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Journal of Function Spaces and Applications
Volume 2012, Article ID 789875, 13 pages
http://dx.doi.org/10.1155/2012/789875
Research Article

G-Convergence of Dirac Operators

Department of Mathematical Sciences, University of Gothenburg, 412 96 Gothenburg, Sweden

Received 2 November 2011; Revised 12 March 2012; Accepted 16 March 2012

Academic Editor: Bjorn Birnir

Copyright © 2012 Hasan Almanasreh and Nils Svanstedt. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the linear Dirac operator with a (1)-homogeneous locally periodic potential that varies with respect to a small parameter. Using the notation of G-convergence for positive self-adjoint operators in Hilbert spaces we prove G-compactness in the strong resolvent sense for families of projections of Dirac operators. We also prove convergence of the corresponding point spectrum in the spectral gap.

1. Introduction

In the present work we study the asymptotic behavior of Dirac operators with respect to a parameter as . We consider Dirac operators =𝐇+𝑉 on 𝐿2(3;4), where 𝐇=𝐇0+𝑊+𝐈 (𝐈 denoting the 4×4 identity matrix) is a shifted Dirac operator. The operators 𝐇0, 𝑊, and 𝑉 are the free Dirac operator, the Coulomb potential (𝑊(𝑥)=(𝑍/𝑥)𝐈, where 𝑍 is the electric charge number), and a perturbation to 𝐇, respectively. We will study the asymptotic behavior the shifted perturbed Dirac operator and the asymptotic behavior of the eigenvalues in the gap of the continuous spectrum of the shifted operator with respect to the perturbation parameter .

G-convergence theory which deals with convergence of operators is well known for its applications to homogenization of partial differential equations, but up to our knowledge it has not yet been applied to the Dirac equation. The concept was introduced in the late 1960s by De Giorgi and Spagnolo [13] for linear elliptic and parabolic problems with symmetric coefficients matrices. Later on it was extended to the nonsymmetric case by Murat and Tartar [46] under the name of H-convergence. A detailed exposition of G-convergence for positive self-adjoint operators is found in Dal Maso [7]. In the present work we will base a lot of our framework on the results in Chapters 12 and  13 in [7]. The Dirac operator is unbounded both from above and below. This means that the theory of G-convergence for positive self-adjoint operators is not directly applicable to Dirac operators. In this work we study self-adjoint projections of Dirac operators which satisfy the positivity so that the theory of G-convergence becomes applicable.

We will consider periodic perturbations, that is, we will assume that the potential 𝑉 is a periodic function with respect to some regular lattice in 𝑁. We are then interested in the asymptotic behaviour of shifted perturbed Dirac operators . This yields homogenization problems for the evolution equation associated with the Dirac operator 𝜕𝑖𝐮𝜕𝑡(𝑡,𝑥)=𝐮𝐮(𝑡,𝑥),(,0)=𝐮0(1.1)

and the corresponding eigenvalue problem𝑢(𝑥)=𝜆𝑢(𝑥).(1.2)

The paper is arranged as follows: in Section 2 we provide the reader with basic preliminaries on Dirac operators, G-convergence, and on the concepts needed from spectral theory. In Section 3 we present and prove the main results.

2. Preliminaries

Let 𝐴 be a linear operator on a Hilbert space. By 𝐑(𝐴), 𝐃(𝐴), and 𝐍(𝐴) we mean the range, domain, and null-space of 𝐴, respectively.

2.1. Dirac Operator

We recall some basic facts regarding the Dirac operator. For more details we refer to the monographs [810].

Let 𝒳 and 𝒴 denote the Hilbert spaces 𝐻1(3;4) and 𝐿2(3;4), respectively. The free Dirac evolution equation reads𝜕𝑖𝜕𝑡𝐮(𝑡,𝑥)=𝐇0𝐮(𝑡,𝑥),(2.1) where 𝐇0𝒴𝒴 is the free Dirac operator with domain 𝐃(𝐻0)=𝒳, which acts on the four-component vector 𝐮. It is a first-order linear hyperbolic partial differential equation. The free Dirac operator 𝐇0 has the form𝐇0=𝑖𝑐𝜶+𝑚𝑐2𝛽.(2.2) Here 𝜶=3𝑖=1𝛼𝑖(𝜕/𝜕𝑥𝑖), is the Planck constant divided by 2𝜋, the constant 𝑐 is the speed of light, 𝑚 is the particle rest mass and 𝜶=(𝛼1,𝛼2,𝛼3) and 𝛽 are the 4×4 Dirac matrices given by𝛼𝑖=0𝜎𝑖𝜎𝑖0,𝛽=𝐼00𝐼.(2.3)

Here 𝐼 and 0 are the 2×2 unity and zero matrices, respectively, and the 𝜎𝑖’s are the 2×2 Pauli matrices𝜎1=0110,𝜎2=0𝑖𝑖0,𝜎3=1001.(2.4)

Note that a separation of variables in (2.1) yields the Dirac eigenvalue problem𝐇0𝑢(𝑥)=𝜆𝑢(𝑥),(2.5) where 𝑢(𝑥) is the spatial part of the wave function 𝐮(𝑥,𝑡) and 𝜆 is the total energy of the particle. The free Dirac operator 𝐇0 is essentially self-adjoint on 𝐶0(3;4) and self-adjoint on 𝒳. Moreover, its spectrum, 𝜎(𝐇0), is purely absolutely continuous (i.e., its spectral measure is absolutely continuous with respect to the Lebesgue measure) and given by𝜎𝐇0=,𝑚𝑐2𝑚𝑐2,+.(2.6)

𝐇0 describes the motion of an electron that moves freely without external force. Let us now introduce an external field given by a 4×4 matrix-valued function 𝑊,𝑊(𝑥)=𝑊𝑖𝑗(𝑥)𝑖,𝑗=1,2,3,4.(2.7)

It acts as a multiplication operator in 𝐿2(3;4), thus the free Dirac operator with additional field 𝑊 is of the form 𝐇=𝐇0+𝑊.(2.8)

The operator 𝐇 is essentially self-adjoint on 𝐶0(3;4) and self-adjoint on the Sobolev space 𝒳 provided that 𝑊 is Hermitian and satisfies the following estimate (see for e.g., [8]):||𝑊𝑖𝑗||𝑐(𝑥)𝑎2|𝑥|+𝑏,𝑥3{0}𝑖,𝑗=1,2,3,4,(2.9) the constant 𝑐 is the speed of light, 𝑎<1, and 𝑏>0. From now on we let 𝑊(𝑥) be the Coulomb potential 𝑊(𝑥)=(𝑍/𝑥)𝐈, (without ambiguity, 𝐈 is usually dropped from the Coulomb term for simplicity). The spectrum of the Dirac operator with Coulomb potential is given by𝜎(𝐇)=,𝑚𝑐2𝜆𝑘𝑘𝑚𝑐2,+,(2.10)

where {𝜆𝑘}𝑘 is a discrete sequence of eigenvalues in the “gap” and the remaining part of the spectrum is the continuous part 𝜎(𝐇0).

In the present paper we consider a parameter-dependent perturbation added to the Dirac operator with Coulomb potential. The purpose is to investigate the asymptotic behavior of the corresponding eigenvalues in the gap and the convergence properties. To this end we introduce a 4×4 matrix-valued function 𝑉=𝑉(𝑥) and define the operator as=𝐇+𝑉.(2.11) We recall that a function 𝐹 is called homogeneous of degree 𝑝 if for any nonzero scalar 𝑎, 𝐹(𝑎𝑥)=𝑎𝑝𝐹(𝑥). The next theorem is of profound importance for the present work.

Theorem 2.1. Let 𝑊 be Hermitian and satisfy the bound (2.9) above. Further, for any fixed , let 𝑉 be a measurable (1)-homogeneous Hermitian 4×4 matrix-valued function with entries in 𝐿𝑝loc(3), 𝑝>3. Then is essentially self-adjoint on 𝐶0(3;4) and self-adjoint on 𝒳. Moreover 𝜎=,𝑚𝑐2𝜆𝑘𝑘𝑚𝑐2,+,(2.12)where {𝜆𝑘}𝑘 is a discrete sequence of parameter dependent eigenvalues corresponding to the Dirac eigenvalue problem 𝑢(𝑥)=𝜆𝑢(𝑥).Proof. See [9, 10].

We will as a motivating example consider perturbations which are locally periodic and of the form 𝑉(𝑥)=𝑉1(𝑥)𝑉2(𝑥). The entries of 𝑉1 are assumed to be (1)-homogeneous. The entries of 𝑉2(𝑦) are assumed to be periodic with respect to a regular lattice in 3. This can also be phrased that they are defined on the unit torus 𝕋3.

The evolution equation associated with the Dirac operator reads𝜕𝑖𝐮𝜕𝑡(𝑡,𝑥)=𝐮𝐮(𝑡,𝑥),(,0)=𝐮0.(2.13) By the Stone theorem, since is self-adjoint on 𝒳, there exists a unique solution 𝐮 to (2.13) given by𝐮(,𝑡)=𝒰(𝑡)𝐮0,𝐮0𝒳,(2.14) where 𝒰(𝑡)=exp((𝑖/)𝑡) is the strongly continuous unitary operator generated by the infinitesimal operator (𝑖/) on 𝒴, see for example, [8] or [11].

In the sequel we will consider a shifted family of Dirac operators denoted by and defined as =𝐇+𝑉, where 𝐇=𝐇+𝑚𝑐2𝐈. Also without loss of generality we will in the sequel put =𝑐=𝑚=1. By Theorem 2.1, for any , we then get 𝜎]̃𝜆=(,0𝑘𝑘[2,).(2.15)

2.2. G-Convergence

For more detailed information on G-convergence we refer to, for example, [12, 13], for the application to elliptic and parabolic partial differential operators, and to the monograph [7] for the application to general self-adjoint operators. Here we recall some basic facts about G-convergence for self-adjoint operators in 𝒴.

Let 𝜆0, by 𝒫𝜆(𝒴) we denote the class of self-adjoint operators 𝐴 on a closed linear subspace 𝒱=𝐃(𝐴) of 𝒴 such that 𝐴𝑢,𝑢𝜆||𝑢||2𝒴forall𝑢𝐃(𝐴).

Definition 2.2. Let 𝜆>0, and let {𝐴}𝒫𝜆(𝒴) then we say that 𝐴 G-converges to 𝐴𝒫𝜆(𝒴), denoted 𝐴𝐺,𝑠𝐺,𝑤𝐴 in 𝒴 if 𝐴1𝑃𝑢𝑠𝑤𝐴1𝑃𝑢 in 𝒴, for all 𝑢𝒴, where 𝑠 and 𝑤 refer to strong and weak topologies respectively, and 𝑃 and 𝑃 are the orthogonal projections onto 𝒱=𝐃(𝐴) and 𝒱=𝐃(𝐴) respectively. Also we say {𝐴}𝒫0(𝒴) converges to {𝐴}𝒫0(𝒴) in the strong resolvent sense (SRS) if (𝜆𝐼+𝐴)𝐺,𝑠(𝜆𝐼+𝐴) in 𝒴forall𝜆>0.
The following result provides a useful criterion for G-convergence of self-adjoint operators. See [7] for a proof.

Lemma 2.3. Given 𝜆>0, {𝐴}𝒫𝜆(𝒴) and the orthogonal projection 𝑃 onto 𝒱. Suppose that for every 𝑢𝒴, 𝐴1𝑃𝑢 converges strongly (resp. weakly) in 𝒴, then there exists an operator 𝐴𝒫𝜆(𝒴) such that 𝐴𝐺,𝑠𝐴 (resp. 𝐴𝐺,𝑤𝐴) in 𝒫𝜆(𝒴).

From now on we will just use the word “converge” instead of saying “strongly converge,” hence 𝐴G𝐴 instead of 𝐴𝐺,𝑠𝐴.

2.3. Some Basic Results in Spectral Theory

For more details see [9, 11, 14]. Given a Hilbert space 𝑋, let (𝓤,𝒜) be a measurable space for 𝒰 and let 𝒜 be a 𝜎-algebra on 𝓤. Let 𝑋=(𝑋) be the set of orthogonal projections on 𝑋, then 𝐸𝒜𝑋 is called a spectral measure if it satisfies the following:(i)𝐸()=𝟎 (this condition is superfluous given the next properties);(ii)Completeness; 𝐸(𝓤)=𝐈;(iii)Countable additivity; if {Δ𝑛}𝒜 is a finite or a countable set of disjoint elements and Δ=𝑛Δ𝑛, then 𝐸(Δ)=𝑛𝐸(Δ𝑛).

If 𝐸 is spectral measure then 𝐸(Δ1Δ2)=𝐸(Δ1)𝐸(Δ2)=𝐸(Δ2)𝐸(Δ1), also 𝐸 is modular, that is, 𝐸(Δ1Δ2)+𝐸(Δ1Δ2)=𝐸(Δ1)+𝐸(Δ2). For an increasing sequence of sets Δ𝑛, lim𝑛𝐸(Δ𝑛)=𝐸(𝑛Δ𝑛), while if Δ𝑛 is a decreasing sequence then lim𝑛𝐸(Δ𝑛)=𝐸(𝑛Δ𝑛). Because of the idempotence property of the spectral measure we have 𝐸𝑛𝑢2=𝐸𝑛𝑢,𝐸𝑛𝑢=𝐸2𝑛𝑢,𝑢=𝐸𝑛𝑢,𝑢𝐸𝑢,𝑢=𝐸𝑢2, which means that the weak convergence and the strong convergence of a sequence of spectral measures 𝐸𝑛 are equivalent.

Let 𝐸𝑢,𝑢(Δ) be the finite scalar measure on 𝒜 generated by 𝐸,𝐸𝑢,𝑢(Δ)=𝐸(Δ)𝑢,𝑢=𝐸(Δ)𝑢2(2.16)

and 𝐸𝑢,𝑣() be the complex measure𝐸𝑢,𝑣(Δ)=𝐸(Δ)𝑢,𝑣,𝑢,𝑣𝑋.(2.17)

By the above notations 𝐸𝑢,𝑣(Δ)𝐸(Δ)𝑢𝐸(Δ)𝑣𝑢𝑣.

Let 𝓤=. The spectral measure on the real line corresponding to an operator 𝑆 is denoted by 𝐸𝑆(𝜆) (where the superscript 𝑆 indicates that the spectral measure 𝐸 corresponds to a specific operator 𝑆)𝐸𝑆(𝜆)=𝐸𝑆(Δ),whereΔ=(,𝜆),for𝜆.(2.18)

Clearly 𝐸𝑆(𝜆) is monotonic (nondecreasing), that is, 𝐸𝑆(𝜆1)𝐸𝑆(𝜆2) for 𝜆1𝜆2. Also lim𝜆𝐸𝑆(𝜆)=𝟎 and lim𝜆𝐸𝑆(𝜆)=𝐈. 𝐸𝑆(𝜆) are self-adjoint, idempotent, positive, bounded, right continuous operator (lim𝑡0+𝐸𝑆(𝜆+𝑡)=𝐸𝑆(𝜆)), and discontinuous at each eigenvalue of the spectrum. If 𝜆 is an eigenvalue, then we define p(𝜆)=𝐸𝑆(𝜆)𝐸𝑆(𝜆0) to be the point projection onto the eigenspace of 𝜆. For 𝜆 being in the continuous spectrum p(𝜆)=𝟎.

Now we state the spectral theorem for self-adjoint operators.

Theorem 2.4. For a self-adjoint operator 𝑆 defined on a Hilbert space 𝑋 there exists a unique spectral measure 𝐸𝑆 on 𝑋 such that (i)𝑆=𝜎(𝑆)𝜆𝑑𝐸𝑆(𝜆);(ii)𝐸(Δ)=𝟎 if Δ𝜎(𝑆)=;(iii)If Δ is open and Δ𝜎(𝑆), then 𝐸(Δ)𝟎.Proof. See for example, [14].

3. The Main Results

Consider the family {} of Dirac operators with domain 𝐃()=𝒳. We will state and prove some useful theorems for operators of the class 𝒫𝜆(𝒴) for 𝜆0, where 𝒳 and 𝒴 are the Hilbert spaces defined above. The theorems are valid for general Hilbert spaces.

The following theorem gives a bound for the inverse of operators of the class 𝒫𝜆(𝒴) for 𝜆>0.

Theorem 3.1. Let 𝐴 be a positive and self-adjoint operator on 𝒴 and put 𝐵=𝐴+𝜆𝐼. Then for 𝜆>0,(i)B is injective. Moreover, for every 𝑣𝐑(𝐵),𝐵1𝑣,𝑣𝜆𝐵1𝑣2𝒴 and 𝐵1𝑣𝒴𝜆1𝑣𝒴.(ii)𝐑(𝐵)=𝒴.

Proof. See Propositions 12.1 and  12.3 in [7].

The connection between the operator and its G-limit of the class 𝒫𝜆(𝒴) for 𝜆0 to their corresponding eigenvalue problems is addressed in the next two theorems. Here we prove the critical case when 𝜆=0, where for 𝜆>0 the proof is analogous and even simpler.

Theorem 3.2. Given a family of operators {𝐴} of the class 𝒫0(𝒴) G-converging to 𝐴𝒫0(𝒴) in the strong resolvent sense. Let 𝑢 be the solution of 𝐴𝑢=𝑓, where {𝑓} is converging to 𝑓 in 𝒴. If {𝑢} converges to 𝑢 in 𝒴, then 𝑢 solves the G-limit problem 𝐴𝑢=𝑓.

Proof. Since 𝐴 G-converges to 𝐴 in the strong resolvent sense 𝐵1𝑃𝑣𝐵1𝑃𝑣,𝑣𝒴,(3.1) where 𝐵 and 𝐵 are 𝐴+𝜆𝐼 and 𝐴+𝜆𝐼, respectively. Note that by Theorem 3.1, 𝐃(𝐵1)=𝐑(𝐵)=𝒴, so the projections 𝑃 and 𝑃 are unnecessary.
Consider 𝐴𝑢=𝑓 which is equivalent to 𝐵𝑢=𝑓+𝜆𝑢; by the definition of 𝐵 we have 𝑢=𝐵1(𝑓+𝜆𝑢). Define 𝒥=𝑓+𝜆𝑢 which is clearly convergent to 𝒥=𝑓+𝜆𝑢 in 𝒴; by the assumptions. Therefore 𝐵1𝒥𝐵1𝒥, this is because 𝐵1𝒥𝐵1𝒥𝒴=𝐵1𝒥𝐵1𝒥+𝐵1𝒥𝐵1𝒥𝒴𝐵1𝒴𝒥𝒥𝒴+𝐵1𝒥𝐵1𝒥𝒴0.(3.2) The convergence to zero follows with help of (3.1) and the boundedness of the inverse operator 𝐵1. Thus, for all 𝑣𝒴𝑢,𝑣=lim𝑢,𝑣=lim𝐵1𝒥=𝐵,𝑣1𝒥,𝑣.(3.3) Hence 𝑢𝐵1𝒥,𝑣=0 for every 𝑣𝒴, which implies 𝐵𝑢=𝒥; therefore 𝐴𝑢=𝑓.

Theorem 3.3. Let {𝐴} be a sequence in 𝒫0(𝒴) which G-converges to 𝐴𝒫0(𝒴) in the strong resolvent sense, and let {𝜇,𝑢} be the solution of the eigenvalue problem 𝐴𝑢=𝜇𝑢. If {𝜇,𝑢}{𝜇,𝑢} in ×𝒴, then the limit couple {𝜇,𝑢} is the solution of the eigenvalue problem 𝐴𝑢=𝜇𝑢.

Proof. The proof is straight forward by assuming 𝑓=𝜇𝑢 (which converges to 𝜇𝑢 in 𝒴) in the previous theorem.

The convergence properties of self-adjoint operators have quite different implications on the asymptotic behavior of the spectrum, in particular on the asymptotic behavior of the eigenvalues, depending on the type of convergence. For a sequence {𝐴} of operators which converges uniformly to a limit operator 𝐴 nice results can be drawn for the spectrum. Exactly speaking {𝜎(𝐴)} converges to 𝜎(𝐴) including the isolated eigenvalues. The same conclusion holds if the uniform convergence is replaced by the uniform resolvent convergence, see for example, [11]. In the case of strong convergence (the same for strong resolvent convergence), if the sequence {𝐴} is strongly convergent to 𝐴, then every 𝜆𝜎(𝐴) is the limit of a sequence {𝜆} where 𝜆𝜎(𝐴), but not the limit of every such sequence {𝜆} lies in the spectrum of 𝐴, (see the below example taken from [15]). For weakly convergent sequences of operators no spectral implications can be extracted. In the present work we frequently write 𝐴 converges to 𝐴 when we mean that the sequence {𝐴} is converging to 𝐴.

Example 3.4. Let 𝐴𝑖, be an operator in 𝐿2() defined by 𝐴𝑖,𝑑=2𝑑𝑥2+𝑉𝑖,(𝑥),for,𝑖=1,2,(3.4)where 𝑉1,𝑉(𝑥)=1,if𝑥+1,0,Otherwise,2,(𝑥)=1,if𝑥,0,Otherwise.(3.5) The operator 𝐴𝑖, converges to 𝐴=𝑑2/𝑑𝑥2 in the strong resolvent sense as for both 𝑖=1,2. One can compute the spectrum for the three operators and obtain 𝜎(𝐴1,)=[0,){𝜇} for 𝜇 being a simple eigenvalue in [1,0] and 𝜎(𝐴2,)=[1,), whereas for the unperturbed limit operator 𝐴 the spectrum consists of just the continuous spectrum, that is, 𝜎(𝐴)=[0,).

Since the uniform convergence is not always the case for operators, the theorem below provides some criteria by which the G-convergence of an operator in the set 𝒫𝜆(𝒴) (and hence the G-convergence in the strong resolvent sense of operators of the class 𝒫0(𝒴)) implies the convergence of the corresponding eigenvalues.

Theorem 3.5. Let {𝐴} be a family of operators in 𝒫𝜆(𝒴), 𝜆>0, with domain 𝒳. If 𝐴 G-converges to 𝐴𝒫𝜆(𝒴), then 𝒦=𝐴1 converges in the norm of (𝒴)((𝒴) is the set of bounded linear operators on 𝒴) to 𝒦=𝐴1. Moreover the 𝑘th eigenvalue 𝜇𝑘 of 𝐴 converges to the 𝑘th eigenvalue 𝜇𝑘 of 𝐴 and the associated 𝑘th eigenvector 𝑢𝑘 converges to 𝑢𝑘 weakly in 𝒳, forall𝑘.

Proof. By the definition of supremum norm 𝒦𝒦(𝒴)=sup𝑣𝒴=1𝒦𝑣𝒦𝑣𝒴=sup𝑣𝒴1𝒦𝑣𝒦𝑣𝒴.(3.6) Also, by the definition of supremum norm there exists a sequence {𝑣}𝒴 with 𝑣𝒴1 such that 𝒦𝒦(𝒴)𝒦𝑣𝒦𝑣𝒴+1.(3.7) It is well known that 𝒦 and 𝒦 are compact self-adjoint operators on 𝒴. Both are bounded operators, by Theorem 3.1, with compact range 𝒳 of 𝒴.
Consider now the right-hand side of (3.7). We write this as𝒦𝑣𝒦𝑣𝒴+1𝒦𝑣𝒦𝑣𝒴+𝒦𝑣𝒦𝑣𝒴+𝒦𝑣𝒦𝑣𝒴+1.(3.8)The first and the third terms converge to zero by the compactness of 𝒦 and 𝒦 on 𝒴 and the second term converges to zero by the G-convergence of 𝐴 to 𝐴. Consequently 𝒦𝒦(𝒴)0.(3.9) Consider the eigenvalue problems associated to 𝐴1 and 𝐴1𝐴1𝑣𝑘=𝜆𝑘𝑣𝑘𝐴,𝑘,1𝑣𝑘=𝜆𝑘𝑣𝑘,𝑘.(3.10) Since 𝐴1 and 𝐴1 are compact and self-adjoint operators it is well known that there exists infinite sequences of eigenvalues 𝜆1𝜆2 and 𝜆1𝜆2, accumulating at the origin, respectively. Define 𝜇𝑘=(𝜆𝑘)1 and 𝜇𝑘=(𝜆𝑘)1 for all 𝑘. Consider now the spectral problems associated to 𝐴 and 𝐴𝐴𝑢𝑘=𝜇𝑘𝑢𝑘,𝑘,𝐴𝑢𝑘=𝜇𝑘𝑢𝑘,𝑘.(3.11) There exists infinite sequences of eigenvalues 0<𝜇1𝜇2 and 0<𝜇1𝜇2, respectively. By the compactness of 𝒦 and 𝒦 the sets {𝜆𝑘}𝑘=1 and {𝜆𝑘}𝑘=1 are bounded in , thus the proof is complete by virtue of the following lemma.

Lemma 3.6. Let 𝒳, 𝒴, 𝒦, 𝒦, 𝜆𝑘 and 𝜆𝑘 be as in Theorem 3.5, and let 𝐴𝒫𝜆(𝒴), 𝜆>0. There is a sequence {𝑟𝑘} converging to zero with 0<𝑟𝑘<𝜆𝑘 such that ||𝜆𝑘𝜆𝑘||𝜆𝑐𝑘𝜆𝑘𝑟𝑘sup𝜆𝑢𝒩𝑘,𝒦𝑢𝒴=1𝒦𝑢𝒦𝑢𝒴,(3.12) where 𝑐 is a constant independent of , and 𝒩(𝜆𝑘,𝒦)={𝑢𝐃(𝒦);𝒦𝑢=𝜆𝑘𝑢} is the eigenspace of 𝒦 corresponding to 𝜆𝑘.Proof. See Theorem 1.4 and Lemma  1.6in [16] Chapter 3.

We can now complete the proof of Theorem 3.5. By the G-convergence of 𝐴 to 𝐴 we obtain, by using Lemma 3.6 and (3.9), convergence of the eigenvalues and eigenvectors, that is, 𝜇𝑘𝜇𝑘 and 𝑢𝑘𝑢𝑘 weakly in 𝒳 as .

Let us now return to the shifted and perturbed Dirac operator . We will throughout this section assume the hypotheses of Theorem 2.1. We further assume that the 4×4 matrix-valued function 𝑉 is of the form 𝑉(𝑥)=𝑉1(𝑥)𝑉2(𝑥) where 𝑉1 is (−1)-homogeneous and where the entries of 𝑉2(𝑦) are 1-periodic in 𝑦, that is,𝑉2𝑖𝑗(𝑦+𝑘)=𝑉2𝑖𝑗(𝑦),𝑘3.(3.13)

We also assume that the entries of 𝑉2 belong to 𝐿(3). It is then well known that𝑉2𝑖𝑗(𝑉𝑥)𝑀2𝑖𝑗=𝕋3𝑉2𝑖𝑗(𝑦)𝑑𝑦,(3.14) in 𝐿(3) weakly*, where 𝕋3 is the unit torus in 3. It easily also follows from this mean-value property that𝑉𝑉1𝑀𝑉2,(3.15)

in 𝐿𝑝(3) weakly for 𝑝>3, compare the hypotheses in Theorem 2.1.

We are now interested in the asymptotic behavior of the operator and the spectrum of the perturbed Dirac operator . We recall the spectral problem for , that is,𝑢̃𝜆(𝑥)=𝑢(𝑥),(3.16)

where there exists a discrete set of eigenvalues {̃𝜆𝑘}, 𝑘=1,2, and a corresponding set of mutually orthogonal eigenfunctions {𝑢𝑘}. We know, by Theorem 2.1, that the eigenvalues (or point spectrum) 𝜎𝑝()(0,2). We also know that has a continuous spectrum 𝜎𝑐()=(,0][2,). This means that the Dirac operator is neither a positive or negative (semidefinite) operator and thus the G-convergence method introduced in the previous section for positive self-adjoint operators is not directly applicable. In order to use G-convergence methods for the asymptotic analysis of we therefore use spectral projection and study the corresponding asymptotic behavior of projections which are positive so that G-convergence methods apply.

Let 𝒜 be a fixed 𝜎-algebra of subsets of , and let (,𝒜) be a measurable space. Consider the spectral measures 𝐸𝐇 and 𝐸 of the families of Dirac operators and 𝐇, respectively, each one of these measures maps 𝒜 onto 𝒳, where 𝒳 is the set of orthogonal projections on 𝒳. By the spectral theorem=𝜎𝜆𝑑𝐸(𝜆).(3.17) By the spectral theorem we can also write𝜎(𝐇)𝐇(𝜆𝑑𝐸𝜆)+𝑉,(3.18) since 𝑉 is a multiplication operator.

We recall that 𝐃()=𝒳, let now𝒩𝑘=𝑢𝒳;𝑢=𝜆𝑘𝑢,(3.19)

that is, the eigenspace of corresponding to the eigenvalue 𝜆𝑘. Further define the sum of mutual orthogonal eigenspaces𝒳𝑝=𝑘𝒩𝑘,(3.20)

where 𝒳𝑝 is a closed subspace of 𝒴 invariant with respect to .

It is clear that for 𝑢𝒳𝑝 we have𝑢,𝑢=𝜆𝑘||𝑢||2>0,𝑘=1,2,.(3.21)

Let us now consider the restriction 𝑝 of to 𝒳𝑝 which can be written as𝑝=𝜆𝜎𝑝𝜆𝐸,𝑝(𝜆),(3.22)

where the spectral measure 𝐸,𝑝 is the point measure, that is, the orthogonal projection onto ker(𝜆𝐈). With this construction 𝑝 is a positive and self-adjoint operator on 𝒳𝑝 with compact inverse (𝑝)1. By Lemma 2.3, see also Proposition  13.4 in [7], we conclude that there exists a positive and self-adjoint operator 𝑝 such that, up to a subsequence, 𝑝 G-converges to 𝑝, where 𝑝 has domain 𝐃(𝑝)=𝒳𝑝 where𝒳𝑝=𝑘𝒩𝑘,(3.23)

is a closed subspace of 𝒴 and where𝒩𝑘=𝑢𝒳;𝑝𝑢=𝜆𝑘𝑢.(3.24)

Moreover, by Theorem 3.5, the sequence of 𝑘th eigenvalues {𝜆𝑘} associated to the sequence {𝑝} converges to the 𝑘th eigenvalue of 𝜆𝑘 of 𝑝 and the corresponding sequence {𝑢𝑘} converges to 𝑢𝑘 weakly in 𝒳. The limit shifted Dirac operator restricted to 𝒳𝑝 is explicitly given by𝑝=𝐇+𝑉1𝑀𝑉2|||𝒳𝑝.(3.25)

This follows by standard arguments in homogenization theory, see for example, [17].

We continue now to study the asymptotic analysis of the projection to the closed subspace of 𝒴 corresponding to the positive part [2,+) of the continuous spectrum of .

We denote by 𝒳𝑐 the orthogonal complement in 𝒳 to the eigenspace 𝒳𝑝. Thus, 𝒳𝑐 is the closed subspace invariant with respect to corresponding to the absolutely continuous spectrum 𝜎𝑐()=(,0][2,). We now define the two mutually orthogonal subspaces 𝒳𝑐,+ and 𝒳𝑐, with𝒳𝑐=𝒳𝑐,+𝒳𝑐,,(3.26)

where 𝒳𝑐,+ corresponds to the positive part [2,+) and 𝒳𝑐, corresponds to the negative part (,0], respectively. Next we define the restriction 𝑐,+ of to 𝒳𝑐,+ which can be written as𝑐,+=𝜆𝜎𝑐,+𝜆𝑑𝐸,𝑐,+(𝜆),(3.27)

where the spectral measure 𝐸,𝑐,+(𝜆) is the continuous spectral measure corresponding to 𝑐,+. By construction 𝑐,+ is a positive and self-adjoint operator on 𝒳𝑐,+. Therefore by Proposition  13.4 in [7], there exists a sequence {𝑐,+} which G-converges to a positive and self-adjoint operator 𝑐,+𝒳𝑐,+. Moreover, since 𝜆 is not an eigenvalue, the corresponding sequence {𝐸,𝑐,+(𝜆)} of spectral measures converges to the spectral measure 𝐸,𝑐,+(𝜆) corresponding to 𝑐,+.

Let us consider the evolution equation𝑖𝜕𝐮𝜕𝑡=(𝑡,𝑥)𝑐,+𝐮𝐮(𝑡,𝑥),(,0)=𝐮0.(3.28) By the Stone theorem, there exists a unique solution 𝐮=𝐮(𝑥,𝑡) to (3.28) given by𝐮(,𝑡)=𝒰(𝑡)𝐮0,𝐮0𝒳𝑐,+,(3.29)

where 𝒰(𝑡)=exp(𝑖𝑐,+𝑡) is the strongly continuous unitary group of transformations generated by the infinitesimal operator 𝑖𝑐,+ on 𝒴. By the G-convergence of the sequence {𝑐,+} it follows that the associated sequence {𝒰𝑐,+(𝑡)} of unitary groups of transformations converges to a unitary group of transformations 𝒰𝑐,+(𝑡) which for every 𝐮0𝒳𝑐,+ defines the solution 𝐮(,𝑡)=𝒰(𝑡)𝐮0 to the limit evolution equation𝑖𝜕=𝜕𝑡𝐮(𝑡,𝑥)𝑐,+𝐮(𝑡,𝑥),𝐮(,0)=𝐮0.(3.30)

Finally, by considering the operator 𝑐, where 𝑐, is the restriction to 𝒳𝑐,, that is, the closed subspace corresponding to the negative part (,0] of the continuous spectrum we can repeat all the arguments from the positive part of the continuous spectrum.

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