Eigenvalue branches of the perturbed Maxwell operator M+λD in a gap of σ(M)

Eigenvalue branches of the perturbed Maxwell operator M + λ D in a gap of σ ( M )Dong Miao Citation: Journal of Mathematical Physics 49, 113508 (2008); doi: 10.1063/1.3013121 View online: http://dx.doi.org/10.1063/1.3013121 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/49/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in New trace formulae for a quadratic pencil of the Schrödinger operator J. Math. Phys. 51, 033506 (2010); 10.1063/1.3327835 The fractional Schrödinger operator and Toeplitz matrices J. Math. Phys. 50, 103524 (2009); 10.1063/1.3237146 On the eigenvalues of the twisted Dirac operator J. Math. Phys. 50, 063513 (2009); 10.1063/1.3133944 The discrete spectrum of the periodic Maxwell operator perturbed by a decreasing potential J. Math. Phys. 49, 063511 (2008); 10.1063/1.2948895 On eigenvalues in gaps for perturbed magnetic Schrödinger operators J. Math. Phys. 39, 63 (1998); 10.1063/1.532308

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Eigenvalue branches of the perturbed Maxwell operatorM+�D in a gap of �„M…

Dong Miaoa�

Institute of Mathematics, Jilin University, Changchun 130012, People’s Republic of China

�Received 18 April 2008; accepted 13 October 2008; published online 14 November 2008�

The propagation of guided waves in photonic crystal fibers �PCFs� is studied. APCF can be regarded as a perfectly two dimensional photonic crystal with a linedefect along the axial direction. This problem can be treated as an eigenvalueproblem for a family of noncompact self-adjoint operators. Under the assumptionthat the background spectrum has a gap, we prove that a line defect can create aneigenvalue of any given fixed value in the gap, provided that the defect is strongenough. Based on a decoupling of regions in R2 by means of Dirichlet and Neu-mann boundaries, then using the trace ideal estimates, we study asymptotic distri-bution of eigenvalues and bounds on the number of eigenvalue branches. In par-ticular, we show that if the defect is weak enough, no eigenvalues can be createdinside the gap. © 2008 American Institute of Physics. �DOI: 10.1063/1.3013121�

I. INTRODUCTION

Photonic crystals are periodically structured dielectric media, which are designed to favorband gaps, i.e., monochromatic electromagnetic waves of certain frequencies cannot propagatethrough these structures. The fact that photonic crystals exhibit band gaps that bear a resemblanceto semiconductors has great importance in physics.35,38 Since the first proposals of a photonic bandgap effect by Yabolonovitch53 and John,36 lots of applications have been studied. Among theseapplications, photonic crystal fibers �PCFs� as fundamental transmission medium to guide elec-tromagnetic wave have been intensively studied, see, e.g., Refs. 3, 4, 18, 21, 41–43, and 49 andreferences therein. PCFs consist of a periodic array of two different optical transparent materialsrunning through the length of the fibers with a central line defect which serves as the core for lightguiding. Physically, guided waves �or guided modes� can be created in these structures, i.e.,electromagnetic waves of certain frequencies propagating along the line defects of these structuresare exponentially decaying in the transversal directions.

From the mathematical point of view, the study of the propagation of electromagnetic wavesin PCFs can be treated as an eigenvalue problem for a family of noncompact self-adjoint opera-tors. It is worth noting that these operators can be regarded as the perturbations of the periodicMaxwell operators �according to the background medium, i.e., the perfectly two dimensionalphotonic crystals�. It is well known that the spectrum of a self-adjoint differential operator withperiodic coefficients exhibits band structure. Sometimes bands can be separated by gaps. Theexistence of gaps in the spectrum of the periodic Maxwell operator has been studied by manyauthors. We refer to Figotin and Kuchment26 and Hempel and Lienau33 for scalar model, Figotinand Kuchment27 for two dimensional photonic crystals, and Filonov28 for the full vectorial Max-well operators. As an application of the high contrast results on the existence of gaps developed in,e.g., Refs. 26, 27, and 33 to the scalar model of the PCFs, it was shown in Ref. 52 that anappropriate choice of the wavenumber in the axial direction leads to an effective high contrast,thus a gap can be created. �Here, no attempt is made to give a complete survey on this subject.�

a�Electronic addresses: [email protected] and [email protected]. Presently at Department of Mathematics, Michi-gan State University, USA.

JOURNAL OF MATHEMATICAL PHYSICS 49, 113508 �2008�

49, 113508-10022-2488/2008/49�11�/113508/33/$23.00 © 2008 American Institute of Physics


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http://dx.doi.org/10.1063/1.3013121

http://dx.doi.org/10.1063/1.3013121

http://dx.doi.org/10.1063/1.3013121

In this paper, under the assumption that the background spectrum has gaps, we analyze theeffects of the line defect in a photonic crystal on the background spectrum. It is well known insolid state physics that if a defect �or impurity� is introduced in a periodic background medium�such as a crystal�, then energy levels �i.e., eigenvalues� can be created in the gaps of the back-ground spectrum, with the corresponding eigenmodes localized exponentially around the defect.

Although it is shown in many works that electromagnetic waves, or more generally, classicalwaves, share great similarities with electron waves, there are also some significant differencesboth on the physical aspect and on the mathematical aspect. It will be clear in the text that standardarguments of the Schrödinger operators �and sometimes, even the divergence-type operators� failto work for the Maxwell operators. On the physical aspect, it is well known that classical waves�and particularly, electromagnetic waves� are harder to localize �see, e.g., Refs. 22, 24, and 37 formore details�.

In this paper, we continue the study of guided waves due to a line defect introduced in a twodimensional photonic crystal. In a previous paper,45 we showed that line defects do not change theessential spectrum of the background spectrum of the two dimensional operator generated on thecross section. Thus the effect of the line defect inside a gap of the background spectrum can onlybe the creation of isolated eigenvalues of finite multiplicity. In Ref. 46, we gave a sufficientcondition on the parameters of the background medium and of the line defect that ensures the riseof eigenvalues in a specified subinterval of the given gap in the background spectrum. Physically,this means that if the background spectrum has gaps, it is possible to guide electromagnetic waveswith suitable cores. Moreover, we proved that the corresponding eigenmodes decay exponentiallyaway from the line defect. In this paper, we use the operators M and M +�D to describe the“background” medium �i.e., the perfectly two dimensional photonic crystal� and the perturbedmedium �i.e., the PCF�, respectively. The main questions concern the set of �’s so that some givenfixed � in the gap of ��M� is an eigenvalue of M +�D, and we also study the dependence of thenumber of eigenvalues on � as �→�.

The study of eigenvalue bounds of Schrödinger operators below a certain energy level goesback to Birman,9 see also Refs. 14, 16, and 13 and references therein. For eigenvalues in thespectral gaps of Schrödinger operators, there are at least two different approaches. One is due toDeift and Hempel,20 see also Refs. 2 and 29–31. The other is Birman’s approach, see, e.g., Refs.5–12 and references therein. �However, no attempt is made to give a complete survey on the vastliterature in this research field.� For divergence-form operators, see e.g., Refs. 1, 7, and 50.However, there are very few works on the Maxwell operators, we refer to Ref. 25.

The methods which we use in this paper are a development of certain ideas of Alama et al.,1,2

Figotin and Klein,23 and Hempel.30 However, we will see in the text that the vector nature ofelectromagnetic waves poses additional problems. It should be pointed out that although similarresults were obtained in Refs. 1 and 44, some of the techniques used in this paper are different.

A. Maxwell’s equations and guided waves

Consider a lossless linear isotropic dielectric medium occupying the whole space R3. Weadopt the following notation:

x� = �x�,x3�� R3, x = �x1,x2�� R2.

The function ��x�� which describes the medium is called electric permittivity. We suppose that ��x��is invariant under any translation in the third normal direction x3, i.e., ��x�� is independent of x3,

��x�� = ��x� . �1.1�

In particular, if ��x� is a periodic function satisfying

113508-2 Dong Miao J. Math. Phys. 49, 113508 �2008�


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��x + n� � = ��x� for all x � R2, n� � � �1.2�

�where � is a lattice in R2�, this structure is called a �two dimensional� photonic crystal �or bandgap material�.38 Furthermore, a PCF can be created if a line defect along x3-direction is introduced.We describe the defect strip by

�l ª �x� = �x�,x3�� R3:x ��l, x3 � R� , �1.3�

where �lª l�, 0 l�, is the support of the defect in the cross section. We suppose that � is ameasurable subset of R2. Without loss of generality, we also suppose that 0 is an inner point of �.Inside the defect, the medium can be different from the background medium.

It is well known that the propagation of time-harmonic electromagnetic waves in any non-magnetic, lossless, linear isotropic dielectric medium in the absence of free charges and currentslooks as follows:

�� E�x�� + i�H�x�� = 0, � · ��x��E�x�� = 0,

�1.4�� H�x�� − i��x��E�x�� = 0, � · �H�x�� = 0.

If the medium is a PCF, guided waves �or guided modes� can be expected to propagate along thex3-direction. The rigorous definition of guided waves is as follows.

Definition: Guided waves/guided modes are solutions of Eq. �1.4� on the form

E�x�� = �E1�x�,E2�x�,E3�x��e−i x3,

�1.5�H�x�� = �H1�x�,H2�x�,H3�x��e−i x3,

with

�R2

��x��E�x��2 + �H�x��2�dx� ,

where

E�x� = �E1�x�,E2�x�,E3�x��, H�x� = �H1�x�,H2�x�,H3�x��,

and �0 is the wave number in the x3-direction.We introduce the following notation:

� = �1

�2

0 − i 0

0

1 = �1

�2

− i ,

where �1=� /�x1, �2=� /�x2. Furthermore, we formally define

� � = ��1�,�2�,− i ��,

� � u� = ��2u3 + i u2,− �1u3 − i u1,�1u2 − �2u1��,

� · u� = �1u1 + �2u2 − i u3,

where u� = �u1 ,u2 ,u3�� and �=��x� is a scalar function.Plugging formulas �1.1� and �1.5� into Eq. �1.4� and eliminating E, one can obtain

113508-3 Eigenvalue branches of the perturbed Maxwell J. Math. Phys. 49, 113508 �2008�


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� � �−1� � H = �2H . �1.6�

The object of this work is the study of the eigenvalue problem �1.6�.

B. Statement of the results

We consider perturbations of spectra of the Maxwell operators M =� �p�x�� , where p�x��describes the properties of the background medium. Since the medium is assumed to be invariantunder any translation in the third normal direction x3, we have p�x��= p�x�. We always assume thatp�x� is a measurable function satisfying

0 p− p�x� p+ � a.e. �1.7�

for some constants p�. Typically, one may think of M as a periodic Maxwell operator accordingto a two dimensional photonic crystal, namely,

p�x + n� � = p�x� x � R2, n� � � , �1.8�

where � is a lattice in R2. However, the above periodicity condition is unnecessary in this paper.As we mentioned above, we shall suppose in this paper that M has at least one spectral gap. Weshall also use q�x��=q�x� to describe the properties of a line defect �which serves as the core forlight guiding� along the x3-direction. Note that the line defect is also invariant under any transla-tion in the x3-direction. We suppose that q�x� is a measurable function satisfying

q�x� → 0 as�x� → � �1.9�

and

0 q�x� q+ � a.e. �1.10�

for some constant q+. Then, we formally introduce the perturbed Maxwell operator,

M�ª M + �D = � � p��x�� ,

where p��x�ªp�x�+�q�x�, ��0. It is well known that the perturbations which decay at infinitydo not change the essential spectrum of the background spectrum �Weyl’s theorem48�, namely,�ess�M��=�ess�M�. Thus the effect of the line defect inside a gap of ��M� can only be the creationof isolated eigenvalues of finite multiplicity. In Ref. 46, we suppose that �a ,b� is a spectral gap ofM and let the interval ��−d ,�+d�� a ,b�. We also suppose that p��x� is constant in �x�R2 : �x� l��. Then we proved that the perturbed operator M has at least one eigenvalue in the interval��−d ,�+d�, provided that the defect is strong enough �in the sense that the size of the supportand/or the strength of the defect is large enough�. Using the modified Combes–Thomas arguments,we also proved that the corresponding eigenfunctions decay exponentially away from the linedefect. Note that the periodicity condition �1.8� for p�x� is unnecessary for deriving the aboveresults.

For some given fixed level � in a gap of ��M�, the main questions in this paper concern theset of �’s so that � is an eigenvalue of M +�D and the growth of the number of eigenvalues as�→�. We can also think of this problem in terms of the generalized eigenvalue problem: givenD=� �q�x�� and � in a gap of ��M�, we solve �� ,u��R+�L2�R2 ;C3� such that

�M − ��u� = − �Du� . �1.11�

As in Ref. 30, in mathematical language � is called a generalized eigenvalue of the problem �1.11�with weight D, while � is an eigenvalue of the self-adjoint operator M�; in physics, � is a couplingconstant and � is an energy level. Note that � lies in the spectral gap, M −� is not a positiveoperator, thus the existence and asymptotic distribution of generalized eigenvalues no longerfollow directly from standard technique in the theory of Schödinger operators. �For details seeTheorem XIII.10 of Ref. 48.�



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Now we give the precise definition of the Maxwell operator. We consider the Maxwell op-erator,

M = � � p�x�� ,

on L2�R2 ;C3�. It is uniquely defined by the closure of the non-negative densely defined quadraticform,

A�u� ,v�� = �� u� ,p�x�� v��, u� ,v� � C01�R2;C3� .

�See Ref. 45 for more details about the basic properties of the operator M.� The operator M� canbe defined similarly. Note that q�x� might have compact support, thus the self-adjoint operatorD=� �q�x�� on L2�R2 ;C3� defined by quadratic forms might have Dom�D�(Dom�M�.However, in this paper we do not use D as an operator in its own right, and in the combinationM +�D the operator D is always restricted to Dom�M�.

In the sequel, we suppose G= �a ,b�, 0ab�, is a gap of ��M�. Namely, there exists aninterval I= �a� ,b�� , 0a�abb�, such that ��M�� I= �a� ,a�� b ,b��.

Inspired by the Schrödinger case, one might want to use the Birman–Schwinger principle tostudy the eigenvalues created by a line defect. But, unfortunately, the kernel ��q� ��M−��−1�� q� is not compact. To avoid this difficulty, in Sec. III we use the strategy proposed byFigotin and Klein,25 where almost the same results can be found. �Here we give all the details forthe sake of completeness and clarity.� To be more precise, we turn to study the resolvents.

Set

R�z� = �M − z�−1 for z � ��M� , �1.12�

R��z� = �M� − z�−1 for z � ��M�� , �1.13�

and

V = R��− 1� − R�− 1� .

Clearly, V0. We will prove in Theorem 3.1 that V is a Hilbert–Schmidt operator, then we canuse the Birman–Schwinger principle for the equation

R��− 1� = R�− 1� + V .

More precisely, the eigenvalue problem

M�u� = �u� , �� G , �1.14�

can be rewritten as

R��− 1�u� = �M� + 1�−1u� = �� + 1�−1u� , �� G .

On the other hand, we can also rewrite the eigenvalue problem

R��− 1�u� = R�− 1�u� + Vu = su� , s � ��R�− 1�� ,

as

u� = − �R�− 1� − s�−1Vu� .

We further set

R�s� = �− V�R�− 1� − s�−1�− V , �1.15�

then we get the eigenvalue problem



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R�s�v� = v� , where v� = �− Vu� . �1.16�

We will show in Proposition 2.1 that �1.16� is equivalent to �1.14�. We will also show in Theorem3.1 that R�s� is a self-adjoint compact operator �in fact, it is a Hilbert–Schmidt�. Hence we canstudy the eigenvalue problem

R��v� ª − R�� + 1�−1�v� = − v� , �� G ,

instead of studying the original eigenvalue problem �1.14� directly. Here R�� is a Hilbert–Schmidt operator on the form

R�� = �� + 1��− V�M − ��−1�M + 1��− V .

Since �M −��−1 is a monotonically increasing norm-continuous operator of ��G, we have thefollowing.

Proposition 1.1: The map ��G�R�� is norm continuous and operator monotone increas-ing.

Note that R�� is a self-adjoint Hilbert–Schmidt operator, its spectrum consists of eigenvaluesof finite multiplicity with 0 being the only possible point of accumulation. Let �1��2��¯0 be the sequence of the negative eigenvalues of R��, counting multiplicity. Clearly, wehave the following.

Proposition 1.2: �� j�� j=1� are monotonically increasing continuous functions for ��G.

Finally, we have the following theorem, which is proven in Sec. III.Theorem 1.3: Let p�x� and q�x� satisfy �1.7� and �1.10�, respectively. Suppose that G

= �a ,b� is a gap of ��M� . Then the only possible point of accumulation of ��M��G is a (i.e., thelower edge of G). Moreover, the eigenvalues �1�2¯ of the operator M� in G coincide withthe set of the solutions of the equations,

� j�� = − 1, j = 1,2, . . . , �1.17�

where �� j�� j=1� are the negative eigenvalues of the operator R�� . Furthermore, if v� i is an

eigenfunction of the operator R��i� according to the eigenvalue � ji��i�=−1 , then

u� i = − �M − �i�−1�M + 1��− Vv� i

is an exponentially localized eigenfunction of M� according to the eigenvalue �i.Remark:

�i� Note that for line defects strong enough, eigenvalues in G created by line defects alwaysexist �cf. Ref. 46�. As a consequence, we can guarantee the existence of a solution for someof the Eq. �1.17�. In fact, as we shall see later, we shall give a stronger conclusion that canguarantee the existence of eigenvalues in G created by line defects �for details, see Sec.IV�.

�ii� Physically speaking, Theorem 1.3 tells us that one can push localized states up from thelower edge of the continuous spectrum by increasing the dielectric parameter p�x� in someregion. Similarly, one can also pull localized states down from the upper edge of thecontinuous spectrum by decreasing the dielectric parameter p�x� in some region.

Note that Theorem 1.3 is just a very elementary result for the study of perturbed spectra.To state our further results precisely, we need some notation. We introduce the following

eigenvalue distribution counting function:

N��,q,�� ª �0s�

dim Ker�M + sD − �� = card�0 s �:�� Ms�� , �1.18�

where ��G with G be a gap in ��M� and Ms=M +sD.



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In order to see the effect of the line defect on the gap G, we also define another countingfunction,

N��,q,�� ª card�� b:�� M�� . �1.19�

Clearly, N�� ,q ,�� describes the number of eigenvalues of M� in �� ,b� created by the line defect.Since these eigenvalues are continuous in s �0s�� by regular perturbation theory �cf. Ref. 40�and are strictly monotone, they must pass through � as s moves from 0 to �. This implies that

N��,q,�� N��,q,�� . �1.20�

In Sec. IV, we shall prove the “completeness,” more precisely, we have the following.Theorem 1.4: Suppose that a level � lies in G . Then there exists a number l�0 such that if

q�x��0 on the disk �x�R2 : �x� l� , then ��M�� for some �=��0.Remark: It is worth noting that �i� this theorem generalizes the result in Ref. 46, where the

perturbed medium inside the support of the defect is assumed to be homogeneous; �ii� the resultobtained in Ref. 46 is weaker than Theorem 1.4; �iii� the proof of this theorem is more simple andis different from Ref. 46. It is also quite different from the methods in Refs. 1, 24, and 44, wheresimilar results were obtained. Note that their methods do not work here, for details, see the remarkat the end of Sec. IV.

After Sec. IV, we further suppose that

�p L��R2�, �q L��R2� C �1.21�

for some constant C�0.In Sec. V, we shall prove that for any given fixed coupling constant ��0, line defects with

sufficiently small support in the cross section �x1 ,x2� do not create eigenvalue branches that crossany given fixed level ��G for all 0s�. More precisely, we have the following.

Theorem 1.5: Suppose that a level � lies in G . Let q�x� satisfy �1.21�. For any given fixedcoupling constant ��0 , there exists l0= l0��0 small enough such that if supp q�x�� x�R2 : �x� l0� , then

N��,q,�� = 0,

where the counting function N�� ,q ,�� is defined in �1.18�.In Sec. VI, we shall estimate the number of eigenvalue branches crossing any given level �

�G. We find that the upper bound of N�� ,q ,�� can be controlled by the measure of the diskcontaining suppq�x� and is independent of the coupling constant �, provided that q�x� has compactsupport.

Theorem 1.6: Let q�x� satisfy �1.21�. Suppose that supp q�x�� x�R2 : �x� l� for some l�0 . Then we have

N��,q,�� C�� − p− 2�l2 for � � p−

2,

where p− is defined in �1.7� and the constant C is independent of � and q�x�.Remark: We will see in Sec. VI that it is reasonable to require ��p−

2, since the case �p−

2 leads to the trivial result: N�� ,q ,��=0.Finally, in Sec. VII, we study various asymptotic behaviors of the perturbations. In Sec. VII A

we suppose that

lim�x�→�

��q�x� = 0 for all � = ��1,�2�, �� 1, �1.22�

and



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q�x��C

�1 + �x��s for some C, s� 0. �1.23�

The main result in this subsection is as follows.Theorem 1.7: Let q�x� satisfy �1.21�–�1.23�. Then there exists a positive constant C1 such that

N��,q,�� C1�1 + 2��2/s

for � large enough.In Sec. VII B we suppose that

0 q�x�C

�1 + �x��s for some C, s� 0. �1.24�

The main result in this subsection is as follows.Theorem 1.8: Let q�x� satisfy (1.21) and (1.24). Then there exists a positive constant C2 such

that

N��,q,�� C2�� − p−−1 2��2/s �1.25�

for � large enough, where p− is defined in �1.7�.In Sec. VII C we suppose that for some constants q� and s�0,

lim�x�→�

�x�sq�x�/p�x� = q�. �1.26�

We also suppose that the density of states � for M exists and is independent of the Dirichlet andNeumann boundary conditions. �See Sec. VII C, for details.�

Our main result in this subsection is given by the following.Theorem 1.9: Let q�x� satisfy (1.26) and (1.21). Suppose that the density of states �� ,M�

exists and is independent of boundary conditions. Then

lim�→�

�−2/sN��,q,�� = lim�→�

�−2/s�0

�

d��R2

dx��1+�q��x�−s��+�q��x�−s��

= �0

�

d��meas�x:� � ��1 + q��x�−s�−1� ,

where �� denotes the characteristic function of the set � .Our method of proof is based on a decoupling of regions in R2 by means of Dirichlet

boundary conditions for Maxwell operators and the trace ideal estimates.At the end of this subsection, we shall compare some other related problems with ours.

�i� We shall first compare our problem with the analogous Schrödinger problem in the colortheory of crystals �cf. Refs. 2 and 20�. In that case, H=−�+V �with V real valued, bounded,and measurable� describes the pure crystal, and the potential W0 describes the “impu-rity.” Also, W�x� decays and E is in a gap of ��H�. Then one may ask for some ��0 suchthat E��H−�W�. This question has been studied in Refs. 2, 29, 7, and 30. However, ifwe just want to find some ��R such that E��H−�W�, then the problem will becomemuch easier. Because it is well known that this problem can be reduced to show that theBirman–Schwinger kernel �W�1/2�H−E�−1�W�1/2 has a nonzero eigenvalue. But the existenceof such an eigenvalue follows from the classical existence theory �cf. Ref. 48� as�W�1/2�H−E�−1�W�1/2 is a nonzero compact self-adjoint operator.

However, in this paper we also notice that the operator D is not relatively compact with respectto the Maxwell operator M. �Note that the perturbation term is of the same order as theunperturbed operator M.� Thus we can find ourselves in a more difficult case.



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�ii� Next, it is worth noting that an important trick of Hempel’s approach to Schrödingeroperators and divergence-form operators �cf. Refs. 20, 1, and 30� is to use �u as anapproximation of some scalar function u�L2��, where � is defined in �1.27� of Sec. I C.Clearly, they have �u�L2��. However, for the Maxwell analog here, the vector natureposes nontrivial difficulty. More precisely, for some reason that will be clear in the text, wehave to consider the Maxwell operator on the space Sª �v� �L2�� ;C3� :� ·v� =0�, but forany nonzero vector function u� �S, we cannot also expect that �u� �S. Thus we cannot applyHempel’s method directly. Moreover, as we have already mentioned in the remark belowTheorem 1.4 that the methods in Refs. 1, 24, 44, and 46 do not work in the proof ofTheorem 1.4, for details see the remark at the end of Sec. IV.

�iii� Finally, we shall compare our problem with the traditional dielectric waveguide �cf. Refs.17 and 39�. The traditional dielectric waveguide is a cylindrical medium infinite in eachdirection, invariant with respect to the geometry as well as the physical characteristics ofthe medium �here, the dielectric permittivity�, under any translation in a privileged direc-tion, for instance, the x3-direction. It is also assumed that each cross section �i.e., parallel to�x1 ,x2� plane� of the medium is homogeneous outside some bounded domain. In this case,the essential spectrum according to the medium shall occupy the whole semiaxis, say,�a ,�� for some a�0. Joly and Poirier39 showed that there is no embedded eigenvalue inthe essential spectrum, except possibly for the lower edge a of the essential spectrum. Theyalso proved that the essential spectrum is stable, i.e., local perturbations in the cross sectiondo not change the essential spectrum. Thus this implies that all these eigenvalues can becharacterized via the min-max principle. However, we can see in this paper that the opera-tor M −� is not a positive operator when � lies in a gap of ��M�. Hence the existence andasymptotic distribution of generalized eigenvalues of the generalized eigenvalue problem

�M − ��u� = − �Du�

no longer follow directly from classical methods.

C. Notation

We introduce the following notation.

• PS= PS�A� denotes the spectral projection of A, where S is a Borel set.• Bl denotes the disk �x�R2 : �x� l� for some l�0.

• �l=�l�x� denotes the square �y�R2 :0yj −xj l , j=1,2� for some l�0; �l is the closureof �l. The outer normal vector on ��l is denoted by n� .

• �l�x� , l�0 denotes a C0��R2� cutoff function with the properties

�l�x� = 1 for �x� l ,

�l�x� = 0 for �x�� l + 1,

�1.27�0 �l�x� 1 for x � R2,

��l L� C�, uniformly for l� 0.

• ��x� denotes a C0��R2� cutoff function satisfying

��x� = 1 for �x� 1,

��x� = 0 for �x�� 2,



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0 ��x� 1 for x � R2,

�� L�, �� L� C� .

We also introduce

�l�x� = �� x

l� for l� 0, �1.28�

then, clearly

��l L� C

l, ��l L�

C

l2 ,

where C is a constant, independent of l.• R�z� denotes �M −z�−1.• Let A and B be self-adjoint operators that are non-negative on a Hilbert space H. We say

A�B�0 if Dom�A��Dom�B� and �Au ,u�� Bu ,u��0 for any u�Dom�A�.• A compact operator A is of class Ts, 1s�, if A sª �tr�A�s�1/s�, where �A�= �A�A�1/2; Ts

is a Banach space. Particularly, A is a Hilbert–Schmidt operator if A�T2.• For any self-adjoint operator A,

NA��, � ª card�� s :s � ��A�� = dim P��, ��A� .

1. Notational disclaimer

In this paper, the letter C, with or without subscript, denotes generic constants that may not bethe same at different occurrences.

II. PRELIMINARIES

The purpose of this section is to formulate some auxiliary results that will be used in theremainder of this paper.

A. The Birman–Schwinger principle

We recall the well-known Birman–Schwinger principle �cf. Refs. 1, 2, 27, and 34�.Proposition 2.1: Let H0 be a self-adjoint operator with G= �a ,b� be its spectral gap. Suppose

V�0 is bounded and is relatively compact with respective to H0 . Let ��G . Then the Birman–Schwinger kernel V1/2��−H0�−1V1/2 is compact and the followings are equivalent.

�i� � is an eigenvalue of H0+�V of multiplicity m.�ii� �−1 is an eigenvalue of V1/2��−H0�−1V1/2 of multiplicity m . Particularly,

NH0+�V��,b� card�0 s �:�� H0 + sV�� . �2.1�

Proof: The proof is standard and we just sketch the proof of the inequality �2.1�. Suppose t� �� ,b� is an eigenvalue of H0+�V. Then there exists u�0 so that

�H0 + �V�u = tu .

Since t�G, we have Vu�0. Furthermore, Kato–Rellich perturbation theory40 tells us that theeigenvalues t�s� of the operator H0+sV are continuous in s and are strictly monotone,



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dt�s�ds

= ��Vu,u�� 0,

thus eigenvalues of H0+�V in �� ,b� must pass through � �from above� as s moves from � to 0.This implies �2.1�. Note that some eigenvalues of H0+�V might be b, thus we can only have theinequality in �2.1�. �

B. A modified Weyl’s decomposition

For technical reasons, we need a modified Weyl’s decomposition �cf. Refs. 39 and 45� whichis analogous to the classical one �for details, see, e.g., Refs. 15, 23, and 25�.

The space L2�R2 ;C3� can be decomposed to the direct sum of the spaces S and G,

L2�R2;C3� = S � G , �2.2�

where

S = �u� � L2�R2;C3�:� · u� = 0�

and

G = �� :�� H1�R2�� .

The sum �2.2� is orthogonal with respect to the inner product �· , ·�. The spaces S and G are leftinvariant by M. A straightforward calculation gives Ker M =G and Im M �S �cf. Ref. 45�. Wedefine M as the restriction of M to S, namely, Dom�M�=Dom�M��S and M=M �Dom�M��S. Notethat M =M � 0 �G, ��M�� p−

2 , +��, where p− is defined in �1.7� �see Ref. 45 for details�, and0��M�, so we have

��M� = ��M� � �0� .

Thus we can study either one of them at different occurrences for convenience.

C. A modified Combes–Thomas estimate

The Maxwell operator satisfies a resolvent decay estimate which is often called the Combes–Thomas estimate in mathematical physics.

Proposition 2.2: �Reference 46� Let R�z�= �M−z�−1 , z��M� . Then for any z� �a , b��G ,there exist constants C and ��0 such that

�1 − �n�� R�z��m Ce−��n−m�, 0 m n ,

for any fixed interval �a , b��G , and the constants C and � are all independent of z , n , and m.

D. The Maxwell operator with Dirichlet/Neumann boundary conditions

Our goal in this subsection is to obtain qualitative information about point spectrum for theMaxwell operator in a square �l of side l in R2 with Dirichlet/Neumann boundary conditions. Wedenote by M�

D the restriction of M to � with Dirichlet boundary condition. It is a non-negativeself-adjoint operator on L2�� ;C3�, uniquely defined by the closure of the quadratic form

A�D�u� ,v�� = �� u� ,p�x�� v��, u� ,v� � C0

1��;C3� . �2.3�

In particular, we write �lD=� �� for the case p�x��1 and �=�l.

Similar as in Refs. 15 and 23 �lD has an operator core consisting of functions which are C2 up

to ��l and whose tangential component vanishes on ��l. In fact, �lD describes a homogeneous

waveguide �whose cross section is �l� with perfectly conducting boundary. As in Ref. 23, we use



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this description to find all the eigenvalues for �lD. For u� �C��l ;C3�, we use ��u� and ��u� to denote

its outer normal and tangential components on ��l, respectively. Rigorously speaking, ��u� meansthat for u� �L2��l ;C3�, � ·u� �L2��l�,

�u� ,� �� = − �� · u� ,�� for all �� H1�� .

Similarly, ��u� =0 means that u� belongs to the domain of the closure in L2��l ;C3� of the operator� � introduced originally on fields of class C0

��l ;C3�. Indeed, the operator Rot given by theexpression � � on the domain

Dom�Rot � = �u� � L2��l;C3�:� � u� � L2��l;C

3�,��u� = 0�

corresponds to the minimal operator generated by the expression � �. �For nonsmooth case ofboundary conditions for Maxwell operators in bounded domains, we refer to Ref. 15�.

We also introduce the Maxwell operator M�N , the restriction of M to � with Neumann bound-

ary condition,

�� u�� ª � � �� u�� = 0.

It is a non-negative self-adjoint operator on L2�� ;C3�, uniquely defined by the closure of thequadratic form

A�N�u� ,v�� = �� u� ,p�x�� v��, u� ,v� � C1��;C3� . �2.4�

In particular, we write �lN=� �� for the case p�x��1 and �=�l.

The following proposition is a modified version of Theorem 30 in Ref. 23.Proposition 2.3: Let �l=�l�x�= �y�R2 :0yj −xj l , j=1,2� be a square in R2.

�i� The dense linear subset

ClD = �u� � C2��l;C

3�:��u� = 0�

is an operator core for �lD , with �l

Du� =� �� u� for u� �ClD.

�ii� The operator �lD has an orthogonal basis of eigenfunctions

�u��,j � ClD:��

lN�

lN� �− i

l �� 0��

lN� �− i

l ��

� � lN� �0�� − i

l ��, j = 0,1,2� , �2.5�

with

� � u��,0 = 0, u��,0 = � !�,0, with !�,0 � C0��l� , �2.6�

and

� � � � u��,j = ��2u��,j, � · u��,j = 0, j = 1,2. �2.7�

More precisely, if �l=�l�x0�, we have

u��,j�x� = v��,j�x − x0� , �2.8�

where



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v��,j�x� = a1��,j� cos��1x1�sin��2x2�

a2��,j� sin��1x1�cos��2x2�

a3��,j� sin��1x1�sin��2x2�

,

with each �� / l�N� � / l�N� �−i� / l� �� 0�� / l�N� �−i� / l� �� / l�N� �0�� −i� / l� ��. Here we set a�,0=� and we also choose a�,j , j=1,2 such that the vectors � ,

a1��,1�

a2��,1�

− a3��,1� ,

a1��,2�

a2��,2�

− a3��,2� ,

form an orthogonal basis for R3 .

�iii� Eigenfunctions of �lN have the form � �u��,j�x� , where u��,j�x� are given by �2.8�.

Proof: It is routine to prove this proposition as Proof of Theorem 30 in Ref. 23 with somemodifications. We give it in details for the sake of clarity. Let the operator �l be defined by �lu�=� �� u� for u� �Cl

D. To see that it is a symmetric operator on ClD, notice that for u� ,v�

�C1��l ;C3� we have

�� u� ,v�� − �u� ,� � v�� = ��l

� · �u� � v��dx = ��l

�u� � v�� · n�ds , �2.9�

where the inner products are in L2��l ;C3�, ds is the curve measure, n� = � �0

� with � be the unit outernormal vector on ��l, and we used Green’s formula. Note that if ��u� =0, i.e., n��u� =0, we musthave ��u��v��=n� · �u��v��=v� · �n��u��=0, so we conclude that the integral in the right hand side of�2.9� is equal to 0.

To prove that the symmetric operator �l is essentially self-adjoint, it suffices to construct anorthogonal basis of eigenfunctions in Cl

D. Note that �cos��n / l�x� :n� �0��N� and�sin��n / l�x� :n�N� are both orthogonal bases for L2��0, l��, it follows that �u��,j� given in �2.5� isan orthogonal basis for L2��l ;C3�. Since

u��,0 = � �sin��1x1�sin��2x2�� ,

we clearly have �2.6�. It is straightforward to check that �u��,j��ClD and �u��,j� also satisfies �2.7�,

so it is an orthogonal basis of eigenfunctions for the operator �l.To prove the first two conclusions of this proposition, it suffices to show that �l is the closure

�l of �l. To do so, note that C02��l ;C3��Cl

D�Dom��l�. As quadratic forms, we clearly have

�l�u��= �l�u�� for u� �C02��l ;C3�, which is a form core for �l

D, hence Dom��lD��Dom��l�. Since

ClD is a form core for �l as a quadratic form, to finish the proof of the proposition, it suffices to

show that ClD�Dom��l

D�, so Dom��l��Dom��lD�.

Thus, given u� �ClD, it suffices to find u� j �C0

1��l ;C3� such that

u� − u� j + � � �u� − u� j� → 0, as j → � . �2.10�

Translating and scaling, if necessary, we can assume that �l=�1�0�= �0,1�2. For each j=1,2 , . . . , we choose a function j �C2��0,1��, such that



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j�t� = 1 for 0 tj

j + 1,

j�t� = 0 forj + /12

j + 1 t 1,

0 j�t� 1 forj

j + 1 t

j + 1/2j + 1

.

Then we set w� j�x�ª" j�x�v� j�x�, where " j�x�ª j�x1� j�x2� and

v� j�x� ª�u�� j + 1

jx� , if 0 x1,x2

j

j + 1

u��x1,1� , if 0 x1 j

j + 1,

j

j + 1 x2 1

u��1,x2� , if 0 x2 j

j + 1,

j

j + 1 x1 1

0, otherwise.

�Clearly we have w� j�x��C0��1 ;C3� and w� j is piecewise C1 with bounded partial derivatives, thus� �w� j �L2��1 ;C3�. Moreover,

� � w� j = �� " j�� v� j +" j�� v� j� =" j�� v� j� ,

since �� " j��v� j =0 by our construction as ��u� =0. If each w� j was a C1-function, instead of onlypiecewise C1, we should be done, since u� j =w� j clearly satisfies �2.10�. To repair that we set u� j

=# j �w� j, where �# j� is a suitably chosen approximate identity, e.g., # j�x�= j2#�jx� for some positiveC� function #�x� on R2 with the support on �1 and �R2#�x�dx=1, so u� j �C0

1��l ;C3�, � �u� j

=# j � �� u� j�, and �2.10� is satisfied.An elementary calculation shows �iii�. �

As a consequence of Proposition 2.3, we give the modified Weyl’s decomposition correspond-ing to Dirichlet boundary condition which reads

L2��l;C3� = S�l

� G�l,

where S�land G�l

are the closure in L2��l ;C3� of the subspaces spanned by �u��,j , j=1,2� and�u��,0�, respectively. Here �u��,j , j=1,2� is given by �2.8�. Indeed,

S�l= �u� � L2��l;C

3�:� · u� = 0,��u� = 0�

and

G�l= �L2� − clos�C0

1��l;C3�:u� = � ! with !� C0

1��l;C�� .

The spaces S�land G� are left invariant by M�l

D . It is also easy to see that Ker M�l

D =G�l,

Im M�l

D �S�l. Thus we define M�l

D as the restriction of M�l

D to S�l, namely, Dom�M�l

D �=Dom�M�l

D ��S�land M�l

D =M�l

D �Dom�M�l

D ��S�l. Note that

M�l

D = M�l

D �S�l� M�l

D �G�l= M�l

D� 0G�l

and



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��M�l

D � � �p− 2, + �� , �2.11�

where p− is defined in �1.7�. Thus we have ��M�l

D �=��M�l

D �� 0�.We write �l

D for M�l

D when p�x��1. By �2.11�, we clearly have �disc��lD�=��l

D�� 2 ,+�� and �disc�M�l

D �=��M�l

D �� p− 2 , +��.

Similarly, Weyl’s decomposition corresponding to Neumann boundary condition reads

L2��l;C3� = S�l

� � G�l� ,

where S�l� and G�l

� are the closure in L2��l ;C3� of the subspaces spanned by �� u��,j , j=1,2�and �� ! :!�C��l��, respectively.

The Maxwell operator on general bounded domain � with Dirichlet/Neumann boundary con-ditions can be defined similarly.

Again, similar as Corollary 31 in Ref. 23, we have the following.Corollary 2.4:

�i� M�l

D has compact resolvent; in fact

tr��M�l

D + 1�−s�� for any s� 1. �2.12�

�ii� For any number E�p− 2 , let NM

�l

D �0,E� denote the number of eigenvalues of M�l

D less

than E , counting multiplicity,

NM�l

D �0,E� = dim P�0,E��M�l

D � = card�0 t E:t � ��M�l

D �� . �2.13�

Then there exists a finite constant C , independent of �l and p�x� , such that

NM�l

D �0,E� Cp−−1meas��l��E − p−

2�, E� p− 2, �2.14�

where p− is defined in �1.7�.

Proof: Since M�l

D �p−�lD, it suffices to prove the corollary for �l

D. Note that

�lD = � � � � = � �� ·� − � � I3 = − � � I3,

where � =�12+�2

2− 2 and I3 is the identity operator on C3. Moreover, �2.12� is the direct conse-quence of Theorem 4.5 in Ref. 45.

It follows from Proposition 2.3 �ii� that the spectrum of �l consists of eigenvalues whosemultiplicity can be read from �2.7�, so an explicit calculation gives �2.14�. �

Remark: Note that ��M�l

D �� 0�=��MlD�. Thus NM

�l

D �0,E� defined in �2.13� is also equal to

the number of strictly positive eigenvalues �counting multiplicity� of M�l

D less than E.Based on Corollary 2.4, we can further formulate Weyl’s law for Dirichlet Maxwell operators,

which is analogous to Weyl’s law for Dirichlet Laplacian �cf. Sec. XIII 15 in Ref. 48�. We writeMl

D=MBl

D in the case Bl= �x�R2 : �x� l� for notational convenience.Corollary 2.5: For E�p−

2, we have

NMlD�0,E� Cp−

−1 meas�Bl��E − p− 2� ,

where p− is defined in �1.7�.Proof: Note that the disk �x�R2 : �x� l� is Jordan contented �cf. Sec. XIII 15 in Ref. 48 for

the definition�; we can use the monotonicity under addition of Dirichlet boundaries and theirdecoupling properties to reduce the problem to one about squares which is then solved by Corol-lary 2.4. So it is routine to prove this corollary in the same way as Theorem XIII. 78 in Ref. 48.



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�

Similarly, we also have the Neumann analog.Corollary 2.6: For E�p−

2, we have

NMlN�0,E� Cp−

−1 meas�Bl��E − p− 2� ,

where p− is defined in �1.7�.

III. THE EFFECTS OF THE “POSITIVE” LINE DEFECTS

Our goal in this section is to prove Theorem 1.3. A defect is said to be positive if q�x��0.Theorem 1.3 follows from Theorem 3.1 and the considerations right before it. It is just an

elementary result for studying the effects of positive line defects.Theorem 3.1: Let R�s� be given by �1.15�. Then for any s��R�−1�� , the Birman–Schwinger

operator R�s� is a T2 operator (i.e., a Hilbert–Schmidt operator), where R�−1� is defined in�1.12�.

Proof: It suffices to prove that

V = R��− 1� − R�− 1�

is a Hilbert–Schmidt operator, where R��−1� is defined in �1.13�. We argue as Proposition 5.1 inRef. 25. Note that

�D = M� − M = �� q�x�� ,

as quadratic forms on the quadratic form domain Dom�M�, where the function q�x� satisfies�1.10�. It is also easy to see that

V = R��− 1��M − M��R�− 1� = − �R��− 1�� q�x�� R�− 1� = − �K��q�x�K ,

where K=�q� �R�−1�, K�=�q� �R��−1� with �q is the characteristic function of the setsupp q�x�.

Since K and K� are T2 operators by Theorem 4.2 in Ref. 45, we conclude that V�T2. �

IV. COMPLETENESS

Our goal in this section is to prove Theorem 1.4. Roughly speaking, this theorem tells us thatany given fixed level ��G can be an eigenvalue of the perturbed Maxwell operator M�, providedthat the support of the defect supp q�x� is large enough.

A. A commutation formula

Proposition 4.1: Suppose that T is a closed (unbounded) operator, S=T� and ST=T�T isdefined by the quadratic form B�f ,g�= �Tf ,Tg� . Then

�

ST + �+ S

1

TS + �T = 1,

in the sense that if ��0 and −��TS� , then −��ST� and �−1�1−S�TS+��−1T� provides theinverse for ST+� , and vice versa.

When S and T are bounded, the above formula is well known �see, e.g., Ref. 51�. However,the formula is also true when T is a closed �unbounded� operator �see, e.g., Refs. 19 and 1�.Particularly, for T= p�

1/2� �, T�= �� p�1/2, we have

− �

M� − �+ ��

1

� � � � − �p�−1 �� = 1,

where we have used the identity



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�p�1/2� � � � p�

1/2 − ��−1 = p�−1/2�� − �p�

−1�−1p�−1/2.

Again this formula is to be interpreted in the sense that if 0�� −�p�−1�, then ��M��.

Revising the role of T and T�, we get the following.Lemma 4.2: ��M��G if and only if 0��T�� , where

T� = � � � � − �p��x�−1.

Lemma 4.3: Suppose ��G . Then 0��T0� , i.e., T0 is invertible, where

T0 ª � � � � − �p�x�−1. �4.1�

By this lemma, there exists 0#1 such that

�− #,#� � ��T0� = 0” .

B. Proof of the completeness

We rewrite p��x�−1 as

p��x�−1 = �p�x� + �q�x��−1 = p�x�−1�1 + �q�x�p�x�

�−1

.

Then we define

p��x� ª lim�→+�

p��x�−1.

Note that p�x� satisfies �1.7�, and we also supposed in Theorem 1.4 that q�x��0 on Bl, thus it isclear that

p��x� = 0 for x � Bl

and

�p��x�� p−−1, a.e.

It follows that

T� → T�ª � � � � − �p��x�, as �→ + � ,

in the strong resolvent sense �cf. Refs. 40 and 47�.Furthermore one can study not only the strong resolvent limit T� of T� as �→� but one can

also look at the limit of T� as l→�, when the size of supp q�x� goes to �. We can also prove thatthe limit exists in the strong resolvent sense �cf., Ref. 48 Theorem VIII.25� and coincides with thefree operator � �� , whose spectrum is � 2 ,��. This implies the following.

Lemma 4.4: There exist � and l large enough so that

��T�� 2/2, + �� .

On the other hand, set

�0 = inf ��M� .

Clearly, �0�p− 2. Since G��M� is an inner gap, we can choose �1 so that

�0 �1 a � .

By Proposition 4.1, 0�� −�1p�x�−1�, but Lemma 4.3 shows that 0��T0�, where T0

ª� �� −�p�x�−1. These facts together with �p�x�−1��1p�x�−1 imply the following.



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Lemma 4.5: Let T0 be given by �4.1�, then

�− �,0� � ��T0� � 0” .

Proof of Theorem 1.4: Lemmas 4.4 and 4.5 together with Kato–Rellich theory imply that as �changes gradually from 0 to +�, the operator T� develops eigenvalues that pass through the point0, i.e., there exists some � so that 0��T��. Hence the conclusion of the theorem follows fromLemma 4.2. �

Remark: It is interesting to see that the main idea in the proof of Theorem I in Ref. 1 is notvalid for our case, which seems strange at first glance. Their methods illumine us that we shouldprove that there exists �0�0 so that for all ��0, ��T�� 0,#��0” . But this is wrong, since aswe have seen that the spectrum of the free operator � �� is bounded below by 2�0.

V. ABSENCE OF EIGENVALUES FOR WEAK DEFECTS

In the remainder of this paper we further suppose that p�x� and q�x� satisfy �1.21�.In this section we shall prove Theorem 1.5.Let M and M�

D be as in Sec. II B. In the case of �=Bm, we use the notation MmD=MBm

D forconvenience. Let G be a spectral gap of M. Now we are confronted with the difficulty that Mm

D

may have eigenvalues inside G, which are created by the boundary condition on �Bm. To avoidthis difficulty, we use the idea proposed in Ref. 2.

For any given ��G, we can fix numbers a1 ,b1 such that

�a1,b1� � �a,b� = G � R \ ��M� ,

and �� a2 ,b2�, where

a2 ªa + a1

2, b2 ª

b + b1

2.

Then in view of Corollary 2.5, the number of the normalized eigenfunctions u�m,n, u�m,n =1,associated with the eigenvalues rm,n of Mm

D in the interval �a2 ,b2� is finite. Moreover, we concludethat these eigenfunctions are exponentially localized near the boundary �B3m/4. More precisely,based on the modified Combes–Thomas estimate �Proposition 2.2�, we have the following.

Proposition 5.1: Suppose that u�m,n are the normalized eigenfunctions associated with theeigenvalues rm,n of Mm

D in the interval �a2 ,b2� , then

�m/4u�m,n Ce−�3m/4��, �5.1�

where � and �m/4 are defined in Proposition 2.2 and �1.27�, respectively, and the constant C isindependent of m .

Proof: To simplify notation we will suppress the index n, i.e., we write u�m=u�m,n and rm

=rm,n.We introduce the quadratic form

C�u�m,v�� = �Mu�m,v�� − rm�u�m,v�� 5.2�

for all v� �Dom�M�. Let v� =R�rm��m/4u�m, where R�z�= �M−z�−1. Then we have

C�v� ,u�m� = ��M − rm�u�m,v�� = �u�m,�m/4u�m� = �m/4u�m 2. �5.3�

On the other hand, using the identity MmDu�m=rmu�m, one can obtain

C�v� ,u�m� = ��M − rm�u�m,v�� = ��M − MmD�u�m,v�� = ��M − Mm

D�u�m,R�rm��m/4u�m�

= �pm� � u�m,�1 − �m−1�� R�rm��m/4u�m� ,

where pm= pm�x� is defined by pm= p�x� for x�R2 \Bm and pm=0 otherwise.



On: Thu, 27 Nov 2014 16:38:10

Note that

� � u�m 2 p−−1�� u�m,p� � u�m� = p−

−1�MmDu�m,u�m� = p−

−1rm.

Thus we can use the modified Combes–Thomas estimate �Proposition 2.2� and the fact supp�1−�m−1��supp pm to obtain

C�v� ,u�m� pm� � u�m �1 − �m−1�� R�rm��m/4u�m C �1 − �m−1�� R�rm��m/4

Ce−��m−1−�m/4�� = Ce−�3m/4��. �5.4�

Then �5.1� follows from �5.3� and �5.4�. �

Now we introduce the �finite dimensional� projection

Pm ª P�a2,b2��MmD� = �

n

�um,n, ·�um,n �5.5�

and the operators

MmD = Mm

D + �b2 − a2�Pm acting on SBm�5.6�

and

MmD = Mm

D + �b2 − a2�$mPm$m acting on SBm, �5.7�

where $m=1−�m/2 and SBmis defined in Sec. II B. It is easy to see that Mm

D is self-adjoint on

Dom�MmD�=Dom�Mm

D�=Dom�MmD� and Mm

D�MmD. Clearly, we have

��MmD� � �a2,b2� = 0” . �5.8�

Moreover, MmD has the following spectral property

Proposition 5.2: Let MmD be given by �5.7�. Then there exists an integer N such that

��MmD� � �a2,b2� = 0”

for all m�N.Proof: Using Proposition 5.1, we can argue as Proof of Lemma 4.3 in Ref. 2 to prove it. �

We also need another pair of operators with Dirichlet boundary conditions

M�,mD = � � p��x�� on SBm

�5.9�

and

M�,mD = M�,m

D + �b2 − a2�$mPm$m.

Proof of Theorem 1.5: Suppose by contradiction that there exist a sequence �qj� and normal-ized eigenfunctions u� j �Dom�Msj

�, u� j =1 such that

supp qj�x� � B j−1

and

Msju� j = �u� j, j � N ,

where Msj=� � �p+sqj�� .

Using cutoff functions �n of type �1.28�, we will show that

�nu� j → 1, as n → � , �5.10�

and



On: Thu, 27 Nov 2014 16:38:10

�Msj− ��nu� j� → 0, as n → � , �5.11�

uniformly for all j�N.Setting fn�x�=�n�x�−1. Similar to �5.2�, we introduce the quadratic form

D�u� j,v�� = �Mu� j,v�� − ��u� j,v�� for v� � Dom�M� .

Clearly,

D�u� j,v�� = ��M − Msj�u� j,v�� = − �sDj�u� j�,v�� .

Let v� = �M−��−1�fn2u� j�, then it follows from Proposition 2.2 that

fnu� j 2 = ��M − ��u� j,v�� = − �sDj�uj�,�M − ��−1�fn2u� j�� = − �sqj� � u� j,� � �M − ��−1�fn

2u� j��

sq+ � � u� j �supp qj� � �M − ��−1�supp fn

fn2u� j sq+Ce−��n−j−1� � � u� j ,

where �supp qjand �supp fn

denote the characteristic functions of supp qj and supp fn, respectively.Since

� � u� j 2 1

p− + sq−�psj

� � u� j,� � u� j� �p−−1,

we can conclude that for any given fixed coupling constant �, there exists N0�0 such that

fnu� j % for n� N0, independently of j .

Thus we have

1 − % �nu� j 1 for n� N0, independently of j . �5.12�

Then we turn to prove �5.11�.Since

supp qj�x� � B j−1 � B1 � �x � R2:�n�x� = 1� ,

we clearly have

sDj��nu� j� = s�nDju� j .

Then a straightforward calculation shows that

�Msj− ��nu� j� = �Msj

− ��fnu� j� = M�fnu� j� + sDj�fnu� j� − �fnu� j

= � � p� � �fnu� j� + sfnDj�u� j� − �fnu� j

= � � p�� fn � u� j + fn� � u� j� + sfnDj�u� j� − �fnu� j

= � � p� fn � u� j + � fn � p� � u� j + fn�� p� � u� j + sDj�u� j� − �u� j�

= � p� � fn � u� j + p� � � fn � u� j + � fn � p� � u� j + fn�Msju� j − �u� j� .

Using the identity

a� � b� � c� = �a� · c��b� − �a� · b��c�

and the fact � ·u� j =0 �since Msjis defined on S�, we get

p� � � fn � u� j = p�� · u� j�� fn − p�� fn�u� j = − p�� fn�u� j .

Thus we have



On: Thu, 27 Nov 2014 16:38:10

�Msj− ��nu� j� = � p� � fn � u� j − p�� fn�u� j + � fn � p� � u� j . �5.13�

Note that supp fn=R2 \Bn and

� fn L� +C

n + C, � fn L� 2 +

C

n2 2 + C .

Thus we have

�Msj− ��nu� j� � p L� � fn L� �supp fn

u� j + p+ � fn L� �supp fnu� j + � fn � p� � u� j

C �supp fnu� j + � fn � p� � u� j ,

where �supp fndenotes the characteristic function of the set supp fn=R2 \Bn.

From �5.10� we know that there exists N1�0 such that

C �suppfnu� j C fn−1u� j Ce−��n−1−j−1�

%

2for n� N1, independently of j .

�5.14�

Thus to obtain �5.11�, it remains to prove

� fn � p� � u� j → 0, as n → �, independently of j .

Note that ��n L�C /n, thus we also have �fn L�C /n.Let w� denote p� �u� j, a straightforward calculation shows that

� fn � p� � u� j = � fn � w� = �2fnw3 + i fnw2

− �1fnw3 − i fnw1

�1fnw2 − �2fnw1 = �2fnw3

− �1fnw3

�1fnw2 − �2fnw1 + i fn w2

− w1

0

¬ v�1 + v�2.

Since

� � u� j 2 1

p− + sjq−�psj

� � u� j,� � u� j� =1

p− + sjq−�Msj

u� j,u� j� p−−1�

and

� j fn L� �fn L� C

n,

we can conclude that there exists N2, such that for n�N2, v�1 % /2, independently of j. On theother hand, we need to prove that there exists N3 such that

v�2 = � fn w2

− w1

0� %

2for n� N3. �5.15�

But clearly, to prove �5.15�, the following inequality suffices:

fn� � u� j %

2 for n� N3.

Note that supp fn=R2 \Bn, and



On: Thu, 27 Nov 2014 16:38:10

fn� � u� j 2 1

p− + sq−�fn

2psj� � u� j,� � u� j�

=1

p− + sq−�� fn

2 � psj� � u� j,u� j� + �fn

2� � psj� � u� j,u� j��

p+ + sq+

p− + sq− � fn

2 � � u� j �supp fnu� j +

�

p− + sq− �supp fn

u� j 2 %

2 �by �5.14��

for n�N3, independently of j. Now we can conclude that there exists N4=max�N1 ,N2 ,N3� suchthat for n�N4,

�Msj− ��nu� j� %, independently of j .

This proves �5.11�.We set n1=N4 and m1=4n1=4N4. Note that

$m1�B2n1

= 0 and supp �n1� B2n1

,

thus we get

$m1Pm1

$m1�n1

u� j � 0.

Then we can conclude from �5.11� that

�Msj;m1

D − ��n1u� j� = �Msj

+ �b2 − a2�$m1Pm1

$m1− ��n1

u� j� = �Msj− ��nu� j� % .

�5.16�

It is reasonable that we can require % to be small enough such that

� �%

1 − %� �a2 + %,b2� .

Since for any self-adjoint operator Y,

dist�z,��Y�� = infu�Dom�Y�

u�0

�Y − z�u u

,

we conclude from �5.12� and �5.16� that

dist��,��Msj;m1

D ��%

1 − %,

which implies that Msj;m1

D has at least one eigenvalue in the interval ��−% / �1−%� ,�+% / �1−%��.To arrive at a contradiction, we argue as in Refs. 29 and 1. We first introduce the auxiliary

operators with Dirichlet boundary conditions,

M j−1,m1

DD = � � p�x��

on SBm1\Bj−1 and

M j−1,m1

DD = M j−1,m1

DD + �b2 − a2�$m1Pm1

$m1.

It follows from the min-max principle that the ith eigenvalue of the operator Msj;m1

D is sandwichedbetween the ith eigenvalue of Mm1

D and the ith eigenvalue of M j−1,m1

DD , i.e.,



On: Thu, 27 Nov 2014 16:38:10

ri�Mm1

D � ri�Msj;m1

D � ri�M j−1,m1

DD � .

However, as argued in Ref. 1 �Proof of Proposition 2.20�, standard results from capacity theory,combined with min-max principle, imply that the ith eigenvalue of M j−1,m1

DD converges �fromabove� to the ith eigenvalue of Mm1

D for j→�. But by Proposition 5.2,

��Mm1

D � � �a2,b2� = 0” ,

and therefore it follows that ��Msj;m1

D �� a2+% ,b2�=0” , which is a contradiction.This completes the proof of Theorem II. �

VI. BOUND OF EIGENVALUE BRANCHES FOR LARGE DEFECTS

Our goal in this section is to prove Theorem 1.6. First, we introduce the operator

M�;lD

ª M�;lD + �b2 − a2�Pl, on SBl

,

where M�;lD is defined in �5.9�.

Since ��Ms�=��Ms�� 0� for all s�0, it is clear that we also have

N��,q,�� = card�0 s �:�� Ms�� .

Note that ��Ms�� p− 2 , +�� for all s�0, then �p−

2 implies that N�� ,q ,��=0. So we onlyconsider the nontrivial case ��p−

2 in the sequel.In analogy with N�� ,q ,��, we also define

N��,q,�;MlD� = card�0 s �;�� Ms;l

D ��

for ��0.Lemma 6.1:

N��,q,�;M� N��,q,�;MlD�, � � 0,

for all l sufficiently large.Proof: This lemma is proven in the same way as Lemma 2.31 in Ref. 1, with the obvious

modifications to take into account that in this lemma we have � � instead of �. �

In the sequel, IBland IBl

denote the identity operators on L2�Bs ;C3� and SBs, 0s�,

respectively. However, we shall always suppress the subscripts of IBland IBl

for notational con-venience, since they can be understood appropriately in the context.

For technical reasons, we shall also introduce the following operators with Dirichlet boundaryconditions:

Ml,nDD = � � p�x�� acting on SBn\Bl

,

M�;l,nDD = � � p��x�� acting on SBn\Bl

,

and

Ml,nDD = Ml,n

DD + �b2 − a2�Pn,

M�;l,nDD = M�;l,n

DD + �b2 − a2�Pn

for 0 ln. The following lemma can help us to control the effect of the additional Dirichletboundary condition on �Bl.

Lemma 6.2: Let l�1 and n�2l . Then for any &�N , s=2& , we have



On: Thu, 27 Nov 2014 16:38:10

�MlD

� Ml,nDD + I�−1 − �Mn

D + I�−1 Ts

s Cl

for some constant C , independent of l and n .First of all, we introduce a cutoff function "l="l�x� such that

"l�x� = �1 for ��x� − l� 3

0 for ��x� − l�� 3.� �6.1�

To prove Lemma 6.2, we need Lemmas 6.3 and 6.4 below. Lemma 6.3 is due to Hempel30

�Lemma1.2�.Lemma 6.3: Let A and B be bounded with AB compact. Then for any &�N , s=2& ,

AB Ts

s B s−2 B�AsB T1.

Lemma 6.4: For any &�N , s=2& , we have

tr�"l�MlD + I�−s"l� Cl

for some constant C , independent of l .Proof: Similar as in Ref. 23, we introduce a modified Maxwell operator M which is elliptic.

Formally,

M ª M + Y ,

with Y =−� �p�x�� ·�, i.e., Yu� =−� �p�x�� ·u��. M is rigorously defined as the nonnegativeself-adjoint operator on L2�R2 ;C3�, given by the closure of the nonnegative quadratic form

F�u�� = �R2

p�x�� u� �2dx + �R3

p�x�� · u� �2dx, u� � C01�R2;C3� .

In particular, for p�x��1 we set

M0 ª �� + �− � � ·� = − � � I3,

where � =�12+�2

2− 2 acts on L2�R2 ;C3� and I3 is the identity operator on C3.By Weyl’s decomposition �2.2�, we have M=M � Y for appropriate operator Y on G. Indeed,

in the sense of quadratic forms,

�M + IS�−s� 0G �M + IS�−s

� �Y + IG�−s

= �M + I�−s �p−M0 + I�−s = �− p−� � I3 + I�−s¬ Zs � I3

for s�1. Thus, we can argue as Proof of Lemma A.1 in Ref. 30 that Zs is the convolution with abounded continuous function fs, given as the inverse Fourier transform of the function �p−� · �2+ p−

2+1�−s, and we see that

tr�"lZs � I3"l� 3fs�0�meas�x � R2:��x� − l� 3� = Cl .

This completes the proof of Lemma 6.4. �

Remark: Indeed, Lemma 6.4 is valid for all s�1.Now we give the proof of Lemma 6.2. It is essentially based on Lemmas A.1 and A.2 in the

Appendix of Ref. 30.Proof of Lemma 6.2: We consider the case where ln−3, the proof in the case where n−3

ln is similar.Step 1. We shall show in this step that, for any &�N, s=2&,



On: Thu, 27 Nov 2014 16:38:10

"l��MlD

� Ml,nDD + I�−1 − �Mn

D + I�−1�"l Ts

s Cl �6.2�

for some constant C, independent of l and n.Indeed, in the sense of quadratic forms,

0 �MnD + I�−1 − �Ml

D� Ml,n

D + I�−1 �MnD + I�−1.

Thus we get

"l�MnD + I�−1 − �Ml

D� Ml,n

DD + I�−1"l Ts "l�Mn

D + I�−1"l Ts.

Note that �b2−a2�Pn�0, we get

�MnD + I�−1 �Mn

D + I�−1.

Furthermore, we also have

�MnD + I�−1

� �Mn,�D + I�−1 �M + I�−1,

and for n� l+3,

"l��MnD + I�−1

� �Mn,�D + I�−1�"l ="l�Mn

D + I�−1"l.

Since all the steps decrease eigenvalues by min-max principle, and therefore increase the Ts-norm,namely,

"l�MnD + I�−1 − �Ml

D� Ml,n

DD + I�−1"l Ts "l�M + I�−1"l Ts

.

Note that "l�M+I�−s"l�0 and 0"l1. Thus we can use Lemma 6.3 to obtain

"l�M + I�−1"l Ts

s "l�M + I�−s"l T1= tr�"l�M + I�−s"l� .

Thus it follows from Lemma 6.4 that, for s�1,

"l��MlD

� Ml,nDD + I�−1 − �Mn

D + I�−1�"l Ts

s Cl

for some constant C, independent of l and n.Step 2. In this step, we shall prove that for any &�N, s=2&,

�1 −"l��MlD

� Ml,nDD + I�−1 − �Mn

D + I�−1� Ts

s Cl . �6.3�

Set

W � �MnD + I�−1 − �Ml

D� Ml,n

DD + I�−1.

For any u� �SBn, let v� = �Mn

D+I�−1u� . Since � · �MnDv��=0 and

� · �Pnv�� = �m

��u�m,n,v�� · u�m,n�� = 0,

where Pn is defined in �5.5�, it follows from the definition of MnD that

0 = � · u� = � · �MnDv� + �b2 − a2�Pnv� + v�� ,

which implies that � ·v� =0, i.e., v� �SBn. Similar arguments show that Wu� �SBn

for all u� �SBn.

For any u� �SBnwe further set



On: Thu, 27 Nov 2014 16:38:10

Xu� � �MnD + I��Wu�� .

It is clear that Xu� = �MnD+I��Wu��=0 in Bn \�Bl, in the distributional sense. In addition we have

�MnD + I�W �Mn

D + I��MnD + I�−1 + �Mn

D + I��MlD

� Ml,nDD + I�−1 2.

Thus X is bounded, independent of l and n. Since

�1 −"l�Wu� = �1 −"l��MnD + I�−1"lXu�

and MnD�Mn

D, we can further use Lemmas 6.3 and 6.4 to obtain

�1 −"l�W Ts

s �1 −"l��MnD + I�−1"l Ts

s X s C �MnD + I�−1"l Ts

s C "l�MnD + I�−s"l T1

C "l�M + I�−s"l T1�by min − max principle� Cl .

This proves step 2.Finally, we can complete the proof of Lemma 6.2.Note that

W ="lW"l +"lW�1 −"l� + �1 −"l�W .

The statement of Lemma 6.2 follows if we apply �6.2� to the first term and �6.3� to the second andthird terms on the right hand side. �

Following Hempel30 and Alama et al.,1 we use a trace ideal estimate to show that the result of

adding a Dirichlet boundary condition to MnD on �Bl�ln� is bounded by the boundary term

�Bl� l.Lemma 6.5: �Reference 30� Let A, B be compact symmetric operators and suppose that B

�T� for some �� 1,��. Also, let ��0, ��A�. Then

�dim P��,��A� − dim P��,��A + B�� A − ��−1 � B T�� .

Lemma 6.6: There exists a constant C such that

dim P�−�,��MnD� − Cl dim P�−�,��Ml

D� Ml,n

DD� dim P�−�,��MnD�, 1 l

n

2.

Proof: This result is the direct consequence of Lemmas 6.2 and 6.5. Note that we apply

Lemma 6.5 with �= ��+1�−1��MnD+I�−1�. �

Finally we can complete the proof of Theorem 1.6. It is routine as the proof of Proposition2.29 in Ref. 1. However, we will give the proof for clarity.

Proof of Theorem 1.6: By Lemma 6.1 and the regular perturbation theory, for n sufficientlylarge,

N��,q,�� N��,q,�;MnD� dim P�−�,��Mn

D� − dim P�−�,��M�;nD � . �6.4�

Since supp q�x��Bl, we have

M�;nD M�;l

D� M�;l,n

DD = M�;lD

� Ml,nDD.

By Lemma 6.6,

dim P�−�,��M�;nD �� dim P�−�,��M�;l

D � + dim P�−�,��Ml,nDD�

� dim P�−�,��Ml,nDD�� dim P�−�,��Mn

D� − dim P�−�,��MlD� − Cl . �6.5�

Combining �6.4� with �6.5�, we get



On: Thu, 27 Nov 2014 16:38:10

N��,q,�� dim P�−�,��MlD� + Cl .

But it follows from Weyl’s law for Maxwell operators �Corollary 2.5� that

dim P�−�,��MlD� C�� − p−

−1 2�l2,

hence,

N��,q,�� C�� − p−−1 2�l2.

This completes the proof. �

VII. PERTURBATIONS DECAYING AT INFINITY

A. Perturbations bounded below by a negative power of �x�

In addition to �1.21� we suppose in this subsection that q�x� satisfies �1.22� and �1.23�.The main result in this subsection is the following.Theorem 7.1: Let q�x� satisfy �1.21�–�1.23�. Then there exists a positive constant C1 such that

N��,q,�� C1�1 + 2��2/s,

for � large enough.Clearly, it follows from Theorem 1.4 that N�� ,q ,��0 for any q�x� satisfying �1.23�. Theo-

rem 7.1, however, gives a more precise description on the growth of N�� ,q ,��.By −�

D ,−� N we denote −� on the disk B1 with Dirichlet and Neumann boundary conditions

on �B1, respectively. By Weyl’s law �cf. Ref. 48� there exist constants CDCN, so that for �� 2,

dim P�−�,��− � D�� CD�� − 2� ,

dim P�−�,��− � N� CN�� − 2� .

Lemma 7.2: Let q�x� satisfy �1.23�. For given fixed '�1 so that CN /CD' . Let

l = C��1/s, �7.1�

where the constant C� is chosen in such a way that for some constant �0�1 ,

�q�x�� 'p+ for all x � Bl and � � �0,

where p+ is defined in �1.7�. Then there exist constants C� and l0 such that for ��0 ,

dim P�−�,��MlN� − dim P�−�,��M�;l

N �� C��1 + 2�l2.

Proof: By monotonicity,

dim P�−�,��MlN�� dim P�−�,��− p+� ;l

N �� dim P�−�,��− p+� ;lD �� dim P�−�,�l2/p+��− �l ;1

D �

� CD��l2/p+ − l2 2�

and

dim P�−�,��M�;lN � dim P�−�,��− 'p+� ;l

N � dim P�−�,�l2/'p+��− �l ;1N � CN��l2/'p+ − l2 2� .

Thus we have

dim P�−�,��MlN� − dim P�−�,��M�;l

N �� CD�/p+ − CN�/'p+�l2 + l2 2�CN − CD�� C��1 + 2�l2.

�



On: Thu, 27 Nov 2014 16:38:10

Lemma 7.3: Let l�1 and n�2l . Then for any &�N , s=2& , we have

�MlN

� Ml,nND + I�−1 − �Mn

D + I�−1 Ts

s Cl �7.2�

for some constant C , independent of l and n .Proof: Let "l�x� be given by �6.1�. Set

J � �MlN

� Ml,nND + I�−1 − �Mn

D + I�−1.

In the sense of quadratic forms,

0"lJ"l "l�MlN

� Ml,nND + I�−1"l "l�Ml

N� Ml,n

ND + I�−1"l.

Add Neumann boundary conditions on �Bl�3 and therefore increase the Ts-norm. But it is known�cf. Ref. 30� that

�− �t−1,tNN + I�−1 Ts

s Ct, t� 2, �7.3�

where −�t−1,tNN denotes the Neumann Laplacian on Bt \Bt−1. Hence it follows that

"lJ"l Ts

s Cl . �7.4�

On the other hand, similar as the argument of step 2 in the Proof of Lemma 6.2, we have Ju��SBn

for all u� �SBn. Moreover, For any u� �SBn

we further set

Vu� � �MnN + I��Ju�� .

It is clear that Vu� = �MnN+I��Ju��=0 in Bn \�Bl, in the distributional sense. In addition we have

�MnN + I�J �Mn

N + I��MnD + I�−1 + �Mn

N + I��MlN

� Ml,nND + I�−1 2.

Thus V is bounded, independent of l and n. Since

�1 −"l�Ju� = �1 −"l��MnN + I�−1"lVu� .

Again, similar as the argument of step 2 in the Proof of Lemma 6.2, we can further use Lemma 6.3and �7.3� to obtain that for &�N, s=2&,

�1 −"l�J Ts

s �1 −"l��MnN + I�−1"l Ts

s V s Cl �7.5�

for some constant C, independent of l and n. Thus �7.2� follows from �7.4� and �7.5� and thefollowing equality:

J ="lJ"l +"lJ�1 −"l� + �1 −"l�J .

�

By Lemmas 7.3 and 6.5, we have the following.Lemma 7.4: There exists a constant C such that

dim P�−�,��MnD� dim P�−�,��Ml

N� Ml,n

ND� dim P�−�,��MnD� + Cl . 1 l

n

2.

Lemma 7.5: Let q�x� satisfy �1.23�. Then there exist constants C and �0 such that

N��,q,�;MnD� = card�0 t �:�� Mt;n

D �� C�1 + 2��2/s.

Proof: By Kato–Rellich perturbation theory, it suffices to prove that



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dim P�−�,��MnD� − dim P�−�,��M�;n

D �� C�1 + 2��2/s �7.6�

for ��0 and n=n��. Let l=C��1/s be given by �7.1�. For any given fixed t� �� ,b1�,

dim P�−�,��M�;nD � dim P�−�,t��M�;l

N� M�;l,n

ND � dim P�−�,t��M�;lN � + dim P�−�,t��Ml,n

ND� .

�7.7�

By Lemma 7.4,

dim P�−�,t��M�;l,nND � dim P�−�,t��Mn

D� − dim P�−�,t��MlN� + Cl . �7.8�

It follows from �7.7� and �7.8� together with Lemma 7.2 that

dim P�−�,t��M�;nD � dim P�−�,t��Mn

D� + Cl − C��1 + 2�l2.

Since the constants C� , l0 can be chosen uniformly for all t� �� ,b1�, we can let t↓� to obtain �7.6�.�

Proof of Theorem 7.1: Using Lemma 7.5, it suffices to prove that for any ��,

N��,q,�� lim infn→�

N��,q,�;MnD� .

The above inequality can be proven in the same way as Theorem III of Ref. 1, with the obviousmodifications to take into that in this theorem we have � � instead of gradients. �

B. Perturbations bounded above by a negative power of �x�

In this subsection we suppose that p�x� and q�x� satisfy �1.21� and �1.24�. We shall proveTheorem 1.8 in this subsection.

Proof of Theorem 1.8: We argue as in Ref. 1. It follows from Lemma 6.1 and regular pertur-bation theory that

N��,q,�� N��,q,�;MnD� dim P�−�,��Mn

D� − dim P�−�,��M�;nD � . �7.9�

For any ��0 small enough such that � / �1+��G= �a ,b�. Then choose C3 so that

�q�x� � for all �x�� l = l��

for l=C3�1/s. Add Dirichlet boundary condition on �Bl,

dim P�−�,��M�;nD �� dim P�−�,��M�;l

D � + dim P�−�,��M�;l,nDD �

� dim P�−�,��M�;l,nDD �� dim P�−�,��1 + ��Ml,n

DD�

= dim P�−�,�/�1+��Ml,nDD�

� dim P�−�,�/�1+��MnD� − dim P�−�,�/�1+��Ml

D� − Cl �by Lemma 6.6�

� dim P�−�,��MnD� − dim P�−�,��Ml

D� − Cl �by �5.8��

� dim P�−�,��MnD� − C4l2�� − p−

−1 2� − Cl �by Corollary 2.5� . �7.10�

Thus �1.25� is an immediate consequence of �7.9� and �7.10�. �

C. Perturbations with power law asymptotics

In addition to �1.21�, we suppose that p�x� and q�x� satisfy �1.26� in this subsection. We alsosuppose that the density of states for M,



On: Thu, 27 Nov 2014 16:38:10

��,M� = limn→�

� 1

meas��n�dim P�−�,��M�n

D/N�� , �7.11�

exists and is independent of the Dirichlet and Neumann boundary conditions on the square �n,where M�n

D/N denotes the Dirichlet or Neumann Maxwell operator on �n. It is well known that ifp�x� is periodic, then by Floquet–Bloch theory �� ,M� exists and is independent of the Dirichletand Neumann boundary conditions.

In this subsection we shall prove Theorem 1.9.Based on the preparing work established above, we are fortunate to see that the idea in Ref. 1

can be adopted here with appropriate modifications, and therefore we only give a sketch of theproof.

As in Refs. 1 and 2 we define the bounded self-adjoint operator,

L ª�q�x�� M − ��−1� � �q�x� ,

on L2�R2 ;C3�.Lemma 7.6:

�ess�L� � �0, + �� .

Proof: Note that

M�M − ��−1 M�M + 1�−1 + �1 + �� M�M − ��−1�M + 1�−1 1 +1 + �

dist��,��M��

and ��M�� p− 2 , +�� see Sec. II B�. Thus we can argue similarly as in Ref. 1 to conclude that

the operator �q�x�� M −��−1� ��q�x�−�q�x�� M−1� ��q�x� is compact, which im-plies the conclusion of the lemma. �

It follows from Proposition 2.1 that

N��,�,q� = P�−�,−1/��L� . �7.12�

Notational disclaimer: For the sake of simplicity of notation, we still use �l in the following todenote a cutoff function satisfying �1.27� except that the norm �x� is replaced by max1j2�xj� �e.g.,�x� l is replaced by max1j2�xj� l�.

Lemma 7.7: Let q�x� satisfy �1.26� and �1.21�. Then there exists constant C�0 so that

L − Ll C�1 + l�−s,

where Llª�ql� � �M −��−1� ��ql with ql=�l

2q.Proof: A straightforward calculation shows that is a direct consequence of �1.26� and �1.21�

and the Combes–Thomas estimate �Proposition 2.2�. �

An elementary application of min-max principle �cf. Ref. 32� shows the following.Lemma 7.8: Let L ,L� be bounded self-adjoint operators with ��L� ,��L�� 0, +��. Sup-

pose that for any 0�1 and ��0,

L − L� �

2� .

Then

dim P�−�,−�2+�/2��L�� dim P�−�,−�1/��L� dim P�−�,−�2−�/2��L�� .

Lemmas 7.7 and 7.8 together with �7.12� imply the following.Lemma 7.9: For any given 0�1, there exists constant C��0 such that for l

�C�� /��1/s,



On: Thu, 27 Nov 2014 16:38:10

N� 2�

2 + �,ql,�� N��,q,�� N� 2�

2 − �,ql,�� .

Note that MnD�p−�−��n

D �S�n� 0�G�n

�. Similar as in Refs. 29 and 1 Weyl’s law implies that MnD

has at most C1n2 eigenvalues in �a1 ,b1�, thus we can choose �na� �a1 ,��, �n

b� �� ,b1�, so that

dist��na,b,��Mn

D��b1 − a1

C1n2 = C2n−2

for C2 uniformly of n. Using the Combes–Thomas estimate �Proposition 2.2�, the following twolemmas can be proven in a similar way as in Ref. 1 with appropriate modifications.

Lemma 7.10: Suppose that �� a1 ,b1�� a1 ,b1�� a ,b��G. Then there exist �na� �a1 ,��,

�nb� �� ,b1�, with �n

a,b��MnD� and

�l� � ��M − �na,b�−1 − �Mn

D − �na,b�−1�� n Cn2e−��n−l�

for some constant C which is independent of l ,n .Lemma 7.11: Let q�x� be as in �1.26� with �1.21�. Let C� be as in Lemma 7.9 and let n

=2C�� /��1/s , l=C�� /��1/s . Then

N�� 2

2 + ��2

�,ql,�nb;Mn

D� N��,q,�� N�� 2

2 − ��2

�,ql,�na;Mn

D�for � large enough.

Proof of Theorem 1.9: First we derive a lower bound using Neumann bracketing. The upperbound can be obtained similarly using Dirichlet bracketing. Fix 0�1 and #�0 so that�a1 ,b1�� −# ,�+#��G. Let n , l be given by Lemma 7.11, i.e., n=2C�� /��1/s, l=C�� /��1/s. Letm=��1/s. Since lim�x�→��x�sq�x� / p�x�=q�, we can choose �x� large enough, say, �x�� l1 so that

�q� − ��p�x��x�s

q�x��q� + ��p�x�

�x�s,

then we choose �1 large enough so that m=��1/s��2l1. Lemma 7.11 shows that

N��,q,�� N�� 2

2 + ��2

�,ql,�nb;Mn

D� = dim P�−�,�nb��Mn

D� − dim P�−�,�nb��M�2/�2 + ��2�;n

D �ql��

� dim P�−�,��MnD� − dim P�−�,�+#��M�2/�2 + ��2�;n

D �ql�� . �7.13�

We can introduce Neumann boundary conditions on ��m and ��l to obtain

dim P�−�,�+#��M�2/�2 + ��2�;nD �ql�� dim P�−�,�+#��M�2/�2 + ��2�;l,n

ND �ql�� + dim P�−�,�+#�

��M�2/�2 + ��2�;m,lNN �ql�� + dim P�−�,�+#��M�2/�2 + ��2�;m

N �ql�� .

But by Weyl’s law

dim P�−�,�+#��M�2/�2 + ��2�;mN �ql�� p− dim P�−�,�+#��− � ;�m

N � Cm2�� + # − p−−1 2� ,

and

lim sup�→�

�−2/s dim P�−�,�+#��M�2/�2 + ��2�;l,nND �ql�� lim sup

�→��−2/s dim P�−�,�+#��Ml,n

ND�

meas��2C��−1/s \ �C��−1/s�� + #� .



On: Thu, 27 Nov 2014 16:38:10

Next we try to estimate in the region �l \�m. As in Ref. 1, partition �l \�m into finitely manysmall squares �i , i=1, . . . ,K, so that maxi=1,. . .,K diam��i��1/st. Define vi=min�i

��x�−1/s�. Clearly,�i

0ª�−1/s�i, i=1, . . . ,K, partitions �C��−1/s \��. Define vi

0=�−1/svi. Introduce Neumann boundaryconditions on �i. A straightforward calculation shows that

lim sup�→�

�−2/s dim P�−�,�+#��M�2/�2 + ��2�;m,lNN �ql��

i

��i0�� + #

1 + 4�q� − ��vi0�/�2 + ��2� .

Inserting the above estimates in �7.13� and letting t→0 first, then #→0, then �→0, we finallyobtain

lim inf�→�

�−2/sN��,q,�� = �R2

dx�� − ��

1 + q��x�−s�� = �0

�

d��meas�x:� � ��1 + q��x�−s�−1� .

�

Remark: Let N�� ,q ,�� be given by �1.19�, we can also obtain the bounds of N�� ,q ,�� by theobservation �1.20�.

ACKNOWLEDGMENTS

The author thanks the hospitality of the Department of Mathematics, Michigan State Univer-sity, where part of this work was done. He also thank the referees for suggestions on how toimprove the presentation of the text.

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Eigenvalue branches of the perturbed Maxwell operator M+λD in a gap of σ(M)

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Transcript of Eigenvalue branches of the perturbed Maxwell operator M+λD in a gap of σ(M)