(161)

(162)

W - [ ala2 - pbl b2 p2-lal b2+ PI-

Ia2bl ] h- b P bIb b ' were- P2al 2 - Ia2 I al a2 - p- I 2

PI = vD8-1 + e, P2 = vft, P = PIP2-1

,

al = cos(PI8), bl = sin(PI 8),

a2 = cos (P2(1 - 8)), b2 = sin (P2(1 - 8)) .

The eigenvalues of the problem (157) can be found from the equation TrW = 2 cos K,

Le.,

(165)

1 I(163) al a2 - 2" (p + p- )bl b2 = cos K, or

(164) COS(P1 8) cos (P2 (1 - 8)) - p +:-1 sin(P18) sin (P2 (1 - 8)) = COSA:.

If'l/J(x) is an eigenfunction of the problem (157), then WU(O) = ei~U(O), and using(163) one can easily obtain

U(O) = ['l/J(O)] = [ p2-lal b2+ PI- Ia2bl ].

'l/J' (0) (p - p-l )bl b2/2 + i sin K

(166)

Note that (158) and (165) imply

U(8) = ['l/J(8)] = [P2-lb~ + Pl-Iblei~] .'l/J' (8) e~~al - a2

Using (160), we can rewrite (164) in the form

(167) cos Te cos ( vft(1- 8)) - (~ + (,8) Si~;e sin (v'~ - 8)) = cos A:,

We introduce now the function

(~ ) (D ) sinTe sin (~(1- 8))

Y(("D,8)=cosTecos y(,(1-8) - 2"+(,8 ~ ~ .

Then (167) can be rewritten as Y(e,D,8) = cosk. Note that Y(e,D,8) is an entirefunction of variable ewhich takes on real values for all real e. The set of eigenvalueseO(K,D,8) ~ el(K,D,8) ~ ... coincides with the set of solutions of (167) for eE R.For D = 0 the functions eZ(K, D, 8) do not depend on 8 and thus the formulas (73) are


applicable to them. Besides, eZ(K, D, 8) are decreasing functions of D (which followsfrom an obvious inequality for the corresponding quadratic forms; compare with theproof of the Lemma 4.1). From the last observation we obtain a priori estimates:

(168) l 2:: o.

The elementary but rather lengthy analysis shows the validity of the following estimates.

LEMMA 5.1. If D = 5 and 0 < 8 ~ 0.1, then (167) has a unique negative solutionwhich coincides with eo (K, D, 8). If

(169) D 2:: 5, D8 ~ 1/4,

then

(170)

Using the substitution e= J-l2 we can represent (164) in the form

(171) cos TJL cos (J-l(1 - 8))

(D 2) sin TJL sin(J-l(l - 8))- - + J-l 8 -- _.....:..-_-....;..... = cos K,2 TJL J-l

In particular, if J-l = iv, v > 0, then (171) takes the form

(172) h( (1 ~)) (D 2~)sinTvsinh(v(1-8))cos Tv cos v -u - - -v u -- =COSK,2 Tv V

To view (171) and (172) as perturbations of (71) and (72) (or (87)) we rewrite themas follows:

hJL = - (1 - cos TJL ) cos j1- J-l8 (T;l sin TJL) sin j1,

sJL = (1 - 8)T;1 sin TJL.

(173)

(174)

(175)

__ DSJL sinj1cosJ-l- -2- Ii + hJ1- = COSK" j1 = J-l(1 - 8),

Correspondingly, (172) will look like

v= v(l - 8).(176)(D - 2v28)T;1 tan Tv tanh v

v = -----------.2{1-COSK,[COSTv coshVj-l}'

5.1. Positive eigenvalues. To estimate the solutions of (173) we will need thefollowing inequalities for hJL and sJL which can be derived elementary from the definitions of those quantities.

LEMMA 5.2. Suppose that

(177)


and hJ-L and sJ-L are defined respectively by (174) and (175). Then

(178)

(179)

We note now that using the notation from Lemma 4.7 we can rewrite (173) (andhence (171)) as follows:

(180) A(DsJ-L' ii) + hJ-L == cos K, sJ-L == (1 - 8),;;1 sin 'J-L' ii == 1L(1 - 8).

The following lemma is the key tool in the approximations of the eigenvaluesILn(K, D, 8) by ILn(K, D).

LEMMA 5.3. Let L be a natural number and a and Co be positive constants suchthat

(181) o< a < 1, Co ~ 1.

Suppose that K == 0, 7r and ILl (K, D, 8) :::; ... :::; ILL (K, D, 8) are the first L solutions of(180) (and hence of (171)) and ILl (K, D) :::; ... :::; ILL(K, D) are the first L solutions of(106). Suppose that the parameter 8 satisfies the inequality

(182) 18J(4Co+ 3)7r(L + 1)8 :::; a < min(4Co/11, 1)

and the parameter D varies in the range

(183)

Then

(184)

In particular,

(185)

11a7r(L + 1) :::; D :::; 4C07r(L + 1).

IILL (K, D, 8) - ILL (K, D) I ::; (33Co + 26)7r(L + 2)8/a.

IlLl(K, D, 8) - ILl(K, D)I ::; 0.041a :::; 0.041.

Proof The proof of this lemma is based on the results of Lemma 4.10, Theorem4.11, and Lemma 5.2. Hence, we have to check the conditions (124)-(126) and (177).Note first of all that (85) yields

(186)

Let us denote v == (4Co + 3)7r(L + 1)8 and N == 7r(L + 1). Using (182) and (183) weobtain for any IL-E [2, N]

,~ == D8 + 1L282 :::; 4C07r(L + 1)8 + [7r(L + 2)8]2 ::; (a/18)2 + a2/(9· 18) < 1.

This enables us to apply Lemma 5.2 for IL E [2, N + 1]. From (178) and (179) it easilyfollows that

(187) IL E [2,N + 1].


(190)

This gives (124) in Lemma 4.10. Then from (178), (179), (182), and (183) we obtain

I01'hI' I ::; ~ ((1/2) + D~,J p,8 ::; ~ ((1/2) + (9/8)a-1) 1r(L + 2)8

(188) ::; ~ ((1/2) + (9/8)a-1

) (~r = 2(1~)2p, (~2 + 9;)D D

::; (18)2P, < 16p,'

Now we have to make some little adjustments since (180) involves variable ji. Namely,(189) easily yields

( )(1 - 8)2 1

18;; 81'11'=;(1-6)-1 I::; (18)2ji ::; 18ji'

One can check now the inequality (126). The inequalities (187), (188), and (190)along with (182) and (183) allow us to apply here Theorem 4.11, which implies theinequality

which along with (186) easily yields the desired inequality, (184). The inequality (185)follows from (184) and (182). 0

We will also need the following corollaries of Lemma 5.3.LEMMA 5.4. Let No be a natural number, and suppose that the parameter 8 is

small enough to satisfy the inequality

(192)

Then for any 1 ~ l ~ No and any D such that 1 ~ D ~ 41r(No + 1)

(193)

Proof The statement of this lemma follows immediately from Lemma 5.3 if wepick a == [111r(No + 1)]-1 and Co == 1. 0

LEMMA 5.5. Let No be a natural number and Co 2:: 1 be a constant, and supposethat the parameter 8 is small enough to satisfy the inequality

(194) 18J(4Co + 3) (No + 1)8 ~ [111r(No + 1)]-1 .

Then for any 1 ~ l ~ No and any D such that 1 ~ D ~ 41rCo

(195)

In particular, for any 1 ~ l ~ No and for any 0 ~ D ~ 41rCo

(196)


Proof The estimates (195) follow from Lemma 5.3 if we pick a = [41r(No + 1)]-1.The inequalities (196) follow easily from (195) and (85). Note that we may exteI).d therange of values of D to 0 since !1z(f\;, D, 8) are decreasing functions of the argumentD. 0

LEMMA 5.6. Suppose that all conditions of Lemma 5.3 are satisfied. Then forany 0 ~ D ~ 4Co1r(L + 1)

(197) 1rl- (33Co + 26)1r(L + 2)8/a ~ !1z(f\;, D, 8) ~ 1r(L + 1).

In particular,

(198) !1z(f\;,D,8) 2::1rl- 0.041a 2::1rl- 0.041.

Proof The right-hand inequality in (197) is a consequence of (168). The left-handinequality in (197) follows from (184) and (85). It is true for any D ~ 4C01r(L + 1)since !1L(f\;, D, 8) is a decreasing function of the argument D. The inequalities (198)follow in similar fashion from (185) and (85). 0

5.2. Negative eigenvalues. Now, in order to complete our comparison of eigenvalues of the operators Q(f\;, D, 8) with those of the operators Q(f\;, D), it remains toanalyze the lowest eigenvalue ~o(f\;, D, 8), which becomes negative for large D. Basedon Lemmas 4.3 and 5.1 the following properties ~o(f\;, D, 8) can be established.

LEMMA 5.7. The following statements are true:(i) ~o(f\;, D, 8) is a decreasing function of D 2:: 0 and increasing function of f\;

such that ~o(f\;, 0, 8) = f\;, and ~o(f\;, D, 8) 2:: -D8-1.(ii) If ~o(f\;,D,8) ~ 0 and we define vo(D,8) = (-~o(0,D,8))1/2, vl(D,8) =

(-~o (1r, D, 8)) 1/2, then Vo (D, 8) 2:: VI (D, 8) .(iii) If C and c are positive constants, then the following limits hold uniformly

for 0 ~ D ~ C and 0 ~ f\; ~ 1r:

(199)

If c ~ D ~ C, then

(200) ~o(f\;, D, 8) = ~o(f\;, D) + 0(8), 8 ~ o.(iv) Suppose that D 2:: 5, D8 ~ 1/4, then for 8 ~ 0.02 and j = 0,1 the following

representations hold:

(201)u·D

Vj(D, 8) = J ,

1 + VI + 2uJD8o~ Uj - 1 ~ (1/2)D8 + 5e- D

/8

.

In particular,

(202) Vj (D, 8) = (1 + pj)D /2, Ipj I ~ 9D8 + 5e- D/

8.

LEMMA 5.8.' Suppose that D 2:: 40, D8 ~ 10-2 . Then

(203)

(204)

0.3D ~ vj(D, 8) ~ 0.7D, j = 0,1,

18D vj(D, 8) - (1/2)1 ~ 4D8 + 30e-O.3D

•

In addition, for D 2:: 30, D8 ~ 10-2 we have

(205) o< v (D 8) - v (D 8) < 4e-O.59D ._ 0, 1,_


5.3. Eigenfunctions. In this section we establish some properties of eigenfunctions of the operators Q(f\;, D, 8). We shall need these properties in order to estimatethe deviation of the spectrum of the operator 8€ from the spectrum of the operatoro8,,8 with separate variables.

LEMMA 5.9. Let f(y) be a complex-valued function with square integrable derivative on the interval [0, 1]. Then for any positive a ~ 1/2

(206) o~ y ~ 1.

Proof Without loss of generality we shall assume that y E [0,1/2]. We note that

o~ y, z ~ 1.(207) If(y) - f(zW :::; Iy - zl11 If'(t)12 dt,

Integrating the inequality If(y)1 2 ~ 2 [If(z)1 2 + If(y) - f(z)1 2] with respect to z E

[y, Y + a] and using the inequality (207) we obtain (206). 0LEMMA 5.10. Let 0 ~ -D2 and 'ljJ()(f\;,D,8;y) = 'ljJ(y) be, respectively, an eigen

value and the corresponding eigenfunction of the operator Q(f\;, D, 8). Then

(20S) 17/J(y)12:::; y'SCD,o11

17/J(t)1 2 dt, 0 :::; Y :::; 1,

(209) 16

17/J(y)12 dy :::; 8y'SCD,o11

17/J(t)12 dt, where

( ) {2Dr if 0 < 0, {}

210 CD,o=4 2Dr+O ifO?O, D 1 =maxD,1.

For 0 negative we also have

(211)

Proof The function 'ljJ satisfies the equation

-'ljJ"(y) = (qD,8(y) + O)'ljJ(y) , 0 ~ y ~ 1,

'ljJ(1) = eik'ljJ(O) , 'ljJ' (1) = eik'ljJ' (0).

Multiplying both sides of the equation by 'ljJ(y) and integrating the result with respectto y E [0, 1], we get the identity

Since the left side of this inequality is positive, we obtain (211). This and the corollaryof the inequality (206),

1592

imply

A. FIGOTIN AND P. KUCHMENT

This, in turn, implies for any a < D- 1

(212) dO'. = 2Da-1 + () .

1-Da

(213)

o~ y ~ 1.

Now we can estimate the coefficient da as follows. We observe first that the coefficientda is an increasing function of D when 0 2:: -D2 and a < D-1. Using this anddenoting D1 == max{D, I}, D1a == u < 1 and then using the condition 0 2:: - D2 weobtain

d 2Dr + uO 1 { 2Dr if 0 < 0,a == < X D2 ll'f II 0u(l - u) - u(l - u) 2 1 + 0 I 0 2:: .

This combined with (210) yields da la=D;.-l /2 ~ CD, (). This inequality and (210) imply

11

I'I/J'(y) 12 dy :::; CD,(J11

I'I/J(y) 12 dy.

Finally, combining the inequalities (206) and (213) we obtain

I'I/J(y) 12 :::; (2a -1 + aCD , (J)11

I'I/J(t)12 dt,

Now we take a == J2/CD ,(}. One can check that a < D-1. Since CD ,(} 2:: 8, a isless than 1/2. We conclude that (208) is true. The inequality (209) follows from(208). 0

The next two lemmas are devoted to properties of the eigenfunctions fz (f\;, D, 8; y),l 2:: 0, of the operators Q(f\;, D, 8). We shall need them later when we consider spacedistribution of the electric field energy.

LEMMA 5.11. Suppose that D -7 00, D8 -7 0 (and hence 8 -7 0). Then theeigenfunctions fz (f\;, D, 8; y), l 2:: 1, satisfy the estimates

(214)

(215)

161 Jz(/',;, D, 8j y)1 2 dx :::; 8 [O(D- 2) + 0 (D8)] ,

111/1(/',;, D, 8; y)1 2 dx = (nl)2(1 - cos/',;) + 0(1).

Proof. Let us fix any naturall 2:: 1 and recall that the initial values fz(', 0), f{(', 0)of the eigenfunction fz associated with the eigenvalue ~z are given by the formula(165), where ~ == ~z == J1f. We shall use the notation U(y) == (fz(·; y), f{('; y)). Usingthe inequality (19~) for Co == D and equality (105), we get the estimate

(216) J1z(f\;, D, 8) == 1fl + 4lD-1 + O(D-2 + D8).

Let us now denote T == J D8 + J1282 . We note that T -7 0 under the conditions of thelemma. Using (165), (160) we obtain for l 2:: 1 and J1 == J1z

(217) U(O) == [fz(f\;,D,8;0)] == [J1-1sin(1-8)J1CO~T+8:-1coS(1-8!~SinT].f{(f\;, D, 8; 0) -D(2J1)-1(T-1sIn T) sln(l - 8)J1 + ~ sIn k


Using the fact that 7,8 « 1, we conclude that this vector can be represented as

[/1-1 sin/1 + 0(8 + 72) ] I

_(2/1)-lDsin/1+0(D8+72 D/1-1 sin/1+/18) J-L=J-Ll·

From (217) and (158) we conclude that for 0 ~ y ~ 8

Iz(Y) = Iz(O) cos(78-1y) + 1{(0)(D8-1 + /12)-1/2 sin(y(D8-1 + /12)1/2)

1593

Now, according to (216), we have for a fixed l ~ 1: sin/1Z = 0(D-1 + D8). Togetherwith the previous inequalities, this gives for 0 ~ y ~ 8: Iz(k,D,8;y) = 0(D-1) +0(8 + 72). This yields the estimate (214).

To verify (215) we consider first U(8). Based on (166) together with (160) weobtain

(218)

Using the parameter 7 (7 -7 0, as has been noted earlier), we come to the representation

U(8) = [ fl-1

sin/1 + 0(8) ] Ie'tK, - cos /1 + 0(72) J-L=J-Ll·

Note now that the vector function U(y + 8) and the vector function Up(Y) (which isassociated with the problem (64)) on the interval [0,1 - 8] correspond to the samedifferential equation,

-Il//'(y) = /1 2 'ljJ(y), o~ y ~ 1- 8,

but to different initial conditions. Namely, using (218) and (70) we may conclude that

U +8 = ['ljJ(y)] 1 = [ fl-1sin/1+0(8) ]1

(y )Iy=o 'l/J'(y) _ e'''' - COSfL + 0(T2 ) _ 'y-o J-L-J-Ll (K"D,8)

From these obse:rvations and (216) we easily get that

max I/z(f\;, D, 8; Y + 8) - IZ(f\;, D; y)1 = 0(1).OSYSl-8

This implies


We recall now (see (143)) that

(220)

(221)

(222)

We obtain from (143) and (105)

11

1Jz(~, D; y)1 2 dx = /-l-2 I(~, /-l)I/L=/Ll(I<,D) = (nl)-2(1- cos~) + 0(1).

The last formula along with (220) (which implies that fZ(f\;,D;y) is bounded) leadsto the representation

11-81Jz(~,D; yW dx = (nl)-2(1- cos~) + 0(1).

This equality and (219) imply the estimate (215). 0LEMMA 5.12. Suppose that 0 ~ D ~ C for some constant C and that 8 ~ O. Then

the eigenfunctions fO(f\;, D, 8; y) satisfy the estimates

18Ifo(~, D, 8; y)1 2 dx = 8 [v-2sinhv2 + 0(1)] ,

11

Ifo(~, D, 8; yW dx = I(~, v) + 0(1),

where I(f\;, v) is defined by (149) and v = J -~(f\;, D) .Proof. Proof of this lemma is similar to the proof of the previous one. Let us

consider first D > 4. In this case ~o(f\;, D) and ~o(f\;, D, 8) are negative for small 8. Weremind the notation ~ = -v2 . Using (165) and (160) again, we get

U(O) = [fo(f\;,D,8;y)] = [v-l.sinhv+o(l)]

f~(f\;, D, 8; y) e'tK, - cosh v .

From this representation we obtain, for 0 ~ y ~ 8, fO(f\;, D, 8; y) = v-I sinh v + 0(1),which implies the estimate (221). Reasoning similar to that used to justify (215) leadsto the equality (222).

If D < 4, then one can check that the statements of the lemma still hold, but inthis case J -~(f\;, D) will be purely imaginary. Of course, all functions and integralsof interest for this case are still real valued. 0

6. Scalar models. In this section we consider some general properties of thespectra of scalar operators associated with the Maxwell operator. (Similar operatorsarise for acoustic waves.)

As will be shown in §7.1, in order to estimate the spectral bands of the Maxwelloperator, we need to consider, in particular, the following scalar operator:

Here we are interested only in the values d = 3 and d = 2.Along with the operator 8€ acting in the space L 2 (Rd , c(x) dx, C), we will con

sider its Floquet components


with the boundary conditions

1595

(224)

(225) j = 1, ... ,d.

The main results of this section are the following statements.THEOREM 6.1. For any natural number No and any positive constant C there

exists a positive constant c such that for any 13 > C and for any W < c

(226) 0"(8c:)n I = [ U [WD~(('8)'WD~(('8)]] n I ,O~n~No

I = [0, 21rw(No - 1)],

where the endpoints D~ ((, 8) can be approximated by the numbers D~ (defined in(133), (134)) as follows:

(227) ID:((,8) - D: I ~ 2 (4w + 103 Ng13- 1) .

In fact, we can describe the asymptotic location of the spectrum in any finiteinterval of the spectral axis.

THEOREM 6.2. For any constant N ~ 1r there exist positive constants C and csuch that for any 132 > C and for any w < c

(228) a(8,,) n[O, N] = [aE(8,,) UaH(8,,)] n[O, N],

where the sets O"E(8c:) and O"H(8c:) are, respectively, of the form (22), (23), (25):

O"E(M) = U[wD~((,8),wD~((,8)],wheren~O

D:((,8) = 21rn(1 + X~), n ~ 1; Do ((,8) = 0, Dt((,8) = 4 + xt,

O"H(M) = U [(1rn)2 + pii, (1rn)2 + pil],n~o

and the quantities x~, TJ~, and p~ satisfy the estimates

(229) Ix~1 ~ 100Nj3-1 + 6e-O.3n + 10N2 j3:;3wn-1,

(230) ITJ:I ~ 5.103 Nj3-1 + 2· 103e-O.3n + 3n-1 + 301w,


1 ~ n ~ N(1rw)-l,

1 ~ n ~ N (1rW)-1 ,

(231)

In particular, the spectrum 0"(8c:) has adjacent bands and gaps of order w.The proof of these theorems is based upon an approximation of the operator 8 k ,c:

by an operator with separable variables. Such an operator can be analyzed on thebasis of our results for one-dimensional models.


6.1. Separated-variables case. It turns out that asymptotic behavior of spectra of the operators 8 k when 8 and (-1 approach 0 in a suitable manner can be

, €

analyzed in terms of spectra of similar operators with separable variables. In thissubsection we consider the case of separated variables when the dielectric constantc(x) has the following special form:

(232) ~ (x) = L P(,8(Xj),l~j~d

Here the auxiliary function Pc" 8(y), Y E R, is defined by the formula

(233) ( )_ {( - (d - 1)d-1 if 0 ~ y < 8,

PC,,8 Y - d- 1 if 8 ~ y < 1,

where p(y) = p(y + n) for nEZ, y E R. We introduce now the notation

o ()-18C,,8= - ~ (x) ~,

o 0

and 8k, c" 8 is the respective Floquet-Bloch component of the operator 8c" 8. We willuse the "circle" symbol 0 in order to indicate quantities associated with the separatevariables case. As before, we are interested primarily in the cases d = 2, 3. We nownote that the eigenvalue problem

(234) _ (E (x)) -1 D.cp = >.cp

with the boundary conditions (224), (225) can be rewritten as

(235) v(x) = L (pc" 8(Xj) - d- 1) .

l~j~d

Thus, to find the eigenvalues of the problem (234) we can do the following. First, wefind the eigenvalues ~ (m, k, g) of the Schrodinger operator satisfying the boundaryconditions (224), (225) and depending on the parameter 9 ~ 0:

(236) Sk cp = -~cp - gv(x)cp,,9

where m is an index which counts the eigenvalues ~. We solve then for 9 ~ 0 theequations

(237) ~(m, k, g) = 9

for all values of the index m. This gives us the set of eigenvalues of the eigenvalueproblem (234), .(224), (225).

We also note here that the dielectric constant ~ (x) does not differ much from theoriginal dielectric constant c(x). Namely, for d = 2 and f 11 = [0,8] x [0,8] we have

(238) o { 0 if x E X - f 11 ,c (x) - c(x) = 2( _ 1 if x Ern.

A similar formula holds for d = 3.


To proceed with the Schrodinger operator S k with separable variables we in-g,

troduce the one-dimensional Schrodinger operator: '

Then the standard Floquet expansion,

holds, and the operator in the right side is defined in (157). (This includes boundaryconditions as well.) Let now ~m(f\;, g, (, 8) and fm(f\;, g, (, 8, y) be, respectively, theeigenvalues and normalized eigenfunctions of Q1'\" g, Le.,

(240) m=0,1, ....

Thus, if I is the identity operator in L2 (R), then we have

(241)

(242)

We can now represent the eigenfunctions and the eigenvalues of the Schrodinger operator Sk as,g

(243) Fm(x) = Fm (k,g,(,8,x) = II fmJ(kj,xj),l~j~d

(244) ~(m) = ~(m,k,g,(,8) = L ~mJ(kj),l~j~d

To analyze (237) we need the following lemmas.LEMMA 6.3. The eigenvalues ~ (m, k, g) are decreasing functions of the parameter

9 2 o. Each equation

(245) ~(m,k,g)=g, mEZ~,

ohas a unique solution 9 =A (m, k, (, 8).

Proof As follows easily from the definition (239) of the operator Qg and from(157), we have QK,,91' ~ QK,,92 for gl 2 g2. This together with (244) impliesthat ~ (m, k, g) are decreasing functions of the parameter g. Then we note that~llg=o = ~lID=O· The last statement along with (73) and monotonicity otthe functions~(m, k, g) implies that each of the equations (245) has a unique solution A(m, k, (, 8) 2o. 0 .

The following statement is standard for operators with separable variables.

LEMMA 6.4. The numbers ~ (m, k, (, 8), m E zi, form the set of all eigenvalueso

of the operator 8k, (,8' taking into account their multiplicity. Moreover, the functions

(246)owm(k,(,8,x)=Fm (k,g,(,8,x)1 0 k '

g=..\(m, ,(,8)


oform the complete set of eigenfunctions of the operator 8k, (, 8 such that

(247)o 0 0 0

8k,(,8Wm ((,8,k,x) =-X (m,k,(,8) Wm ((,8,k,x), mEZi·

oProof The proof readily follows from the definition of the operators 8k, (, 8 and

(232). 0The main results of this section are the next two statements for the case d = 2.THEOREM 6.5. Let d = 2. For any natural number No and any positive constant

C there exists a positive constant c such that for any j3 > C and for any W < c

(248) a(8c:,6)n I = [O:S;~NO[WD~('8),wD;t('8)]] nI,

I = [0, 21rw(No - 1)],

and the endpoints D~ ((, 8) can be approximated by the numbers D~ (which are definedin (133), (134)) as

(249) ID~((,8) - D~I ~ 7w.

In fact, we can describe the location of the spectrum in any bounded portion ofthe spectral axis, as is formulated in the next statement.

THEOREM 6.6. Let d = 2. For any constant N ~ 1r there exist positive constantsC and c such that for any j3 > C and for any w < c

(250)

o 0

where the sets aE(8(,8) and aH(8(,8) are, respectively, of the form (22), (23), (25):

aE(8c:,6) = U[wD~(,8),wD;t(,8)], wheren~O

D~((, 8) = 21rn(1 + x~), n ~ 1; Do ((,8) = 0, Dci((,8) = 4 + xci,

and the quantities x~, TJ~, and p~ satisfy the estimates

(251) Ix~1 ~ 100Nj3-1 + 6e-O.3n

,

(252) ITJ~I ~ 5.103 Nj3-1 + 2· 103e-O.3n + 3n-1 + 300w,


1 ~ n ~ N(1rw)-l,

1 ~ n ~ N(1rw)-l,

(253)

oIn particular, the spectrum a(8(, 8) has adjacent bands and gaps of order w.

The proofs of the theorems are provided below; they are based on the resultsthat we obtained for one-dimensional operators and on appropriate estimates for theeigenvalues of the operator 8k .

,€


6.2. 2D case with separated variables. From now on we shall assume thatd = 2. The relationships (237), (239), (244) and Lemmas 6.3 and 6.4 imply the nextstatement.

o 0

COROLLARY 6.7. For d = 2 the spectrum a(e,,8) of the operator e,,8 can berepresented as

(254) a(8c:,6) = U Bn ((,8),

nEZ~

where each band Bn ((, 8) is the set

(255)

Here g(n, k, (, 8) is the unique solution of the equation

(256)

The bands Bn ((, 8) can overlap, but this possibility is limited by the followingstatements.

LEMMA 6.8. Each band Bn ((, 8), n E Z~, is an interval, i.e.,

Bn ((,8) = [Gil ((,8), Gii ((,8)].

The endpoints Gil ((, 8) < Gii ((, 8) of the interval Bn ((, 8) can be found from theequations

(257)

In addition,

(258) Gil((,8) :S Gm((, 8), Gii((,8) :S Gt!n((, 8) if mj ~ nj, j = 1,2.

In particular, if for some natural N the set Ulnh~NB n ((, 8) has an interior gap,

then the entire set UB n ((, 8) has this gap. (Here Inll = nl + n2.)Proof We note first that ~n(f\;, D, 8), being the eigenvalues of the Schrodinger

operator Q(f\;, D, 8), are monotonic functions of the argument f\; (see [RS]), and according to our definition ~n(f\;, D, 8) ~ ~m(f\;, D, 8) if n ~ m. These observations alongwith (256) imply all the statements of the lemma. 0

In particular, Lemma 6.8 enables us to consider just a finite number of bandsBn (( ,8) in order to prove existence of a gap in the spectrum.

6.2.1. Preliminary analysis of the spectra. The analysis of the spectrumo

a(e" 8) is based on Corollary 6.7 and on our results for the Schrodinger operatorswith point potential. It shows that the spectrum naturally splits into two subspectra.Both of these subspectra have band-gap structure, but the typical sizes of bands andgaps for these spectra are different. Moreover, the eigenmodes associated with thesesubspectra have different spatial energy distribution. We will call these subspectrathe E-subspectrum and the H-subspectrum. They can be defined in terms of theintervals Bn from Corollary 6.7 as follows:

(259) a(8c:,6) = aE(8c:,6) UaH(8c:,6), where

(260) aE(8c:,6) = U B(O,n)((,8), aH(8c:,6) = U B(nl,n2)((,8).nEZ+ nl, n2 >0


Thus, the E-subspectrum can be associated with (256) where at least one index nj

is zero, while the H-subspectrum can be associated with these equations when bothindices nj are positive. To analyze the equations for the E-subspectrum we substituteD = gw- 1 and rewrite these equations as

(261)

oCorrespondingly, the spectrum (TE (8" 8) can be rewritten as

(262) (JE(8t;,Ii) = U illBCO,n)((,8),nEZ+

where the band B co ,n), n ~ 0, is the set described as

Here D(n, k, (, 8) is the unique solution of (261).We note now that if w is small, then (256) can be viewed as perturbations of

(132); hence

Here (TE is the absolute spectrum described by (26) and Lemma 4.13. As far as theintervals Bn , n > 0, are concerned, we note that for small wand n > 0

therefore

(265)

where en are some constants.

6.2.2. E-subspectrum. This subsection is devoted to precise formulation of theapproximate formula (264). Analysis of this subspectrum amounts to investigating(261). These equations can be viewed as perturbations of (132). This analysis is basedon the results of Lemmas 5.3, 5.6, 5.7, and 5.8 and, in particular, on the relationships(184), (202), (204) and (205).

oWe start with the E-subspectrum (TE(8,,8) and (261). It is sufficient to con-

sider the case w!J.en {k 1 ,k2 } = {0,1r}. If '" = 0,1r, then the equations (261) can berewritten as

(266) fLn(t-., D, 8) = JvJ(D, 8) + Dill, j = 0,1, '" = 0, 1r, n > O.

If w (and hence 8) is small, this is a perturbation of the equation

(267) j = 0,1, '" = 0, 1r, n > O.


This is just (132), which has already been well studied. As follows from the generaltheory [RS], depending on parity of n, either J1n(O, D) > J1n(1r, D) or J1n(O"D) <J1n(1r, D). It is convenient to introduce the quantities

(268) J1~(D, 8) = max J1n(f\;, D, 8),K=O,7r

In particular, for 8 = 0 these are the previously defined quantities J1~(D) (see (104)).The endpoints D~((,8) of the bands B CO ,n)((,8), n ~ 0 (see notation in Lemma 6.8)are the unique solutions of the equations

(269)

(270)

f1;;; (D, 8) = Vv3(D, 8) + Dw for D;; ((,8).

fL;;(D,8) = VVf(D,8)+Dw for D;;((,8).

Here the dependence of these endpoints on the parameter ( is hidden in the parameterw. If ill (and hence 8) is small, these equation can be viewed as perturbations of (134).

The two lemmas below are the key statements in describing the E-subspectrumwhich can be established based on Lemmas 4.3, 4.4, 5.8, 4.6, 5.3, 5.4, and 5.7.

LEMMA 6.9. Suppose that 13 > C for some positive constant C. Then for anyinteger n ~ 0

(271)

Moreover, for any natural number No

(272) ID~((, 8) - D~I ~ 6w,

for sufficiently small w.LEMMA 6.10. For any constant N ~ 1r there exist constants C and c such that for

any 13 > C and w < c and for any integer n E [1, Nw- 1] the following representationshold:

(273)

(275)

(276)

177:1 ~ 5.103Nj3-1 + 2· 103 e-O.3n + 3n-1+ 300w,

177~ I ~ 2 . 103N 13- 1 + 800e-O.6n + 80w.

6.2.3. H -subspectrum. In this section we investigate the portion of the spectrum associated with (256) for n = (n1' n2) > O. If we fix the multiindex n, thenaccording to Lemma 6.8 the endpoints Gn((, 8), Gti ((, 8) of the band Bn ((, 8) are thesolutions to the equations

(277)

(278)

where J1~ are defined by (268) and (104).


We start with the following preliminary statement.LEMMA 6.11. For any constant N ~ 1r there exist constants C and c such that

for any (3 > C and w < c and for any n = (n1' n2) such that n1, n2 E [1, VN/1r] thefollowing inequalities hold:

(279)

Proof Let us denote the left sides of (277) and (278), respectively, by Ln(g)and Lti(g). Under the conditions of the lemma we may use the inequalities (196)and (195) from Lemma 5.5, where Co = 2Nw- 1 . Those inequalities hold for anyo~ D ~ 71rNiiJ- 1 . Together with (85) and (97), they imply for sufficiently big C andsmall c

These inequalities along with (277) and (278) imply the inequalities (279) in theindicated range of the index n. 0

Now we can get some more precise estimates for the endpoints g~((,8).LEMMA 6.12. For any constant N ~ 1r there exist constants C, L, and c such

that for any (3 > C and w < c and for any n = (n1' n2) such that n1, n2 E [1, VN/1r]the following inequalities hold:

(280)

Proof Due to of the Lemma 6.11, we can use again the inequalities (195) in orderto estimate g~ ((, 8) more precisely. Setting in (195) No equal to the smallest naturalnumber grater than VN/1r and Co = 2Nw- 1 and using (279), we consequently arriveat the following inequalities for sufficiently small w:

(281) IJl~J (g~((,8)w-\8) - Jl~j (g~((,8)W-l)1 :5: Ld3-\

(282) L~ (g~((,8)W-l) - L [Jl~j (g~((,8)w-l)r :5: L 2{3-\j=1,2

where L 1 and L 2 are some constants. Then using (104), (101), and (279), we obtain

(283)

for some constant L 3 • The last inequalities together with (282) imply the desiredinequalities (280). 0

Proof of Theoremos 6.5 and 6.6. To prove Theorem 6.5 we use the representationof the spectrum, O"E(8C;,8) as the union of intervals provided by the formulas (260),(262), (263) and note that D:;;((,8) ~ D(n,k,(,8) ~ D~((,8). This along withLemma 6.9 and (21) implies the statement of Theorem 6.5.

The proof of Theorem 6.6 is also based on formulas (260), (262), (263) for theE-subspectrum and, in addition to that, on the formulas (255), (256) (where n >0) for the H-subspectrum. Then we note that g~((, 8) ~ g(n, k, (, 8) ~ g~((, 8).This observation along with the asymptotic relationships from Lemmas 6.10 and 6.12implies the statement of Theorem 6.6. 0


6.3. General scalar 2D case. In this section we deal only with the case d = 2.We show that under some appropriate condition on ( and 8, the difference between

the function ~ (x) (for which variables separate) and c(x) is sufficiently small, whichensures closeness of the corresponding spectra.

We denote the eigenvalues of a self-adjoint operator A by Ao(A) ::; Al(A) ::; .We also denote the eigenfunctions of the operator ek, e(X) by ez(c, k, x), l = 0,1, .

LEMMA 6.13. Suppose that c(x) ::; Cl(X) and the eigenfunctions eZ(Cl' k, x) arenormalized as

(284)

Then

(285) n = 0,1, ....

Let the matrix A q with the entries

(286) An(m,I) = L[e1(x) - e(x)] e:'n(e1,k,x)el(e1,k,x) dx,

o::; m, l ::; n,

(287)

have the l2-norm IIAnl1 < 1. Then

Am ( 8k,c(X)) ::; Am ( 8k,Cl(X)) (l-IIAnll) -1,

In particular,

o::; m ::; n.

Am ( 8k,c(x)) ::; Am ( 8k,Cl(X)) +Am ( 8k,Cl(X)}IAnll (l-IIAnll) -1, 0::; m::; n.

(288)

Proof From the min-max principle we have for m = 0, 1, ...

(289) Am (ek e(x)) = min max ( r 1\7<p(xW dX) ( r 1<p(x)12e(x) dX) -1 ,, VrncV <pEVrn } X } X

where V is the domain of the quadratic form of the operator ek, e(X) and V m isa subspace of the dimension m + 1. Since the domain does not depend on c, thisrepresentation implies the inequality (285). To prove (287) we pick the subspaceW n+1 to be equal to the linear span of the vectors ez = eZ(Cl' k, x), 0 ::; l ::; n. Then(289) leads to

(290) Am (ek ( ))::; min max ( r 1\7<p(xW dX) ( r 1<p(xWe(x) dX) -1 .,e x v rn cw n +1 <pEVrn } X } X

Introducing the notation t = (to, ... tn) E C n+1 and <Pt = Ltjej, we obtain from(289)

(291) Am (ek ())::; min max ( r 1\7<pt(xW dX) ( r l<pt(xWe(x) dX) -1 .,e x v rn cw n +1 tEVrn } x } x


Let us introduce the matrix L n with the entries

o :s; m, l :s; n,

(292)

and the identity matrix In acting in the space C n +1 . Then, since ez are the normalizedeigenfunctions of the operator 8k,el(X)' we can rewrite (291) as

( ). (Lnt, t)

Am 8k < mIn max ., e(x) - V71tcWn+1 tEV71t ((In - An)t, t)

Since 1((In - An)t, t)1 ~ (l-IIAII)(t, t), where (.,.) is the Hermitian scalar product inC n + 1 , the inequality (292) implies (287). 0

To estimate the norm IIAnl1 we use the following simple statement, which is theconsequence of the inequality

IIAII = max Aj(A) :s; Tr{A} :s; (n + 1) max {IAm,ml}.O::;m::;n

PROPOSITION 6.14. Let A = {Am, z}, 0 :s; m, l :s; n, be a positive definite matrixand IIAII be its l2-norm. Then

(293) IIAII :s; (n + 1) max {IAm,ml}.0::;m::;n

o 0

We now apply Lemma 6.13 for Cl(X) =c (x) = cp. In this case 8 k 0 =8k;- 8., c , \:"

To use the lemma we need to normalize the eigenfunctions eq(cp, k, x) according to(284). We recall that these eigenfunctions can be represented, due to separation ofvariables, by means of the formulas (243) and (246). Thus, taking in account thatthe eigenfunctions fm(K, y) are normalized according to (240), we now define the

onormalized eigenfunctions \lim (k,(, 8, x):

o 0

(294) \lim (k,(,8,x) = \lim (k,x)

(~dl1P(,6(y)lfmJ(k j ,Y)12dY) -1/2 J-!dfmJ(kj,Xj).

It is now clear that

Le(x) ~:u (k, x) ~n (k, x) dx = omn·

To apply Lemma 6.13 we need to estimate the integrals (286). This is the subject ofthe next lemma.

LEMMA 6.15. For any constant N ~ 1r there exist positive constants C and csuch that for any (3 > C and w < c and for any integer n E [0, Nw- 1] the followingestimates holft

(295) r I ~(o,n) (x)1 2dx ~ 8C10D~((,0)~ 9C10max{21rn,5}.Jr1,1

If n > 0 and (1rn)2 :s; N, then

(296) r I ~n (x)1 2 dx ~ 8C10w-1gri((, 0) ~ 9C10w-1(1rn)2 ~ 9C10w-1N.Jr1,1


(297)

Proof Using the notation of the Lemma 5.10 we normalize the functions'l/JO(K, D, 8; y) by the condition

Consider now two of these functions 'l/Jj ~ 'l/JoJ (K, D j ,8; y) and the constants Cj

JCDJ ,oJ for j ~ 1,2 (see (210)). Then we have

['¢'j(y)[2 ::; Gj , 0::; Y ::; 0; 11['¢'j(Y) [2 dy = 1;

£j = 16

['¢'j(Y) [2 dy ::; Gjo = Cj'

Now we consider the function

I}i(x) = I}i(Xl, X2) = (E2 11 pc, 6(y)I'¢'j(Y)12 dY) -1/2 ILL '¢'j(Xj)

'l/J1 (X1)'l/J2(X2)

From (297) it follows that

r [1}i(xW dx < £1£2 < CIC

2 .

Jr1,1 - ((-1)(L1+L2)+2 - ((-1)(C1+ C2)+2

(We used here that the middle term in these inequalities is an increasing functionof both arguments L 1 and L 2 .) In addition to that, the right-hand side of the lastinequality can be estimated as

Hence,

(298)

Let us consider

where in are the eigenfunctions of the operator Q(K, D, 8) and D(n, k, (, 8) is definedby (263). Then D;;«(, 8) :::; D(n, k, (, 8) :::; D:j;((, 8). Let us apply now the inequality

o(298) for W =W (0, n)· We may use in this inequality CDJ,oJ corresponding to fa·This allows us to take the top line in the representation (210) and to set there D 1 =D:j;((, 8) if n > 0 and D 1 ~ 5 for n ~ O. Now using (271), (273), (133), and (139) toestimate D:j;((, 8) and plugging the results consequently in (210) and then in (298),we arrive at the desired inequality, (295). We also use Lemma 2.1 in order to replace(- 1 by (.


The arguments needed to establish (296) are similar those we used for (295).o

The only difference is that in the formula (299) we now set W =Wn= f n 1fn 2 andD(n, k, (, 8) = g(n, k, (, 8)w- 1 . Here g(n, k, (, 8) is defined according to (255). Thenwe use again the representation (210), where

OJ = J-ln.(kj ,D,8)ID_ C k;- $:)--1'J -9 D, ,\:" v W

Then we note that gIi ((, 8) ~ 9 (n, k, (, 8) ~ gii ((, 8) and use the estimates (280)for g~((, 8) and estimates (281) and (283) for J-l~(D, 8). Plugging the relevant estimates into the representation (210) and then using the inequality (298), we obtainthe inequality (296). 0

Lemmas 6.15 and 6.13 imply the following important statements.LEMMA 6.16. For any natural number No and a positive constant C there exists

a positive constant c such that for any (3 > C and for any w < c and any integerm E [0, No - 1] the following is true:

In particular,

Proof. The proof of the lemma is based on Lemmas 6.13 and 6.15. Note first ofo

all that in view of Theorem 6.5 the first No eigenvalues of the operator 8k 8 belongto the E-subspectrum, and therefore the corresponding eigenfunctions are o~'the formoWCO,m)' Now let us apply Lemma 6.13 and Proposition 6.14 for n = No -1. Then wehave

(302) 1 0 2Am,m = ( I WCO,m) (x)1 dx,

r 1 ,1

o~ m ~ No -1.

This identity, inequality (295), and the estimate of the norm llANo-III based onProposition 6.14 imply that

The last inequalities under the conditions of Lemma 6.13 lead to the estimates (300),which, in turn, imply (301). 0

LEMMA 6.17. For any constant N ~ 1r there exist positive constants C and csuch that for any (31 > C (see (18)), for any w < c, and for any integer m such that

(303)

the following inequalities hold:

In particular,


Proof The proof of this lemma is similar to the previous one, but it is a littlebit more complicated, since now we have to take care of the eigenfunctions associatedwith both the E-subspectrum and the H-subspectrum. In fact, the condition (303)enables us to treat both cases more or less uniformly.

Let us estimate first the total number L of the eigenvalues (taking in account theirmultiplicity) which satisfy the condition (303). We note that in view of Theorem 6.6the total numbers LE and LH of the eigenvalues from the E-subspectrum and the Hsubspectrum, respectively, satisfying the condition (303) can be roughly estimated asLE ~ Nw- 1, LH ~ N. These bounds enable us to estimate the maximal permissibleindices m and n for both subspectra. These numbers can be used in the estimates(295) and (296). This along with (302) and with the similar identity for index n leadsto the inequality maxAz,z ~ 60N8w- 1. This inequality, in view of Proposition 6.14,implies for small w that IIALII ~ 64N28w-2. This bound together with (287) leadsimmediately to the estimates (304). The estimates (305) follow from (304) if we takeinto account (18). 0

Proof of Theorems 6.1 and 6.2. The proof of these theorems is a combination ofo

Theorems 6.5 and 6.6 for the operators 8C;,8 and of the inequalities provided by theLemmas 6.16 and 6.17.

Let us start with the Theorem 6.1. The inequalities (301), Theorem 6.5, andtrivial observation that 8 = (3-1 w lead to the statement of Theorem 6.1.

As far as Theorem 6.2 is concerned, our intention is to use the inequalities (305)in order to modify the results of the Theorem 6.6. We now observe, using (18),that (3-1 = (312w < (322w, which under the condition (32 » 1 of Theorem 6.2 implies(3-1 « w. The last inequality allows us to drop (3-1 in the estimate (253). In additionto that, we note that (312 = (323w, which under the condition (32 » 1 yields (312 « w.Combining these observations with the inequalities (305) and Theorem 6.6, we cometo the statement of Theorem 6.2. 0

7. Spectrum of the Maxwell operator.

7.1. General properties and Floquet-Bloch theory. We will use the notation introduced in the beginning of the paper, in particular (3), (4) for the domainsX and X', standard basis vectors ej, 1 ~ j ~ 3, in the space R 3 , and the followingdomains in R 2 . The dielectric constant €(x) satisfies (5) and (6). We remind that forthis kind of periodic dielectric media we will be interested in the waves propagatingalong the plane (e1' e2), which leads to the condition that the fields Hand E dependon coordinates Xl and X2 only. That is, from now on we shall assume that

Let us now make the definitions of our operators precise. First of all, due to thetwo-dimensionality of the problem we will work with the spaces L2 (R2 , c 3 , €dx) andL 2 (R2 , C 3). Consider in these spaces the linear subspaces

(306)

(307)

J1 = {E(X1' X2) E L2(R2, C 3, €dx) I~ .€E = O},

J2 = {H(X1' X2) E L 2(R2, C 3) I~ . H = O},

J1 C L2(R2,C3,€dx), J2 c L2(R2,C3).

In both cases we understand the divergence in the distributional sense. These subspaces are obviously closed in the corresponding spaces. Our main Hilbert space will

1608

be the direct sum

(308)


It is well known that for the waves traveling along the (el' e2) plane there are twopossible polarizations. Mathematically speaking, they correspond to the splitting ofthe space J into the direct sum of two special subspaces:

(309)

where the subspace SE (of E-modes) consists of vector fields of the form

((0,0, E), (HI, H 2 , 0)),

and SH (the space of H-modes) consists of fields

((E1 , E 2 , 0), (0,0, H)).

Let us now turn to the operators. We define the curl operator as an operator actingbetween the spaces J1 and J2 in the following way: its domain D 1 C J1 , as of anoperator on J1 , is the set

(310)

(311)

(312)

(313)

(315)

(316)

(317)

Here, as before, the differentiations are understood in the distributional sense. Nowcurl = \7x is naturally defined on D 1 and maps this domain into J2 (which is trivial tocheck). It is convenient to look closely at the conditions that functions from D 1 mustsatisfy. Let E(Xl' X2) belong to D 1 . Representing E = (E1 , E2 , E3 ), and denotingE' = (E1 , E 2 ), we get

\7E3 E L2 (R2, C 2

),

div€E' = 0,

8E2 _ 8El E L2(R2 ).

aXl aX2

Here we understand "div" in (312) in the 2D sense. Analogously, we define

(314) D2 = {H(Xl' X2) E J2 1 \7 x H E L2 (R2, C 3

; €dx) (= L2 (R2, C 3

))},

and for H = (HI, H 2 , H 3 ) E D 2 , H' = (HI, H 2 ) we have

\7H3 E L2 (R2 ,C2),

\7. H' = 0,

8H2 _ 8Hl E L2(R2 ).

aXl aX2

The operator-€-I\7x maps D 2 into J1 . We can now define the operator

M = [ 0 Ci€(X)-I\7X]-ci\7x 0 '

where c is the velocity of light. The domain of this operator is

(318)


This is our definition of the relevant Maxwell operator. It is easy to check that theoperators

Ci€(X)-l curl: J2 ~ J1 , -cicurl: J1 ~ J2

are adjoint to each other (see, for instance, [BS] for the thorough discussion of suchoperators in a more general case), so the Maxwell operator M has the structure

M = [1* ~]for some closed operator A and hence is self-adjoint. Looking at (311) and (315),one immediately sees that the direct decomposition (309) leads to the correspondingdecomposition of the domain D (since the conditions in (311) and (315) are imposedindependently on the third and the first two components of magnetic and electricfields):

(319)

Besides, direct computation shows that the operator M leaves the spaces SE and SHinvariant. Hence,

M_[MlsE 0 ]- 0 MlsH '

where MIsE and MlsH are some self-adjoint operators in SE and SH, respectively.We can now conclude that

(320) a(M) = a(MlsE) U a(MlsH)·

It turns out that the spectrum aH(M) :== a(MlsH) is exactly that studied in the firstpart [FK] of the paper. The other part of the spectrum, aE(M) :== a(MlsE)' is themain subject of our current investigation. Let us collect the relevant facts about theoperator MIsE.

(a) It acts in the space

(321)

where H' == (HI, H 2 ), and the divergence is understood in the 2D sense: \7. H' ==

8H1/8x l + 8H2 / 8x2.

(b) The domain of the operator is characterized by the conditions

(322)

(c) The matrix representation of the operator is

(323)

(324)

We will also need the operator

(82 82)ME == _€-1~ == _€-1 _ + _

8xi 8x~

defined in L 2 (R2 , €dx) with the domain equal to the Sobolev space H 2 (R2).


THEOREM 7.1. A real number ,\ belongs to the spectrum a E (M) of the self-adjointoperator MIsE if and only if,\2 belongs to a(ME ).

The proof of this theorem will follow from some results about the Floquet expansions of the relevant operators. (These results will be described later on in thissection.)

LEMMA 7.2. The domain of the operator MIsE can be described as follows: itconsists of all vector functions F = (E, HI, H2) E [HI(R2)]3 such that

(325) \7 . H' = 8H1 + 8H2 = O.8XI 8X2

The HI-norm on the domain is equivalent to the graph norm.Proof First of all, any F that satisfies the conditions of the lemma obviously

belongs to the domain of MIsE. Thus let us consider the converse statement. Conditions E E £2 and \lEE £2 (see (311)) imply, by the definition of Sobolev spaces,that E E H I (R2 ), since

(326)

(327)

Now, conditions (325) (which is one of the conditions on elements of the space J) and(315) imply that H' satisfies the system of equations

{8HI/8xI + 8H2/8x2 = 0,8H2/ 8xI - 8HI/ 8x2 = f

for some f E £2(R2). Ellipticity of this system enables one to gain smoothness ofsolutions. Namely, let us denote by h j (for j = 1,2) the Fourier transform of H j .

After the Fourier transform the system (327) becomes

(328) {~lhl (~) + ~2h2(~) = 0,-~2hl(~) + ~lh2(~) = -il(~).

Solving (328), we get hI = i~21(~)/1~12, h2 = -i~11(~)/1~12, and, hence,

(329) Ihl2 = Ih I 12+ Ih212 = IJl2/1~12.

Expressing now the square of the HI-norm of H' as

r Ih(~)12(1 + IW2d~ + r Ih(~W(1 + IW2d~,llf-I~I llf-I>I

and using (329) we conclude that the following estimate holds:

I 2 2 1 2(1 + 1~1)2 I 2 I 2(330) IIH IIH! ~ 411hll£2 + C IfI 1~12 ~ C{IIH 11£2 + IlcurlH IIL2}'

If-I>I

The estimates (326) and (330) imply that all components of the field belong to theSobolev space H.1 and that the HI norm is equivalent to the graph one. 0

Now we remind the reader of some basics of the Floquet theory (see details in[RS] , [K93]). The operator MIsE is invariant with respect to translations on elementsof the integer lattice Z2. According to the standard scheme (see [RS], [K93]), thisshould lead to some direct integral decomposition of our operator:

(331) MIsE = i ED

M(k)dk,


where M(k) is some measurable self-adjoint operator function on K. This will mean,in particular, that

We are going to describe this decomposition in more detail. We will also show thatthe spectra of operators M (k) are discrete. This will lead to the proof of Theorem 7.1.

Let us consider the following Floquet transforms: for f E L 2 (R2 , c 3 ) we set

(332)

(333)

[(k,x)f(k,x)== L f(x-m)e ikom ,

mEZ2

f(k,x) == e-ikox[(k,x), x EX', k E K.

Here K == [0, 211"] X [0, 211"]. (We also need the (noncompact) set K' == K\211"Z2.) Thesetransforms are correctly defined if we consider the sum as a Fourier series in variablesk with values in the Hilbert space L2 (X', C3). (See [K93] for thorough study andhistory of these transforms.) The reason why we restrict the values of the argumentx only to the set X' is that the function [(k, x) satisfies some natural relations withrespect to translations: [(k,x+n) == eikon[(k,x) for all n E Z2. Therefore, its valuesare determined completely if they are given on any fundamental domain of the groupZ2, in particular, on X or on X'. We get the isometry F : f ~ [between L2 (R2, C 3 )

and J: L 2 (X', C3) (see [K93], [RS]).We have to study the behavior of the subspace BE and of the domain of the

operator MIsE with respect to the transform (332). First of all, the images of thespaces L 2 and HI under the transform (332) are known (see [K93]). We present theseresults in the next two lemmas.

LEMMA 7.3 (see [K93, Thm. 2.2.5]). Transform (332) is an isometry of the spaceL2(R2) onto the space L 2(K, L2(X')), where the latter denotes the Hilbert space ofsquare integrable functions on K with values in the space L2(X').

Remark. It will be convenient for us to identify the space L 2 (K, L 2 (X')) withL 2 (K', L 2 (X')), which is obviously possible.

To describe the image of the Sobolev space H 1 (R2 ) we need to define some newobjects. Consider the Sobolev space H 1(X') and its (closed) subspace £1 (wherek E K) that consists of all HI-functions that satisfy the cyclic (or Floquet) boundaryconditions

(334)u(l + 8/2, X2) == eik1 u(8/2, X2),U(Xl' 1 + 8/2) == eik2 u(xl, 8/2).

This subspace is closed due to the embedding theorems. It was shown in [K93,Thm. 2.2.1] that £1 == ukEK£l is an analytic subbundle of the trivial Hilbert bundle

K x H 1(X') over K. In fact, the bundle £1 is defined over the whole space C 2 as ananalytic (trivial) Bilbert bundle (see [K93]). One of the results of Theorem 2.2.5 in[K93] can be rephrased as follows.

LEMMA 7.4. Transform (332) is an isometry of the space H 1 (R2 ) onto the spaceL 2 (K, £1), where the latter denotes the Hilbert space of square integrable sections overK of the bundle £1 (i.e., the subspace of the space L 2 (K, H 1 (X')) that consists of allfunctions g(k) such that g(k) E £1 for almost all k E K).

Remark. As in the previous lemma, we can identify the space L 2 (K,£I) withL2(K', £1).


Our next task is to describe the image of the space BE (which is a subspace of[L2(R2)]3) under the transform F. According to (321) and Lemma 7.3, it reduces todescribing the image of the subspace of (L2(R2))2 that consists of vector functionsH = (HI, H 2 ) that have zero divergence: divH = o. The distributional meaning ofthe zero divergence condition is that

(335)

for all functions <p E CO(R2 ). The expression in (335) is obviously continuous in<p E H 1 (R2

); hence (335) is still valid on the whole space H 1 (R2). We conclude that

F = (E, HI, H 2 ) belongs to BE if and only if

(336) F = (E, HI, H2) E [L2(R2)]3 and

(H, ~<p) = 0 for all <p E H 1 (R2).

We will rewrite condition (336) in terms of the transform F. Since the operator~ commutes with translations, and due to Lemma 7.4, we conclude that (336) isequivalent to

(337)

for all L2-sections 9 of the bundle £1. Here ~x denotes the gradient operator withrespect to the variable x. Let us consider the action of this operator for different valuesof the "quasi momentum" k. We denote by ~(k) the restriction of the operator ~ ontothe space £1, and consider ~(k) an operator between the spaces £1 and [L2(X')]2.We also consider the restriction of the bundle £l to the space C 2 \21rZ2 (in particular,to K'). This restriction is certainly an analytic Hilbert bundle by itself.

LEMMA 7.5. (i) ~(k) produces an analytic morphism of the bundle £1 over C 2

into the trivial bundle C 2 x [L2(X')]2.(ii) This morphism has zero kernel over C 2 \21rZ2 , and its range is closed in the

every fiber over C 2 \21rZ2 .

Proof Analyticity is standard: using the transform f ~ 1 (333) instead of thetransform F (332), we get the operator family

where T 2 is the 2D torus obtained from the unit cube X' (see simple details in [K93]).This operator function is obviously analytic (and even linear) with respect to k. Usingthe same representation of our operator function we can check the statements aboutthe range and the kernel. Namely, let us expand functions on T 2 into the Fourierseries:

g(x) = L gl e27ril.x.

lEZ2

Then

(~x + ik )g(x) = i L (21rl + k )gl e27ril.x.

lEZ2


If now k is not in the dual lattice 21rZ2, then none of the vector coefficients (21rl + k)can vanish, so the kernel of the operator contains only zero. (On the dual lattice,however, the kernel is not trivial.) Let us note now that the mapping

i L (21rl + k )gZ e27riZ.x ~ L gz e27riZ.x

ZEZ2 ZEZ2

(defined only on the range of the operator) is uniquely defined and continuous fromL 2 to HI topology, provided again that k is not in 21rZ2. Namely, the square of theHI-norm of the right hand side can be estimated as follows:

~ 1 12(1 + Ill)2 == ~ 1 12121rl + k12

(1 + Ill)2LJ gz LJ gz 121rl + kl2

ZEZ2 ZEZ2

~ 2 (1 + Ill)2 ~ 1 2= LJ 191 (21fl + k)1 121fl + kl 2 ::;: C LJ 91 (21fl + k)1 ,

ZEZ2 ZEZ2

which is the L 2-norm of i2::zEZ2 (21rl + k )gZ e27riZ.x. In the last inequality we employed

the fact that for any k f/:. 21rZ2 we have 121rl + kl ~ Ck (l + Ill) for alll E Z2. Now thestatement about the range follows from the estimates above. 0

This lemma implies the following.COROLLARY 7.6. The image of the morphism \7 (k) is an analytic subbundle 1m\7

in the trivial bundle over C 2\21rZ2 with the fiber [L2(X')]2.Proof The statement is local in k. Locally we can assume that the base U of our

bundles is a polydisk (and hence is a Stein manifold). According to Lemma 7.5, fork E U the operator \7(k) is left invertible. Hence, due to the results of [A] (see alsoTheorem 4.4 in [ZK]) there exists an analytic left inverse operator T(k) for k E U.The composition \7(k)T(k) provides an analytic projection operator onto 1m \7(k),which proves the statement. 0

We now introduce another analytic subbundle of C2\21rZ2 x [L2(X')]2 that hasthe fiber over the point k E C 2\21rZ2 equal to (1m \7(k))..L C [L2(X')]2. This newbundle is denoted by £0, and its fibers are denoted by £2. (We understand hereorthogonality in the meaning of the bilinear form JF . F instead of the standardscalar product.)

We are now ready to describe the image of the space SE under the transform(332). Namely, comparing (337) with the statement of Corollary 7.6, we get thefollowing statement.

LEMMA 7.7. The image of the space S E under the transform (332) coincides withthe space of sections

Remark. The reason to consider the set K' instead of the whole fundamentaldomain K is n~w obvious: the bundle £0 has singularities at the points of the duallattice. This is an effect that does not arise in the general theory of elliptic equations (see [K93]) but naturally arises for operators that can be included into ellipticcomplexes. (See the Floquet theory for such complexes in [P].)

Our next step is to describe the image of the domain D(MlsE) of the operatorMIsE under the transform (332). According to Lemma 7.2, D(MlsE) can be describedas the kernel in the space [HI (R2)]3 of the continuous operator

A : [H I (R2)]3 ~ L 2(R2), \vhere A(E, HI, H 2) == divH.


Let us apply now the transform (332). According to Lemmas 7.3 and 7.4 thespaces L2 (R2 ) and [H1(R2 )]3 transform correspondingly into L2(K',L2 (X:)) and[L2 (K', £1 )]3. The operator A transforms into the multiplication by the analytic morphism that is equal to zero on the first component of the vector function (E, HI, H 2 )

and coincides on [£1]2 with the analytic morphism of [£1]2 into L2 (K',L2 (X')) thaton the fiber over k is equal to A(k) = divl[£~]2. Switching to the transform (333)

instead of (332), we can prove analogously to Corollary 7.6 the following statement.---1

LEMMA 7.8. The kernel of the morphism A(k) is an analytic subbundle £ in thebundle [£1 ]2I C 2\27rZ2 over C 2\27rZ2.

Proof First of all, the statement is local in k, so we will consider a small bidiskneighborhood of a point k E K'. Let us switch to the transform (333) and expandinto Fourier series on T 2. After that we get the operator function

Expanding the function 9 into the Fourier series

L gz e27riZ.x,

ZEZ2

we can represent the operator as

(\7 + ik) . 9 = L i(27rl + k) . gz e27riZ.x.

Assuming that k ~ 27rZ2, one can easily show surjectivity of this operator. Namely,let

f = L fz e27riZ.x E L 2 (T2 ).

ZEZ2

Since (27rl + k) =I- 0, we can find a vector gz such that i(27rl + k) . gz = fz and

Igzi = 1(2l{~k)l· Then obviously 9 = 'EZEZ 2 gZ e27riZ.x belongs to [H 1(T2)]2 and solves

the equation (\7+ik)·9 = f, which shows the surjectivity. Now, as in Corollary 7.6, weuse Allan's theorem (this time for the right inverse operators) and get the statementof the lemma. 0

Remark. We notice that for 9 E KerA(k) we have gz -l (27rl+k). We are ready todescribe the image of the domain D(MlsE) of the operator MIsE under the Floquettransform (332).

LEMMA 7.9. The image of D(MlsE) under the transform (332) coincides with

2 ' 1 ---1 2' 1 ---1L(K,£)tBL2 (K,£)=L(K,£ tB£).

Now we have to consider the action of our operator MIsE: D(MlsE) ---+ BE, whereD(MlsE) denotes the domain of the operator MIsE described in Lemma 7.2. Firstof all, the differential expression (323) defines a continuous operator from the space[H1 (R2 )]3 into the space [L2 (R2 )]3 and from the space [H1(X')]3 into [L2 (X')]3.In the first of these two cases, restricting the operator to the space BE, we get theoperator MIsE. In the second one, for k E K' we restrict the operator to the fiber

---1 ,over k of the bundle £ 1

X £ and get some analytic morphism M (k) over K between


1 ---1 , , °the bundles £ EB £ and (K x L 2 (X )) EB £ . This is an analytic morphism, since it isthe restriction to an analytic subbundle of a constant morphism of trivial bundles. Asimple exercise is to check that multiplication by M(k) is the image of the operatorMIsE under the Floquet transform (332):

(338) - -(MF)(k,·) = M(k)(F)(k, .).

This formula is another way of saying that the decomposition (331) holds.We now need to establish some properties of the operator M(k).LEMMA 7.10. Operator M(k) defined in L2(X';cdx) EB £2 by the formula (323)

---1with the domain £1 x £k is self-adjoint and has a discrete spectrum.

Proof First of all, direct calculation shows that M(k) is symmetric on its domain.Its self-adjointness, after switching to the transform (333), amounts to the fact thatany weak solution F = (E, HI, H 2) in [L2(T2)]3 of the system of equations

-icc-1 (8/8x2 + ik2)H1 + icc-1 (8/8xl + ik1 )H2 = 0,

(8/8xl + ik1 )H1 + (8/8x2 + ik2)H2 = 0,

-ic(8/8x2 + ik2)E = 1/;1,ic(8/8x2 + ik2)E = 1/;2,

cp, 1/;1, 1/;2 E L 2 (T2),

belongs to [HI (T2)]3. Expanding into Fourier series and using estimates as in theproof of Lemma 7.2, one can easily show that F E [H1(T2)]3 as soon as k is not onthe dual lattice.

Discreteness of the spectrum follows from the fact that the domain of M (k )equipped with the graph norm (see Lemma 7.2) is compactly embedded into thespace where the operator acts. 0

COROLLARY 7.11. The following equality holds:

(339)

where the bar denotes closure.Proof First we say that A E R is not in UkEK,a(M(k)). (Nonreal values of A are

not interesting due to self-adjointness.) We will show that then A does not belong toa(MlsE). Let p = dist(A, UkEK,a(M(k))). Consider the morphism that on the fiber

over the point k E K' is equal to M(k) - A. Due to the assumption, it is invertiblein each fiber. Hence, there is the inverse analytic morphism (M (k) - A) -1. Due toself-adjointness, we get that the norm of (M(k) - A) -1, as an operator in L 2 (X') EB £2,is at most p-l (uniformly with respect to k). Then we conclude that the operator ofmultiplication by (M(k) - A) is invertible in L2 (K', L2 (X') EB£o), and its inverse is theoperator of multiplication by (M(k) - A)-I. Since the multiplication by (M(k) - A)is the Floquet image of the operator MIsE - A, the latter one is also invertible, soA tt: a(MlsE)·

The converse statement is the following: if A E UkEK,a(M(k)) , then A E

a(MlsE). This part is in general more delicate than the first one, but it follows as inTheorem 4.5.1 in [K93]. 0

We are prepared now to prove Theorem 7.1.Proof of Theorem 7.1. Let A E a(MlsE)' and due to Lemma 7.10, A E

UkEK,a(M(k)). We will have to show that A E a(ME ). Due to the closedness of


the spectra, it is sufficient to assume that A E UkEK' a(M (k)). According to Lemma

7.10, there is k E K' such that A is an eigenvalue of the operator M(k). Hence; there---1

exists a vector function F =(E,H1 ,H2) E £1 X £k such that M(k)F = AF. In otherwords, all three components E,H1 , and H2 belong to the space H 1 (X'), they allsatisfy the cyclic conditions (334), the last two components satisfy the condition

8H1 + 8H2 = 08Xl 8X2 '

and in addition they satisfy the system of equations

(340)

(341)

(342)

If A =I- 0, then we can express HI and H2 from (341) and (342) and plug theminto (340). This will lead to equality -c2c-1 C:1E = A2E. At the same time, fromthe equalities (341) and (342) we conclude, first, that E E H 2 (X') and, second,that not only the function E itself but also its partial derivatives satisfy the Floquetcyclic conditions (334). Since E =I- 0 (otherwise F would also be equal to zero), weconclude that the equation (-c2C:1- A2c)E = 0 has a nonzero solution in H 2(X') withconditions (334) on the function and its partial derivatives. Then Theorem 4.5.1 in[K93] says that zero is in the spectrum in L 2(R2) of the operator (-c2C:1 - A2c) andhence of the operator (-c2c-1C:1- A2). The last remark is that the spaces L 2(R2) andL 2 (R2 , cdx) have the same elements and equivalent norms. We conclude, therefore,that A2 E a(ME)'

Now, let us assume that A2 (=I- 0) E a(ME ). According to the same theorem, 4.5.1in [K93] , applied to the operator (-c2 C:1 - A2c), this means that there exists a pointk E K such that the equation (-C2C:1-A2c)E = 0 and, hence, the equation (-c2c- 1C:1A2 )E = 0 have a nontrivial solution in H 2 (X') that satisfies together with its partialderivatives the conditions (334). Since the spectra of these boundary value problemsME(k) depend continuously on k E K, we conclude that A2 E UkEK,a(ME(k)).Hence, it is sufficient to show that if A E UkEK,a(ME(k)), then A E UkEK,a(M(k)).So let k E K', and A2 and E are an eigenvalue and the corresponding eigenfunctionof the cyclic boundary value problem

(-c2c-1 C:1 - A2 )E = 0 in X',

E(1 + 8/2, X2) = eik1 E(8/2, X2), E(Xl,1 + 8/2) = eik2 E(Xl' 8/2),

\7E(1 + 8/2, X2) = eik1 \7E(8/2, X2), \7E(Xl' 1 + 8/2) = eik2 \7E(Xl' 8/2).

Determining HI and H2 according to (341) and (342), we conclude that (340) andthe condition (325) are also satisfied, so A E a(M(k)) C UkEK,a(M(k)).

We have to consider the case of A = 0 separately. The thing is that zero belongsto both spectra: to a(ME) as well as to aE(M). For the operator ME this is obvious,since zero belongs to the spectrum of the Laplacian in L 2 (R2 ). Now consider theoperator MIsE' We choose a vector k E K' with arbitrarily small norm and a unitvector a that is orthogonal to k. Now consider the vector function F = eik.xa. Then


F belongs to the domain of M(k), and the L2-norm IIM(k)FII is small (although theL2-norm of F is fixed). Hence, 0 E UkEK,a(M(k)). 0

Analogously to Theorem 7.1, the following can be proven.THEOREM 7.12. A real number A belongs to the spectrum aH(M) of the self

adjoint operator MlsH if and only if A2 belongs to the spectrum a(MH) of the operator

where the operator MH is defined in L 2 (R2 ) in an appropriate weak sense (see detailsin [FK]).

The spectrum a(MH) (and hence aH(M)) was described in [FK], so according to(320) and Theorems 7.1 and 7.12 our task is reduced to investigation of a(ME ) only.

7.2. Proof of the main results. All the statements of Theorems 2.2 and 2.3(except for (28)) on the spectrum of Maxwell operator a(M) follow immediately from(13), Theorems 6.1 and 6.2 for the operator 8 e (i.e., for E-polarized fields), andTheorem 1 from our paper [FK] for the operator r e, associated with H-polarizedfields. As far as the estimate (28) for the first band of the operator r e is concerned,we can mention that in [FK] we estimated the width of this band by Cw for someconstant C. Now based on our results for one-dimensional problems, we can makethis estimate more precise and come to (28).

We showed in [FK] that the upper limit of the first band of the operator r e issmaller than the upper limit of the first band of the following operator with separatevariables:

oAnalysis of the spectral structure of the operator reboils down to analysis of the one-dimensional operator -aycl1ay acting in L 2 (R). We note that the spectrum of thatone-dimensional operator is the same as the spectrum of the operator 8 1, e1 = -cl1a;acting in L2 (R, C1 (y )dy). Spectral analysis of the operator 8 1, e1' in turn, can bereduced to the analysis of a Schrodinger operator in the same fashion as we did forthe 2D operator 8 e . In fact, it is easier. Thus, in order to find the spectrum ofthe operator 8 e , we introduce the Schrodinger operator Q~ = - ::2 - g(c1(Y) - 1).This operator is almost the same as the operator Q"",g defined by (239), and we canrepresent it in terms of the operator Q (~, D, 8) as Q~ = Q (~, g(w- 1 - 8), 8), 9 2:: o.Therefore, the upper limit u of first band of the operator Q~ solves the followingequation for g:

If we denote u = Dw- 1 , we obtain the equation for D : ~o(1r, D, 8) = Dw. UsingLemma 4.3(ii) and Lemma 5.7(iii), we conclude that D = 4 + O(w). Hence, theupper limit for the first band of the operator 8 e equals 4 + O(w) and, therefore, the

oupper limit of the operator r e is 8+O(w). This leads straightforwardly to the desiredestimate, (28), which completes the proof of Theorem 2.2.


7.3. Space distribution of the electric field energy. In this section we consider how the electric field energy is distributed in the space for the E-polarizedeigenmodes of the Maxwell operator. We are interested in the portion of the energyresiding in the areas with high dielectric constant (i.e., in thin "walls" in our medium)in comparison with the energy residing in the areas with low dielectric constant (aircolumns).o We consider the cell of peJiod X (see (5)) and the eigenfunctions of theoperator 8(, 8. We pick the operator 8(, 8 with separate variables instead of 8 e , sinceit is simpler. We have already shown that these operators have practically the samespectra, and so we expect that the properties of the eigenfunctions are similar too. Letus denote the portion of the electric field energy associated with an eigenmode \lI andhigh dielectric constant area in X by £d (\lI ,X) and the corresponding energy in lowdielectric constant area filled by air by £a(\lI, X). Let us pick now an eigenfunctiono 0

\lin (k, (, 8, x) of the operator 8k, (, 8. Since the density of the electric field energy is

c(x)E~(x), x = (Xl, X2), we have

(343)

(344)

t'd(in, X) = ({ lin (k,(,8,x)12

dx,lx-os

t'a(in,X) = ( { lin (k,(,8,x)12

dx.los

o 0 0

The ratio £d(\lIn, X)/£a(\lIn, X) significantly depends on whether \lin is associatedwith E -subspectrum or H-subspectrum. Our main tools here will be Lemmas 5.11

oand 5.12. Let us consider first an eigenfunction \lI(O,n) as in formula (299). Then

(345) ( lico,n) (k,(,8,x)12

dxlx-os

~ 1D'fo(k1, D, 8; x)1 2 dx11

Ifn(k1, D, 8, x)1 2 dxlD=D(n, k, (, 8)

(346) ( lico,n) (k,(,8,x)12

dxlos

::; lD'fo(k1, D, 8; x)1 2 dx11

Ifn(k1, D, 8, xW dxlD=D(n, k, (, 8)

Combining formulas (343)-(346) with Lemma 5.12 and with the bound (273) forD(n, k,(, 8), we get

(347)

o

£d(\lI(O n),X) -1 [v-2 sinh v ()]0' >W +01 ,

£ ('TI X) - I(kl,v)a ':l'(O,n),

Since W ---+ 0,. the former relationships say that for the eigenmodes from the Esubspectrum the overwhelming portion of their electric field energy resides in thinwalls with large dielectric constant.

oLet us consider now an eigenfunction \lin, n = (nl' n2) > 0 associated with the

H-subspectrum. It has the following form (see (255) and (256)):


This representation implies

1619

Combining the inequalities (348) and (348) Lemma 5.11 and the inequality (280), wecome to the estimate

(350)

o

[d(~n, X) :::; L (1 - cos kj)-lO (w + ,812 ) .

£a(\lJn, X) j=1,2

Here we took into account that 13w = 13? (see (19)). The last inequality implies that ifo< kj < 7r, j = 1,2, and if w, 1311 ---+ 0, then for the eigenmodes from H -subspectrumthe overwhelming portion of their energy resides in the air columns with dielectricconstant 1.

Acknowledgment. We are very grateful to the reviewers of our paper for theirvaluable suggestions.

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1620

[J87]

[J91][JMW]

[K82]

[K93][LL]

[M]

[MM]

[MPD]

[MBRJ]

[P]

[PSS]

[PM]

[RS]

[RMP]

[SSE]

[vAL]

[VP]

[WVG]

[V]

[YG]

[ZK]

[ZS]


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#' lafigotin/papers/figotin kuchment band...key words. propagation of electromagnetic and...

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