Opening up and control of spectral gaps of the Laplacian in periodic domains

Opening up and control of spectral gaps of the Laplacian in periodic domains

Andrii Khrabustovskyi

Research Training Group ”Analysis, Simulation and Design of Nanotechnological Processes”
Department of Mathematics, Karlsruhe Institute of Technology

andrii.khrabustovskyi@kit.edu

Abstract. The main result of this work is as follows: for arbitrary pairwise disjoint finite intervals , and for arbitrary we construct a family of periodic non-compact domains such that the spectrum of the Neumann Laplacian in has at least gaps when is small enough, moreover the first gaps tend to the intervals as . The constructed domain is obtained by removing from a system of periodically distributed ”trap-like” surfaces. The parameter characterizes the period of the domain , also it is involved in a geometry of the removed surfaces.

Keywords: periodic domains, Neumann Laplacian, spectrum, gaps, asymptotic analysis, photonic crystals.

Introduction

The problem we are going to solve belongs to the spectral theory of periodic self-adjoint differential operators. It is known that usually the spectrum of such operators is a locally finite union of compact intervals called bands. In general the bands may overlap. The open interval is called a gap if it has an empty intersection with the spectrum, but its ends belong to it.

In general the presence of gaps is not guaranteed, for example, the spectrum of the Laplacian in has no gaps: . Therefore one of the central questions arising here is whether the gaps really exist in concrete situations. This question is motivated by various applications, in particular in connection with photonic crystals attracting much attention in recent years. Photonic crystals are periodic dielectric media in which electromagnetic waves of certain frequencies cannot propagate, which is caused by gaps in the spectrum of the Maxwell operator or related scalar operators. We refer to paper [31] concerning mathematical problems arising in this field.

The problem of constructing of periodic operators with spectral gaps attracts a lot attention in the last twenty years. Various examples were presented in [28, 14, 15, 16, 18, 22, 44, 24] for periodic divergence type elliptic operators in , in [21] for periodic Schrödinger operators, in [15, 17] for Maxwell operators with periodic coefficients in , in [12, 41, 20, 13, 27] for Laplace-Beltrami operators on periodic Riemannian manifolds, in [38] for Laplacians in periodic domains in . We refer to overview [23] where these and other related questions are discussed in more details. Also we mention papers [3, 4, 40, 37, 8, 10, 43, 1, 2, 9, 19, 36, 35, 39] devoted to the same problem for the operators posed in unbounded domains with a waveguide geometry (quantum waveguides).

The present paper is devoted to spectral analysis of the Neumann Laplacians in periodic domains. We denote by the set of all domains satisfying the property

 ∃d=d(Ω)>0: Ω=Ω+dk, ∀k∈Zn

(i.e. is periodic and the cube is a period cell). Let and be the Neumann Laplacian in . Operators of this type occur in various areas of physics. For example in the case the operator governs the propagation of -polarized electro-magnetic waves in a periodic dielectric medium with a perfectly conducting boundary. Below (see Remark 0.4) we discuss an application of our results to the theory of photonic crystals.

The example of periodic domain with gaps in the spectrum of the Neumann Laplacian was presented in [38]. Here the authors considered the Neumann Laplacian in perforated by -periodic family of circular holes and proved that the gaps in its spectrum open up when the diameter of holes is close enough to the distance between their centers (the last one is fixed).

In the present work we want not only to construct a new type of periodic domains with gaps in the spectrum of the Neumann Laplacian but also be able to control the edges of these gaps making them close (in some natural sense) to predefined intervals. Let us formulate our main result.

Theorem 0.1 (Main Theorem).

Let be an arbitrarily large number and let () be arbitrary intervals satisfying

 0<α1,αj<βj<αj+1, j=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯1,m−1,αm<βm

Let .

Then one can construct the family of domains with such that the spectrum of the Neumann Laplacian in (we denote it ) has the following structure in the interval when is small enough:

 σ(Aε)∩[0,L]=[0,L]∖(m⋃j=1(αεj,βεj)), (0.2)

where the intervals satisfy

 ∀j=1,…,m:limε→0αεj=αj, limε→0βεj=βj. (0.3)
Remark 0.1.

In work [11] Y. Colin de Verdière proved (among other results) the following statement: for arbitrary numbers () there exists a bounded domain () such that the first eigenvalues of the Neumann Laplacian in are exactly . Our theorem can be regarded as an analogue of this result for the Neumann Laplacians in non-compact periodic domains.

Some preliminary results towards the proof of Theorem 0.1 were obtained by the author and E. Khruslov in [29] where the case was considered. However the general case is much more complicated. Similar results for the Laplace-Beltrami operators on periodic Riemannian manifolds without a boundary and for elliptic operators in the entire space were obtained by the author in [27] and [28] correspondingly.

We now briefly explain how to construct the family . Let , be pairwise disjoint open domains belonging to the unit cube in . We suppose that for any contains a flat part. Within this flat part we make a small circular hole , the obtained set we denote by (see the left picture on Fig. 1):

 Sεj=∂Bj∖Dεj, j=1,…m.

Here is a parameter characterizing the size of the hole, namely we suppose that the radius of is equal to if or if . Here , are positive constants. Finally we set

 Ωε=Rn∖(⋃i∈Znm⋃j=1Sεij),% where Sεij=ε(Sεj+i),

i.e. is obtained by removing from families of periodically distributed ”trap-like” surfaces (see Figure 1, right picture). Obviously, , the cube is the period cell. We denote by the Neumann Laplacian in (the precise definition will be given in the next section).

We will prove (see Theorem 1.1) that for an arbitrarily large interval the spectrum of the operator has exactly gaps in when is small enough. Moreover when these gaps converge to some intervals () depending in a special way on the domains and the numbers and satisfying

 0<σ1,σj<μj<σj+1, j=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯1,m−1,σm<μm. (0.4)

Finally we will prove (see Lemma 1.1) that for an arbitrary intervals , satisfying (0.1) one can choose and in such a way that the equalities

 σj=αj, μj=βj, j=1,…,m

hold. For the volumes of the sets and for the numbers we will present the exact formulae. It is clear that the main theorem follows directly from Theorem 1.1 and Lemma 1.1.

Remark 0.2.

The idea how to construct the domain is close to the idea which was used in [28], where the operator in was studied. In this work the role of ”traps” is played by the family of thin spherical shells which are -periodically distributed in and on which becomes small as . A similar idea was also used in [27] where the periodic Laplace-Beltrami operator was studied.

The analysis of the asymptotic behaviour of spectra was carried out in [27, 28] using the methods of the homogenization theory. The idea to use this theory in order to open up the gaps in the spectrum of periodic differential operators was firstly proposed in [44]. Since the proof in [27, 28] is rather cumbersome, in the present work we carry out the analysis using another method (see the next remark). On the other hand the results of [27, 28] helped us to guess the form of the equation (1.5) below whose roots are the limits of the right ends of the spectral bands.

Boundary value problems in domains with ”traps” were also considered in [5, 6], where the authors studied the homogenization of semi-linear parabolic equations and their attractors. Similar homogenization problems were studied in [34].

Remark 0.3.

Let us briefly describe the scheme of the proof of Theorem 1.1. We enclose the left end (resp. the right end) of the -th band between the -th eigenvalues of the Neumann and periodic (resp. the antiperiodic and Dirichlet) Laplacians posed on the period cell. We prove that both ends of this enclosure converge to if and to infinity if (resp. converge to if and to infinity if ) as .

The most difficult part of the proof is the investigation of the asymptotic behaviour of the eigenvalues of the Neumann Laplacian (see Theorem 2.2). To obtain the asymptotics of eigenvalues we will construct convenient approximations for the corresponding eigenfunctions. The analysis of the eigenvalues of the Dirichlet Laplacian (see Theorem 2.3) is carried out using the same ideas but it is essentially simpler. The analysis of the eigenvalues of the periodic (resp. antiperiodic) Laplacian repeats word-by-word the analysis for the eigenvalues of the Neumann (resp. Dirichlet) Laplacian.

Remark 0.4.

The obtained results can be applied in the theory of photonic crystals. Let us introduce the following sets in :

 Ωε={(x1,x2,z)∈R3: x=(x1,x2)∈Ωε, z∈R},Sε=R3∖Ωε,

where is defined above. We suppose that is occupied by a dielectric medium whereas the union of the screens is occupied by a perfectly conducting material. It is supposed that the electric permittivity and the magnetic permeability of the material occupying are equal to .

The propagation of electromagnetic waves in is governed by the Maxwell operator (below by and we denote the electric and magnetic fields, )

 MεU=(icurlH,−icurlE)

subject to the conditions

 divE=divH=0 in Ωε,Eτ=0, Hν=0 on Sε.

Here and are the tangential and normal components of and , correspondingly. We are interested only on the waves propagated along the plane , i.e. when depends on only.

It is known that if the medium is periodic in two directions and homogeneous with respect to the third one (so-called crystals) then the analysis of the Maxwell operator reduces to the analysis of scalar elliptic operators. Let us formulate this statement more precisely. We denote

 J={(E,H): divE=divH=0 in Ωε, Eτ=Hμ=0 on Sε}, JE={(E,H)∈J: E1=E2=H3=0},JH={(E,H)∈J: H1=H2=E3=0}.

The elements of the subspaces and are usually called - and -polarized waves. The subspaces and are -orthogonal and each can be represented in unique way as where . Moreover and are invariant subspaces of . Thus is a union of (-subspectrum) and (-subspectrum).

We denote by and the Dirichlet and the Neumann Laplacians in , correspondingly. It can be shown on a formal level of rigour (see, e.g, [25]) that iff and iff . Using Friedrichs type inequalities one can easily prove (see [29, Lemma 3.1]) that , (here is a constant) and therefore

 min{λ:λ∈σ(Aε0)}→ε→0∞. (0.5)

Then using Theorem 1.1, Lemma 1.1 and (0.5) we conclude that for an arbitrarily large the Maxwell operator has gaps in when is small enough and as these gaps converge to intervals , which can be controlled via a suitable choice of and .

1. Construction of the family {Ωε}ε and main results

Let be a small parameter and let . Let , be arbitrary open domains with Lipschitz boundaries satisfying the following conditions:

• for ,

• , where

 Y={x=(x1,…,xn)∈Rn: |xi|<1/2,∀i},
• for any the boundary of has a flat subset, namely

 ∃~xj∈∂Bεj, ∃rj>0: the set Brj(~xj)∩∂Bj % belongs to a (n−1)-dimensional hyperplain.

Here by we denote the ball with the center at the point and the radius .

For we denote

• , where is defined by the following formula:

 dεj=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩djε2n−2,n>2,ε−1exp(−1djε2),n=2.

Here , are positive constants. It is supposed that is small enough so that .

• .

Finally we set

 Ωε=Rn∖(⋃i∈Znm⋃j=1Sεij),% where Sεij=ε(Sεj+i).

Let us define precisely the Neumann Laplacian in . We denote by the sesquilinear form in which is defined by the formula

 ηε[u,v]=∫Ωε(∇u,∇¯v)dx (1.1)

and the definitional domain . Here . The form is densely defined closed and positive. Then (see, e.g., [26, Chapter 6, Theorem 2.1]) there exists the unique self-adjoint and positive operator associated with the form , i.e.

 (Aεu,v)L2(Ωε)=ηε[u,v],∀u∈dom(Aε), ∀v∈dom(ηε). (1.2)

We denote by the spectrum of . To describe the behaviour of as we need some additional notations.

In the case we denote by the capacity of the disc

 T={x=(x1,…,xn)∈Rn: |x|<1, xn=0}

Recall (see, e.g, [32]) that it is defined by

 κ=infw∫Rn|∇w|2dx,

where the infimum is taken over smooth and compactly supported in functions equal to on .

We set (below )

 σj=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩κdn−2j4bj,n>2,πdj2bj,n=2, (1.3)

where is the volume of the domain . We assume that the numbers and are such that

 σj<σj+1, j=1,…m−1. (1.4)

Let us consider the following equation (with unknown ):

 1+m∑j=1σjbj(1−m∑i=1bi)(σj−λ)=0. (1.5)

It is easy to show (see [27, Subsect. 3.2]) that if (1.4) holds then equation (1.5) has exactly roots, they are real and interlace with . We denote them , supposing that they are renumbered in the increasing order, i.e.

 σj<μj<σj+1, j=1,…,m−1,σm<μm<∞. (1.6)

Now we can formulate the main result on the behaviour of as .

Theorem 1.1.

Let be an arbitrary number satisfying . Then the spectrum of the operator has the following structure in when is small enough:

 σ(Aε)∩[0,L]=[0,L]∖(m⋃j=1(σεj,μεj)), (1.7)

where the intervals satisfy

 ∀j=1,…,m:limε→0σεj=σj,limε→0μεj=μj. (1.8)

Theorem 1.1 shows that has exactly gaps when is small enough and when these gaps converge to the intervals . Now, our goal is to find such numbers and domains that the corresponding intervals coincide with the predefined ones.

We use the notations . Let

 L:Rm×Rm→Rm×Rm,(d,b)L↦(σ,μ)

be the map with the definitional domain

 dom(L)={(d,b)∈Rm×Rm: dj>0, bj>0,m∑j=1bj<1 and (???) holds}

and acting according to formulae (1.3), (1.5), (1.6) (i.e. are defined by (1.3) and are the roots of equation (1.5) renumbered according to (1.6)).

Lemma 1.1.

The map maps onto the set

 G={(σ,μ)∈Rm×Rm: σj<μj<σj+1, j=1,…,m−1,σm<μm<∞}.

Moreover is one-to-one and the inverse map is given by the following formulae:

 dj=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩⎛⎜ ⎜ ⎜ ⎜⎝4σjρjκ(1+m∑i=1ρi)⎞⎟ ⎟ ⎟ ⎟⎠1n−2,n>2,2σjρjπ(1+m∑i=1ρi),n=2, (1.9) bj=ρj1+m∑i=1ρi, (1.10)

where

 ρj=μj−σjσj∏i=1,…,m|i≠j(μi−σjσi−σj). (1.11)
Proof.

Let be an arbitrary element of . We have to show that

 ∃!(d,b)∈dom(L) such that ∀j=1,…m {(???) holds ,(???) holds with λ=μj,

moreover, this is defined by formulae (1.9)-(1.11).

At first we find . Let us consider the following system of linear equations with respect to unknowns :

 1+m∑j=1σjρjσj−μj=0, j=1,…,m. (1.12)

It is proved in [27, Lemma 4.1] that this system has the unique solution which is defined by formula (1.11). Therefore in view of (1.5) in order to find we need to solve the following system:

 bj(1−m∑i=1bi)−1=ρj, j=1,…,m.

It is clear that it has the unique solution which is defined by (1.10). Since then

 ∀j: μj>σj;∀i≠j: sign(μi−σj)=sign(σi−σj)≠0

and hence . Therefore and .

Finally knowing we express from (1.3) and obtain the formula (1.9). The lemma is proved. ∎

Now, Theorem 0.1 follows directly from Theorem 1.1 and Lemma 1.1. Indeed, let , be arbitrary intervals satisfying (0.1) (and therefore by Lemma 1.1 ). We define the numbers , by formulae (1.9)-(1.10) with instead of , . For the obtained numbers we construct the domains satisfying and such that

 |Bj|=bj for j=1,…,m (1.13)

(it is easy to do, see example below for one of possible constructions). Finally using the domains and the numbers we construct the family of periodic domains . In view of Theorem 1.1 the corresponding family of Neumann Laplacians satisfies (0.2)-(0.3).

Example 1.1.

Let , be arbitrary numbers satisfying

 bj>0, j=1,…,m and m∑j=1bj<1.

We present one of the possible choices of the domains satisfying and (1.13).

We denote:

 l=(12+12m∑i=1bi)1/n,^l=12(n−1)ln−1(1−m∑i=1bi),lj=bjln−1.

Finally we define the domains , by the following formula:

 Bj={x∈Rn:x1∈(−l2+(j−1)^l+j−1∑i=1lj, −l2+(j−1)^l+j∑i=1lj), |xk|

It is easy to show that these domains satisfy conditions and (1.13).

2. Proof of Theorem 1.1

2.1. Preliminaries

We present the proof of Theorem 1.1 for the case only. For the case the proof is repeated word-by-word with some small modifications.

In what follows by we denote generic constants that do not depend on .

Let be an open domain in . By we denote the mean value of the function over the domain , i.e.

 ⟨u⟩B=1|B|∫Bu(x)dx.

Here by we denote the volume of the domain .

If is a -dimensional surface then the Euclidean metrics in induces on the Riemannian metrics and measure. We denote by the density of this measure. Again by we denote the mean value of the function over , i.e , where .

We introduce the following sets:

 Yε=Y∖m⋃j=1Sεj.

By we denote the Neumann Laplacian in . It is clear that

 σ(Aε)=ε−2σ(Aε). (2.1)

It is more convenient to deal with the operator since the external boundary of its period cell is fixed (it coincides with ).

In view of the periodicity of the analysis of its spectrum reduces to the analysis of the spectrum of the Laplace operator on with the Neumann boundary conditions on and quasi-periodic boundary (or -periodic) boundary conditions on . Namely, let

 Tn={θ=(θ1,…,θn)∈Cn: |θk|=1, ∀k}.

For we introduce the functional space consisting of functions from that satisfy the following condition on :

 ∀k=¯¯¯¯¯¯¯¯1,n:u(x+ek)=θku(x) % for x=(x1,x2,…,−1/2,…,xn)↑k-th place, (2.2)

where .

By we denote the sesquilenear form defined by formula (1.1) (with instead of ) and the definitional domain . We define the operator as the operator acting in and associated with the form , i.e.

 (Aθ,εu,v)L2(Yε)=ηθ,ε[u,v],∀u∈dom(Aθ,ε), ∀v∈dom(ηθ,ε).

The functions from satisfy the Neumann boundary conditions on , condition (2.2) on and the condition

 ∀k=¯¯¯¯¯¯¯¯1,n:∂u∂xk(x+ek)=θk∂u∂xk(x) for x=(x1,x2,…,−1/2,…,xn). (2.3)

The operator has purely discrete spectrum. We denote by the sequence of eigenvalues of written in the increasing order and repeated according to their multiplicity.

The Floquet-Bloch theory (see, e.g., [7, 30, 42]) establishes the following relationship between the spectra of the operators and :

 σ(Aε)=∞⋃k=1Lk,% where Lk=⋃θ∈Tn{λθ,εk}. (2.4)

The sets are compact intervals.

Also we need the Laplace operators on with the Neumann boundary conditions on and either Neumann or Dirichlet boundary conditions on . Namely, we denote by (resp. ) the sesquilinear form in defined by formula (1.1) (with instead of ) and the definitional domain (resp. ). Then by (resp. ) we denote the operator associated with the form (resp. ), i.e.

 (A∗,εu,v)L2(Yε)=η∗,ε[u,v],∀u∈dom(A∗,ε), ∀v∈dom(η∗,ε),

where is (resp. ).

The spectra of the operators and are purely discrete. We denote by (resp. ) the sequence of eigenvalues of (resp. ) written in the increasing order and repeated according to their multiplicity.

Using the min-max principle (see, e.g., [42]) and the enclosure one can easily prove the inequality

 ∀k∈N, ∀θ∈Tn:λN,εk≤λθ,εk≤λD,εk. (2.5)

2.2. Number-by-number convergence of eigenvalues of the Dirichlet, Neumann and θ-periodic Laplacians

We denote

 Bm+1=Y∖m⋃j=1¯¯¯¯¯¯Bεj.

By , we denote the operator which acts in and is defined by the operation and the Neumann boundary conditions on . By (resp. , ) we denote the operator which acts in and is defined by the operation , the Neumann boundary conditions on and the Neumann (resp. Dirichlet, -periodic) boundary conditions on . Finally, we introduce the operators , , which act in and are defined by the following formulae:

We denote by (resp. , ) the sequence of eigenvalues of (resp. , ) written in the increasing order and repeated according to their multiplicity. It is clear that

 λN1=λN2=⋯=λNm+1=0, λNm+2>0, (2.6) λD1=λD2=⋯=λDm=0, λDm+1>0, (2.7) λθ1=λθ2=⋯=λθm+1=0, λθm+2>0 if θ=(1,1,…,1), (2.8) λθ1=λθ2=⋯=λθm=0, λθm+1>0 if θ≠(1,1,…,1). (2.9)
Theorem 2.1.

For each one has

 limε→0λN,εk=λNk, (2.10) limε→0λD,εk=λDk, (2.11) limε→0λθ,εk=λθk(∀θ∈Tn). (2.12)

For the case Theorem 2.1 was proved in [29]. For the proof is similar.

2.3. Asymptotics of the first m non-zero eigenvalues of the Neumann Laplacian

We get more complete information about the behaviour of , (it is clear that ).

Theorem 2.2.

For one has

 limε→0ε−2λN,εk=μk−1. (2.13)
Proof.

Let , be the eigenfunctions corresponding to and satisfying the conditions

 (uεk,uεl)L2(Yε)=δkl, (2.14) uεk are real functions

(here is the Kronecker delta). It is clear that .

By the min-max principle (see, e.g, [42]) we get

 ∀k∈N: λN,εk=minu∈H(uε1,…,uεk−1)∥∇u∥2L2(Yε)∥u∥2L2(Yε), (2.15)

where

 H(uε1,…,uεk−1)={u∈H1(Yε): (u,uεl)L2(Yε), l=1,…,k−1}. (2.16)

Using the Cauchy inequality we get the estimate

 m+1∑j=1(⟨uεk⟩Bεj)2≤C

and therefore there exist a subsequence (for convenience still denoted by ) and numbers , , such that

 limε→0⟨uεk⟩Bj=skj. (2.17)

We denote .

During the proof we will use the function defined by the formula (below