# G-Convergence of Dirac Operators

Authors: Hasan Almanasreh; Nils Svanstedt
Journal: Journal of Function Spaces and Applications
 (2012)

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## 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.


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## Body

## 1. Introduction

In the present work we study the asymptotic behavior of Dirac operatorsℋ̃h with respect to a parameter h∈ℕ as h→∞. We consider Dirac operators ℋ̃h=H̃+Vh on L2(ℝ3;ℂ4), where H̃=H0+W+I (I denoting the 4×4 identity matrix) is a shifted Dirac operator. The operators  H0, W, and Vh are the free Dirac operator, the Coulomb potential (W(x)=(-Z/x)I, where Z is the electric charge number), and a perturbation to H̃, respectively. We will study the asymptotic behavior the shifted perturbed Dirac operator ℋ̃h and the asymptotic behavior of the eigenvalues in the gap of the continuous spectrum of the shifted operator with respect to the perturbation parameter h.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 [1–3] for linear elliptic and parabolic problems with symmetric coefficients matrices. Later on it was extended to the nonsymmetric case by Murat and Tartar [4–6] 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 potentialVh is a periodic function with respect to some regular lattice in ℝN. We are then interested in the asymptotic behaviour of shifted perturbed Dirac operators ℋ̃h. This yields homogenization problems for the evolution equation associated with the Dirac operator ℋ̃h(1.1)iℏ∂∂tuh(t,x)=H̃huh(t,x),uh(⋅,0)=uh0and the corresponding eigenvalue problem(1.2)H̃huh(x)=λhuh(x).The paper is arranged as follows: in Section2 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

LetA 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 [8–10].Let𝒳 and 𝒴 denote the Hilbert spaces H1(ℝ3;ℂ4) and L2(ℝ3;ℂ4), respectively. The free Dirac evolution equation reads(2.1)iℏ∂∂tu(t,x)=H0u(t,x),
where H0:𝒴→𝒴 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 form(2.2)H0=-iℏcα⋅∇+mc2β.
Here α·∇=∑i=13αi(∂/∂xi), ℏ 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 by(2.3)αi=(0σiσi0),β=(I00-I).HereI and 0 are the 2×2 unity and zero matrices, respectively, and the σi’s are the 2×2 Pauli matrices(2.4)σ1=(0110),σ2=(0-ii0),σ3=(100-1).Note that a separation of variables in (2.1) yields the Dirac eigenvalue problem(2.5)H0u(x)=λu(x),
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∞(ℝ3;ℂ4) 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 by(2.6)σ(H0)=(-∞,-mc2]∪[mc2,+∞).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,(2.7)W(x)=Wij(x)i,j=1,2,3,4.It acts as a multiplication operator inL2(ℝ3;ℂ4), thus the free Dirac operator with additional field W is of the form
(2.8)H=H0+W.The operatorH is essentially self-adjoint on C0∞(ℝ3;ℂ4) and self-adjoint on the Sobolev space 𝒳 provided that W is Hermitian and satisfies the following estimate (see for e.g., [8]):(2.9)|Wij(x)|≤ac2|x|+b,∀x∈R3∖{0}i,j=1,2,3,4,
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 by(2.10)σ(H)=(-∞,-mc2]∪{λk}k∈N∪[mc2,+∞),where{λk}k∈ℕ 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 a4×4 matrix-valued function Vh=Vh(x) and define the operator ℋh as(2.11)Hh=H+Vh.
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.
LetW be Hermitian and satisfy the bound  (2.9) above. Further, for any fixed  h∈ℕ, let  Vh be a measurable  (-1)-homogeneous Hermitian  4×4 matrix-valued function with entries in  Llocp(ℝ3),  p>3. Then  ℋh is essentially self-adjoint on  C0∞(ℝ3;ℂ4) and self-adjoint on  𝒳. Moreover
(2.12)σ(Hh)=(-∞,-mc2]∪{λhk}k∈N∪[mc2,+∞),where {λhk}k∈ℕ is a discrete sequence of parameter dependent eigenvalues corresponding to the Dirac eigenvalue problem ℋhuh(x)=λhuh(x).Proof.
See  [9, 10].We will as a motivating example consider perturbations which are locally periodic and of the formVh(x)=V1(x)V2(hx). 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 ℝ3. This can also be phrased that they are defined on the unit torus 𝕋3.The evolution equation associated with the Dirac operatorℋh reads(2.13)iℏ∂∂tuh(t,x)=Hhuh(t,x),uh(⋅,0)=uh0.
By the Stone theorem, since ℋh is self-adjoint on 𝒳, there exists a unique solution uh to (2.13) given by(2.14)uh(⋅,t)=Uh(t)uh0,∀uh0∈X,
where 𝒰h(t)=exp(-(i/ℏ)ℋht) is the strongly continuous unitary operator generated by the infinitesimal operator -(i/ℏ)ℋh on 𝒴, see for example, [8] or [11].In the sequel we will consider a shifted family of Dirac operators denoted byℋ̃h and defined as ℋ̃h=H̃+Vh, where H̃=H+mc2I. Also without loss of generality we will in the sequel put ℏ=c=m=1. By Theorem 2.1, for any h∈ℕ, we then get
(2.15)σ(H̃h)=(-∞,0]∪{λ̃hk}k∈N∪[2,∞).


### 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.Letλ≥0, 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,u〉≥λ||u||𝒴2forallu∈D(A).Definition 2.2.
Letλ>0,and let {Ah}⊂𝒫λ(𝒴) then we say that Ah G-converges to A∈𝒫λ(𝒴),denoted Ah→G,sG,wA in 𝒴 if Ah-1Phu→swA-1Pu in 𝒴, for all u∈𝒴,where s and w refer to strong and weak topologies respectively, and Ph and P are the orthogonal projections onto 𝒱h:=D(Ah)¯and𝒱:=D(A)¯ respectively. Also we say {Ah}⊂𝒫0(𝒴) converges to {A}⊂𝒫0(𝒴) in the strong resolvent sense (SRS) if (λI+Ah)→G,s(λI+A)in𝒴for allλ>0.
The following result provides a useful criterion for G-convergence of self-adjoint operators. See [7] for a proof.Lemma 2.3.
Givenλ>0,  {Ah}⊂𝒫λ(𝒴) and the orthogonal projection  Ph onto  𝒱h. Suppose that for every  u∈𝒴,  Ah-1Phu converges strongly (resp. weakly) in  𝒴, then there exists an operator  A∈𝒫λ(𝒴) such that  Ah→G,sA (resp. Ah→G,wA) in  𝒫λ(𝒴).From now on we will just use the word “converge” instead of saying “strongly converge,” henceAh→GA instead of Ah→G,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⊆ℂ and let 𝒜 be a σ-algebra on U. Let ℙX=ℙ(X) be the set of orthogonal projections on X, then E:𝒜→ℙX 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).IfE is spectral measure then E(Δ1∩Δ2)=E(Δ1)E(Δ2)=E(Δ2)E(Δ1), also E is modular, that is, E(Δ1∪Δ2)+E(Δ1∩Δ2)=E(Δ1)+E(Δ2). For an increasing sequence of sets Δn, limn→∞E(Δn)=E(∪nΔn), while if Δn is a decreasing sequence then limn→∞E(Δn)=E(∩nΔn). Because of the idempotence property of the spectral measure we have ∥Enu∥2=〈Enu,Enu〉=〈En2u,u〉=〈Enu,u〉→〈Eu,u〉=∥Eu∥2, which means that the weak convergence and the strong convergence of a sequence of spectral measures En are equivalent.LetEu,u(Δ) be the finite scalar measure on 𝒜 generated by E,(2.16)Eu,u(Δ)=〈E(Δ)u,u〉=‖E(Δ)u‖2andEu,v(∆) be the complex measure(2.17)Eu,v(Δ)=〈E(Δ)u,v〉,∀u,v∈X.By the above notationsEu,v(Δ)≤∥E(Δ)u∥∥E(Δ)v∥≤∥u∥∥v∥.LetU=ℝ. 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)(2.18)ES(λ)=ES(Δ),whereΔ=(-∞,λ),forλ∈R.ClearlyES(λ) is monotonic (nondecreasing), that is, ES(λ1)≤ES(λ2) for λ1≤λ2. Also limλ→-∞ES(λ)=0 and limλ→∞ES(λ)=I.  ES(λ) are self-adjoint, idempotent, positive, bounded, right continuous operator (limt→0+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(λ-0) 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 operatorS 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 Δ∩σ(S)=∅;(iii)
IfΔ⊂ℝ is open and Δ∩σ(S)≠∅, then E(Δ)≠0.Proof.
See for example,  [14].


## 2.1. Dirac Operator

We recall some basic facts regarding the Dirac operator. For more details we refer to the monographs [8–10].Let𝒳 and 𝒴 denote the Hilbert spaces H1(ℝ3;ℂ4) and L2(ℝ3;ℂ4), respectively. The free Dirac evolution equation reads(2.1)iℏ∂∂tu(t,x)=H0u(t,x),
where H0:𝒴→𝒴 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 form(2.2)H0=-iℏcα⋅∇+mc2β.
Here α·∇=∑i=13αi(∂/∂xi), ℏ 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 by(2.3)αi=(0σiσi0),β=(I00-I).HereI and 0 are the 2×2 unity and zero matrices, respectively, and the σi’s are the 2×2 Pauli matrices(2.4)σ1=(0110),σ2=(0-ii0),σ3=(100-1).Note that a separation of variables in (2.1) yields the Dirac eigenvalue problem(2.5)H0u(x)=λu(x),
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∞(ℝ3;ℂ4) 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 by(2.6)σ(H0)=(-∞,-mc2]∪[mc2,+∞).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,(2.7)W(x)=Wij(x)i,j=1,2,3,4.It acts as a multiplication operator inL2(ℝ3;ℂ4), thus the free Dirac operator with additional field W is of the form
(2.8)H=H0+W.The operatorH is essentially self-adjoint on C0∞(ℝ3;ℂ4) and self-adjoint on the Sobolev space 𝒳 provided that W is Hermitian and satisfies the following estimate (see for e.g., [8]):(2.9)|Wij(x)|≤ac2|x|+b,∀x∈R3∖{0}i,j=1,2,3,4,
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 by(2.10)σ(H)=(-∞,-mc2]∪{λk}k∈N∪[mc2,+∞),where{λk}k∈ℕ 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 a4×4 matrix-valued function Vh=Vh(x) and define the operator ℋh as(2.11)Hh=H+Vh.
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.
LetW be Hermitian and satisfy the bound  (2.9) above. Further, for any fixed  h∈ℕ, let  Vh be a measurable  (-1)-homogeneous Hermitian  4×4 matrix-valued function with entries in  Llocp(ℝ3),  p>3. Then  ℋh is essentially self-adjoint on  C0∞(ℝ3;ℂ4) and self-adjoint on  𝒳. Moreover
(2.12)σ(Hh)=(-∞,-mc2]∪{λhk}k∈N∪[mc2,+∞),where {λhk}k∈ℕ is a discrete sequence of parameter dependent eigenvalues corresponding to the Dirac eigenvalue problem ℋhuh(x)=λhuh(x).Proof.
See  [9, 10].We will as a motivating example consider perturbations which are locally periodic and of the formVh(x)=V1(x)V2(hx). 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 ℝ3. This can also be phrased that they are defined on the unit torus 𝕋3.The evolution equation associated with the Dirac operatorℋh reads(2.13)iℏ∂∂tuh(t,x)=Hhuh(t,x),uh(⋅,0)=uh0.
By the Stone theorem, since ℋh is self-adjoint on 𝒳, there exists a unique solution uh to (2.13) given by(2.14)uh(⋅,t)=Uh(t)uh0,∀uh0∈X,
where 𝒰h(t)=exp(-(i/ℏ)ℋht) is the strongly continuous unitary operator generated by the infinitesimal operator -(i/ℏ)ℋh on 𝒴, see for example, [8] or [11].In the sequel we will consider a shifted family of Dirac operators denoted byℋ̃h and defined as ℋ̃h=H̃+Vh, where H̃=H+mc2I. Also without loss of generality we will in the sequel put ℏ=c=m=1. By Theorem 2.1, for any h∈ℕ, we then get
(2.15)σ(H̃h)=(-∞,0]∪{λ̃hk}k∈N∪[2,∞).


## 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.Letλ≥0, 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,u〉≥λ||u||𝒴2forallu∈D(A).Definition 2.2.
Letλ>0,and let {Ah}⊂𝒫λ(𝒴) then we say that Ah G-converges to A∈𝒫λ(𝒴),denoted Ah→G,sG,wA in 𝒴 if Ah-1Phu→swA-1Pu in 𝒴, for all u∈𝒴,where s and w refer to strong and weak topologies respectively, and Ph and P are the orthogonal projections onto 𝒱h:=D(Ah)¯and𝒱:=D(A)¯ respectively. Also we say {Ah}⊂𝒫0(𝒴) converges to {A}⊂𝒫0(𝒴) in the strong resolvent sense (SRS) if (λI+Ah)→G,s(λI+A)in𝒴for allλ>0.
The following result provides a useful criterion for G-convergence of self-adjoint operators. See [7] for a proof.Lemma 2.3.
Givenλ>0,  {Ah}⊂𝒫λ(𝒴) and the orthogonal projection  Ph onto  𝒱h. Suppose that for every  u∈𝒴,  Ah-1Phu converges strongly (resp. weakly) in  𝒴, then there exists an operator  A∈𝒫λ(𝒴) such that  Ah→G,sA (resp. Ah→G,wA) in  𝒫λ(𝒴).From now on we will just use the word “converge” instead of saying “strongly converge,” henceAh→GA instead of Ah→G,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⊆ℂ and let 𝒜 be a σ-algebra on U. Let ℙX=ℙ(X) be the set of orthogonal projections on X, then E:𝒜→ℙX 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).IfE is spectral measure then E(Δ1∩Δ2)=E(Δ1)E(Δ2)=E(Δ2)E(Δ1), also E is modular, that is, E(Δ1∪Δ2)+E(Δ1∩Δ2)=E(Δ1)+E(Δ2). For an increasing sequence of sets Δn, limn→∞E(Δn)=E(∪nΔn), while if Δn is a decreasing sequence then limn→∞E(Δn)=E(∩nΔn). Because of the idempotence property of the spectral measure we have ∥Enu∥2=〈Enu,Enu〉=〈En2u,u〉=〈Enu,u〉→〈Eu,u〉=∥Eu∥2, which means that the weak convergence and the strong convergence of a sequence of spectral measures En are equivalent.LetEu,u(Δ) be the finite scalar measure on 𝒜 generated by E,(2.16)Eu,u(Δ)=〈E(Δ)u,u〉=‖E(Δ)u‖2andEu,v(∆) be the complex measure(2.17)Eu,v(Δ)=〈E(Δ)u,v〉,∀u,v∈X.By the above notationsEu,v(Δ)≤∥E(Δ)u∥∥E(Δ)v∥≤∥u∥∥v∥.LetU=ℝ. 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)(2.18)ES(λ)=ES(Δ),whereΔ=(-∞,λ),forλ∈R.ClearlyES(λ) is monotonic (nondecreasing), that is, ES(λ1)≤ES(λ2) for λ1≤λ2. Also limλ→-∞ES(λ)=0 and limλ→∞ES(λ)=I.  ES(λ) are self-adjoint, idempotent, positive, bounded, right continuous operator (limt→0+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(λ-0) 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 operatorS 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 Δ∩σ(S)=∅;(iii)
IfΔ⊂ℝ is open and Δ∩σ(S)≠∅, then E(Δ)≠0.Proof.
See for example,  [14].


## 3. The Main Results

Consider the family{ℋ̃h}h∈ℕ of Dirac operators with domain D(ℋ̃h)=𝒳. We will state and prove some useful theorems for operators of the class 𝒫λ(𝒴)  for  λ≥0, 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 class𝒫λ(𝒴) for λ>0.Theorem 3.1.
LetA be a positive and self-adjoint operator on  𝒴 and put  B=A+λI. Then for  λ>0,(i)
B  is injective. Moreover, for everyv∈R(B),〈B-1v,v〉〉≥λ∥B-1v∥𝒴2 and ∥B-1v∥𝒴≤λ-1∥v∥𝒴.(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 class𝒫λ(𝒴) for λ≥0 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{Ah} of the class  𝒫0(𝒴) G-converging to  A∈𝒫0(𝒴) in the strong resolvent sense. Let  uh be the solution of  Ahuh=fh, where  {fh} is converging to  f in  𝒴. If  {uh} converges to  u in  𝒴, then  u solves the G-limit problem  Au=f.Proof.
SinceAh G-converges to A in the strong resolvent sense
(3.1)Bh-1Phv⟶B-1Pv,∀v∈Y,
where Bh and B are Ah+λI and A+λI, respectively. Note that by Theorem 3.1, D(Bh-1)=R(Bh)=𝒴, so the projections Ph and P are unnecessary.
ConsiderAhuh=fh which is equivalent to Bhuh=fh+λuh; by the definition of Bh we have uh=Bh-1(fh+λuh). Define 𝒥h=fh+λuh which is clearly convergent to 𝒥=f+λu in 𝒴; by the assumptions. Therefore Bh-1𝒥h→B-1𝒥, this is because
(3.2)‖Bh-1Jh-B-1J‖Y=‖Bh-1Jh-Bh-1J+Bh-1J-B-1J‖Y≤‖Bh-1‖Y‖Jh-J‖Y+‖Bh-1J-B-1J‖Y⟶0. The convergence to zero follows with help of (3.1) and the boundedness of the inverse operator Bh-1. Thus, for all v∈𝒴(3.3)〈u,v〉=lim⁡h→∞〈uh,v〉=lim⁡h→∞〈Bh-1Jh,v〉=〈B-1J,v〉.
Hence 〈u-B-1𝒥,v〉=0 for every v∈𝒴, which implies Bu=𝒥; therefore Au=f.Theorem 3.3.
Let{Ah} be a sequence in  𝒫0(𝒴) which G-converges to  A∈𝒫0(𝒴) in the strong resolvent sense, and let  {μh,uh} be the solution of the eigenvalue problem  Ahuh=μhuh. If  {μh,uh}→{μ,u} in  ℝ×𝒴, then the limit couple  {μ,u} is the solution of the eigenvalue problem  Au=μu.Proof.
The proof is straight forward by assumingfh=μhuh (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{Ah} of operators which converges uniformly to a limit operator A nice results can be drawn for the spectrum. Exactly speaking {σ(Ah)} 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 {Ah} is strongly convergent to A, then every λ∈σ(A) is the limit of a sequence {λh} where λh∈σ(Ah), but not the limit of every such sequence {λh} 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 Ah converges to A when we mean that the sequence {Ah} is converging to A.Example 3.4.
LetAi,h be an operator in L2(ℝ) defined by
(3.4)Ai,h=-d2dx2+Vi,h(x),forh∈N,i=1,2,where
(3.5)V1,h(x)={-1,ifh≤x≤h+1,0,Otherwise,V2,h(x)={-1,ifx≥h,0,Otherwise.
The operator Ai,h converges to A=-d2/dx2 in the strong resolvent sense as h→∞ for both i=1,2. One can compute the spectrum for the three operators and obtain σ(A1,h)=[0,∞)∪{μh} for μh being a simple eigenvalue in [-1,0] and σ(A2,h)=[-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 set𝒫λ(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{Ah} be a family of operators in  𝒫λ(𝒴),  λ>0, with domain  𝒳. If  Ah G-converges to  A∈𝒫λ(𝒴), then  𝒦h:=Ah-1 converges in the norm of  ℬ(𝒴)(ℬ(𝒴) is the set of bounded linear operators on  𝒴) to  𝒦:=A-1. Moreover the  kth eigenvalue  μhk of  Ah converges to the  kth eigenvalue  μk of  A and the associated  kth eigenvector  uhk converges to  uk weakly in  𝒳,   for allk∈ℕ.Proof.
By the definition of supremum norm(3.6)‖Kh-K‖B(Y)=sup⁡‖v‖Y=1‖Khv-Kv‖Y=sup⁡‖v‖Y≤1‖Khv-Kv‖Y.
Also, by the definition of supremum norm there exists a sequence {vh}⊂𝒴 with ∥vh∥𝒴≤1 such that
(3.7)‖Kh-K‖B(Y)≤‖Khvh-Kvh‖Y+1h.
It is well known that 𝒦h 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 as(3.8)‖Khvh-Kvh‖Y+1h≤‖Khvh-Khv‖Y+‖Khv-Kv‖Y+‖Kvh-Kv‖Y+1h.The first and the third terms converge to zero by the compactness of 𝒦h and 𝒦 on 𝒴 and the second term converges to zero by the G-convergence of Ah to A. Consequently
(3.9)‖Kh-K‖B(Y)⟶0.
Consider the eigenvalue problems associated to Ah-1 and A-1(3.10)Ah-1vhk=λhkvhk,k∈N,A-1vk=λkvk,k∈N.
Since Ah-1 and A-1 are compact and self-adjoint operators it is well known that there exists infinite sequences of eigenvalues λh1≥λh2≥⋯ and λ1≥λ2≥⋯, accumulating at the origin, respectively. Define μhk:=(λhk)-1 and μk:=(λk)-1 for all k∈ℕ. Consider now the spectral problems associated to Ah and A(3.11)Ahuhk=μhkuhk,k∈N,Auk=μkuk,k∈N.
There exists infinite sequences of eigenvalues 0<μh1≤μh2≤⋯ and 0<μ1≤μ2≤⋯, respectively. By the compactness of 𝒦h and 𝒦 the sets {λhk}k=1∞ and {λk}k=1∞ are bounded in ℝ, thus the proof is complete by virtue of the following lemma.Lemma 3.6.
Let𝒳,  𝒴,  𝒦h,  𝒦,  λhk and  λk be as in Theorem 3.5, and let  Ah∈𝒫λ(𝒴),  λ>0. There is a sequence  {rhk} converging to zero with  0<rhk<λk such that
(3.12)|λhk-λk|≤cλkλk-rhksup⁡u∈N(λk,K)‖u‖Y=1‖Khu-Ku‖Y,
where c is a constant independent of h, and 𝒩(λk,𝒦)={u∈D(𝒦);𝒦u=λku} is the eigenspace of 𝒦 corresponding to λk.Proof.
See Theorem  1.4 and Lemma1.6in [16] Chapter 3.We can now complete the proof of Theorem3.5. By the G-convergence of Ah to A we obtain, by using Lemma 3.6 and (3.9), convergence of the eigenvalues and eigenvectors, that is, μhk→μk and uhk→uk weakly in 𝒳 as h→∞.Let us now return to the shifted and perturbed Dirac operatorℋ̃h. We will throughout this section assume the hypotheses of Theorem 2.1. We further assume that the 4×4 matrix-valued function Vh is of the form Vh(x)=V1(x)V2(hx) where V1 is (−1)-homogeneous and where the entries of V2(y) are 1-periodic in y, that is,(3.13)V2ij(y+k)=V2ij(y),k∈Z3.We also assume that the entries ofV2 belong to L∞(ℝ3). It is then well known that(3.14)V2ij(hx)⟶M(V2ij)=∫T3V2ij(y)dy,
in L∞(ℝ3) weakly*, where 𝕋3 is the unit torus in ℝ3. It easily also follows from this mean-value property that(3.15)Vh⟶V1M(V2),inLp(ℝ3) 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 operatorℋ̃h. We recall the spectral problem for ℋ̃h, that is,(3.16)H̃huh(x)=λ̃huh(x),where there exists a discrete set of eigenvalues{λ̃hk}, k=1,2,… and a corresponding set of mutually orthogonal eigenfunctions {uhk}. We know, by Theorem 2.1, that the eigenvalues (or point spectrum) σp(ℋ̃h)⊂(0,2). We also know that ℋ̃h has a continuous spectrum σc(ℋ̃h)=(-∞,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 ℋ̃h we therefore use spectral projection and study the corresponding asymptotic behavior of projections ℋ̃h which are positive so that G-convergence methods apply.Let𝒜 be a fixed σ-algebra of subsets of ℝ, and let (ℝ,𝒜) be a measurable space. Consider the spectral measures EH̃ and Eℋ̃h of the families of Dirac operators ℋ̃h and H,̃ respectively, each one of these measures maps 𝒜 onto ℙ𝒳, where ℙ𝒳 is the set of orthogonal projections on 𝒳. By the spectral theorem(3.17)H̃h=∫σ(H̃h)λdEH̃h(λ).
By the spectral theorem we can also write(3.18)∫σ(H̃)λdEH̃(λ)+Vh,
since Vh is a multiplication operator.We recall thatD(ℋ̃h)=𝒳, let now(3.19)Nhk={uh∈X;H̃huh=λhkuh},that is, the eigenspace ofℋ̃h corresponding to the eigenvalue λhk. Further define the sum of mutual orthogonal eigenspaces(3.20)Xhp=⨁k∈N⁡Nhk,where𝒳hp is a closed subspace of 𝒴 invariant with respect to ℋ̃h.It is clear that foruh∈𝒳hp we have(3.21)(H̃huh,uh)=λk|uh|2>0,k=1,2,….Let us now consider the restrictionℋ̃hp of ℋ̃h to 𝒳hp which can be written as(3.22)H̃hp=∑λ∈σp(H̃h)λEH̃h,p(λ),where the spectral measureEℋ̃h,p is the point measure, that is, the orthogonal projection onto ker(ℋ̃h-λI). With this construction ℋ̃hp is a positive and self-adjoint operator on 𝒳hp with compact inverse (ℋ̃hp)-1. By Lemma 2.3, see also Proposition  13.4 in [7], we conclude that there exists a positive and self-adjoint operator ℋ̃p such that, up to a subsequence, ℋ̃hp G-converges to ℋ̃p, where ℋ̃p has domain D(ℋ̃p)=𝒳p where(3.23)Xp=⨁k∈N⁡Nk,is a closed subspace of𝒴 and where(3.24)Nk={u∈X;H̃pu=λku}.Moreover, by Theorem3.5, the sequence of kth eigenvalues {λhk} associated to the sequence {ℋ̃hp} converges to the  kth eigenvalue of λhk of ℋ̃p and the corresponding sequence {uhk} converges to uk weakly in 𝒳. The limit shifted Dirac operator restricted to 𝒳p is explicitly given by(3.25)H̃p=(H̃+V1M(V2))|Xp.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 of𝒴 corresponding to the positive part [2,+∞) of the continuous spectrum of ℋ̃h.We denote by𝒳hc the orthogonal complement in 𝒳 to the eigenspace 𝒳hp. Thus, 𝒳hc is the closed subspace invariant with respect to ℋ̃h corresponding to the absolutely continuous spectrum σc(ℋ̃h)=(-∞,0]∪[2,∞). We now define the two mutually orthogonal subspaces 𝒳hc,+ and 𝒳hc,- with(3.26)Xhc=Xhc,+⨁⁡Xhc,-,where𝒳hc,+ corresponds to the positive part [2,+∞) and 𝒳hc,- corresponds to the negative part (-∞,0], respectively. Next we define the restriction ℋ̃hc,+ of ℋ̃h to 𝒳hc,+ which can be written as(3.27)H̃hc,+=∫λ∈σc,+(H̃h)λdEH̃h,c,+(λ),where the spectral measureEℋ̃h,c,+(λ) is the continuous spectral measure corresponding to ℋ̃hc,+. By construction ℋ̃hc,+ is a positive and self-adjoint operator on 𝒳hc,+. Therefore by Proposition  13.4 in [7], there exists a sequence {ℋ̃hc,+} which G-converges to a positive and self-adjoint operator ℋ̃c,+∈𝒳c,+. Moreover, since λ is not an eigenvalue, the corresponding sequence {Eℋ̃h,c,+(λ)} of spectral measures converges to the spectral measure Eℋ̃,c,+(λ) corresponding to ℋ̃c,+.Let us consider the evolution equation(3.28)i∂∂tuh(t,x)=H̃hc,+uh(t,x),uh(⋅,0)=uh0.
By the Stone theorem, there exists a unique solution uh=u(x,t) to (3.28) given by(3.29)uh(⋅,t)=Uh(t)uh0,∀uh0∈Xhc,+,where𝒰h(t)=exp(-iℋ̃hc,+t) is the strongly continuous unitary group of transformations generated by the infinitesimal operator -iℋ̃hc,+ on 𝒴. By the G-convergence of the sequence {ℋ̃hc,+} it follows that the associated sequence {𝒰hc,+(t)} of unitary groups of transformations converges to a unitary group of transformations 𝒰c,+(t) which for every u0∈𝒳c,+ defines the solution u(·,t)=𝒰(t)u0 to the limit evolution equation(3.30)i∂∂tu(t,x)=H̃c,+u(t,x),u(⋅,0)=u0.Finally, by considering the operator-ℋ̃hc,- where ℋ̃hc,- is the restriction to 𝒳hc,-, 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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*Source: 789875-2012-04-05.xml*