G-Convergence of Dirac Operators

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.

of G-convergence becomes applicable.
We will consider periodic perturbations, i.e. we will assume that the potential V h is a periodic function with respect to some regular lattice in R N . We are then interested in the asymptotic behaviour of shifted perturbed Dirac operatorsH h . This yields homogenization problems for the evolution equation associated with the Dirac operatorH h iℏ ∂ ∂t u h (t, x) =H h u h (t, x) , u h (·, 0) = u 0 h and the corresponding eigenvalue problem 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.

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 [14], [15] and [17].
Let X and Y denote the Hilbert spaces H 1 (R 3 ; C 4 ) and L 2 (R 3 ; C 4 ), respectively. The free Dirac evolution equation reads (1) iℏ ∂ ∂t u(t, x) = H 0 u(t, x) , Here α · ∇ = 3 i=1 α i ∂ ∂x i , ℏ is the Planck constant divided by 2π, the constant c is the speed of light, m is the particle rest mass and α = (α 1 , α 2 , α 3 ) and β are the 4×4 Dirac matrices given Here I and 0 are the 2×2 unity and zero matrices, respectively, and the σ i 's are the 2×2 Pauli matrices Note that a separation of variables in (1) yields the Dirac eigenvalue problem 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 H 0 is essentially self-adjoint on C ∞ 0 (R 3 ; C 4 ) and self-adjoint on X. Moreover, its spectrum, σ(H 0 ), is purely absolutely continuous (i.e. its spectral measure is absolutely continuous with respect to the Lebesgue measure) and given by H 0 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 , It acts as a multiplication operator in L 2 (R 3 ; C 4 ), thus the free Dirac operator with additional field W is of the form The operator H is essentially self-adjoint on C ∞ 0 (R 3 ; C 4 ) and self-adjoint on the Sobolev space X provided that W is Hermitian and satisfies the following estimate (see e.g. [14]) 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, where Z is the elctric charge number (without ambiguity, I is usually dropped from the Coulomb term for simplicity). The spectrum of the Dirac operator with Coulomb potential is given by where {λ k } k∈N is a discrete sequence of eigenvalues in the "gap" and the remaining part of the spectrum is the continuous part σ(H 0 ).
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 h = V h (x) and define the operator H h as (6) H We recall that a function F is called homogeneous of degree p if for any nonzero scalar a, F (ax) = a p F (x). The next theorem is of profound importance for the present work.
Theorem 1. Let W be Hermitian and satisfy the bound (5) above. Further, for any fixed h ∈ N, let V h be a measurable (−1)-homogeneous Hermitian 4× 4 matrix-valued function with entries in L p loc (R 3 ), p > 3. Then H h is essentially self-adjoint on C ∞ 0 (R 3 ; C 4 ) and self-adjoint on X. Moreover where {λ k h } k∈N is a discrete sequence of parameter dependent eigenvalues corresponding to the Proof. See [15,17].
We will as a motivating example consider perturbations which are locally periodic and of the form V h (x) = V 1 (x)V 2 (hx). The entries of V 1 are assumed to be (−1)-homogeneous. The entries of V 2 (y) are assumed to be periodic with respect to a regular lattice in R 3 . This can also be phrased that they are defined on the unit torus T 3 .
The evolution equation associated with the Dirac operator H h reads By the Stone theorem, since H h is self-adjoint on X, there exists a unique solution u h to (7) given by is the strongly continuous unitary operator generated by the infinitesimal operator −(i/ℏ)H h on Y, see e.g. [6] or [14].
In the sequel we will consider a shifted family of Dirac operators denoted byH h and defined asH h =H + V h , whereH = H + mc 2 I. Also without loss of generality we will in the sequel put ℏ = c = m = 1. By Theorem 1, for any h ∈ N, we then get

G-convergence.
For more detailed information on G-convergence we refer to e.g. [4,11] for the application to elliptic and parabolic partial differential operators, and to the monograph [3] for the application to general self-adjoint operators. Here we recall some basic facts about G-convergence for self-adjoint operators in Y.
In the present work we frequently write A h converges to A when we mean that the sequence where s and w refer to strong and weak topologies respectively, and P h and P are the orthogonal projections onto The following result provides a useful criterion for G-convergence of self-adjoint operators.
See [3] for a proof.
From now on we will just use the word "converge" instead of saying "strongly converge", 2.3. Some Basic Results in Spectral Theory. For more details see [2,6,15]. Given a Hilbert space X, let (U , A ) be a measurable space for U ⊆ C and A being a σ−algebra on U . Let P X = P(X) be the set of orthogonal projections on X, then E : A −→ P X is called a spectral measure if it satisfies the following (i) E(∅) = 0 (This condition is superfluous given the next properties).
(iii) Countable additivity; if {△ n } ⊂ A is a finite or a countable set of disjoint elements and Because of the idempotence property of the spectral measure we have E n u 2 = E n u, E n u = E 2 n u, u = E n u, u → Eu, u = Eu 2 , which means that the weak convergence and the strong convergence of a sequence of spectral measures E n are equivalent.
Let E u,u (△) be the finite scalar measure on A generated by E, Let U = R. The spectral measure on the real line corresponding to an operator S is denoted by E S (λ) (where the superscript S indicates that the spectral measure E corresponds to a specific operator S) Now we state the spectral theorem for self-adjoint operators. Theorem 2. For a self-adjoint operator S defined on a Hilbert space X there exists a unique spectral measure E S on X such that Proof. See e.g. [2].

The main results
Consider the family {H h } h∈N of Dirac operators with domain D(H h ) = X. We will state and prove some useful theorems for operators of the class P λ (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 class P λ (Y) for λ > 0.
Theorem 3. Let A be a positive and self-adjoint operator on Y and put B = A + λI. Then for Proof . See Propositions 12.1 and 12.3 in [3].
The connection between the eigenvalue problems of the operator and its G-limit of the class P λ (Y) for λ ≥ 0 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. Proof . Since A h G-converges to A in the strong resolvent sense where B h and B are A h + λI and A + λI respectively. Note that by Theorem 4, D(B −1 h ) = R(B h ) = Y, so the projections P h and P are unnecessary.
The convergence to zero follows with help of (9) and the boundedness of the inverse operator Hence u − B −1 J , v = 0 for every v ∈ Y, which implies Bu = J , therefore Au = f . Proof . The proof is straight forward by assuming f h = µ h u h (which converges to µu in Y) in the previous theorem.
The convergence properties of self-adjoint operators has 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 h } of operators which converges uniformly to a limit operator A nice results can be drawn for the spectrum. Exactly speaking {σ(A h )} converges to σ(A) including the isolated eigenvalues. The same conclusion holds if the uniform convergence is replaced by the uniform resolvent convergence, see e.g. [6]. In the case of strong convergence (the same for strong resolvent convergence), if the sequence {A h } is strongly convergent to A, then every λ ∈ σ(A) is the limit of a sequence {λ h } where λ h ∈ σ(A h ), but not the limit of every such sequence {λ h } lies in the spectrum of A, (see the below example taken from [16]). For weakly convergent sequences of operators no spectral implications can be extracted.
Example. Let A i,h be an operator in L 2 (R) defined by 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 P λ (Y) (and hence the Gconvergence in the strong resolvent sense of operators of the class P 0 (Y)) implies the convergence of the corresponding eigenvalues. Theorem 6. Let {A h } be a family of operators in P λ (Y), λ > 0, with domain X. If A h Gconverges to A ∈ P λ (Y), then K h := A −1 h converges in the norm of B(Y) (B(Y) is the set of bounded linear operators on Y) to K := A −1 . Moreover the k th eigenvalue µ k h of A h converges to the k th eigenvalue µ k of A and the associated k th eigenvector u k h converges to u k weakly in X, ∀k ∈ N.

The operator
Proof . By the definition of supremum norm Also, by the definition of supremum norm there exists a sequence It is well-known that K h and K are compact self-adjoint operators on Y. Both are bounded operators, by Theorem 3, with compact range X of Y.
Consider now the right hand side of (11). We write this as The first and the third terms converge to zero by the compactness of K h and K on Y and the second term converges to zero by the G-convergence of A h to A. Consequently Consider the eigenvalue problems associated to A −1 h and A −1 Since A −1 h and A −1 are compact and self-adjoint operators it is well-known that there exists infinite sequences of eigenvalues λ 1 h ≥ λ 2 h ≥ · · · and λ 1 ≥ λ 2 ≥ · · · accumulating at the origin, respectively. Define µ k h := (λ k h ) −1 and µ k := (λ k ) −1 for all k ∈ N. Consider now the spectral problems associated to A h and A There exists infinite sequences of eigenvalues 0 < µ 1 h ≤ µ 2 h ≤ · · · and 0 < µ 1 ≤ µ 2 ≤ · · · respectively. By the compactness of K h and K the sets {λ k h } ∞ k=1 and {λ k } ∞ k=1 are bounded in R, thus the proof is complete by virtue of the following lemma.
Lemma 2. Let X, Y, K h , K, λ k h and λ k be as in Theorem 6, and let A h ∈ P λ (Y), λ > 0. There is a sequence {r k h } converging to zero with 0 < r k h < λ k such that where c is a constant independent of h, and N (λ k , K) = {u ∈ D(K) ; Ku = λ k u} is the eigenspace of K corresponding to λ k .
We can now complete the proof of Theorem 6. By the G-convergence of A h to A we obtain, by using Lemma 2 and (12), convergence of the eigenvalues and eigenvectors, i.e. µ k h → µ k and u k h → u k weakly in X as h → ∞.
Let us now return to the shifted and perturbed Dirac operatorH h . We will throughout this section assume the hypotheses of Theorem 1. We further assume that the 4 × 4 matrix-valued where V 1 is (-1)-homogeneous and where the entries of V 2 (y) are 1-periodic in y, i.e.
We also assume that the entries of V 2 belong to L ∞ (R 3 ). It is then well-known that in L ∞ (R 3 ) weakly*, where T 3 is the unit torus in R 3 . It easily also follows from this mean-value property that in L p (R 3 ) weakly for p > 3, cf the hypotheses in Theorem 1.
We are now interested in the asymptotic behavior of the operator and the spectrum of the perturbed Dirac operatorH h . We recall the spectral problem forH h , i.e.
where there exists a discrete set of eigenvalues {λ k h }, k = 1, 2, . . . and a corresponding set of mutually orthogonal eigenfunctions {u k h }. We know, by Theorem 1, that the eigenvalues (or point spectrum) σ p (H h ) ⊂ (0, 2). We also know thatH h has a continuous spectrum . This means that the Dirac operator is neither a positive or negative (semi-definite) 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 ofH h we therefore use spectral projection and study the corresponding asymptotic behavior of projectionsH h which are positive so that G-convergence methods apply.
Let A be a fixed σ-algebra of subsets of R, and let (R, A ) be a measurable space. Consider the spectral measures EH and EH h of the families of Dirac operatorsH h andH respectively, each one of these measures maps A onto P X , where P X is the set of orthogonal projections on X. By the spectral theorem By the spectral theorem we can also write (20) since V h is a multiplication operator.
We recall that D(H h ) = X, let now i.e. the eigenspace ofH h corresponding to the eigenvalue λ k h . Further define the sum of mutual orthogonal eigenspaces It is clear that for u h ∈ X p h we have (H h u h , u h ) = λ k |u h | 2 > 0, k = 1, 2, . . . .
Let us now consider the restrictionH p h ofH h to X p h which can be written as where the spectral measure EH h ,p is the point measure, i.e. the orthogonal projection onto ker(H h − λI). With this constructionH p h is a positive and self-adjoint operator on X p h with compact inverse (H p h ) −1 . By Lemma 1, see also Proposition 13.4 in [3], we conclude that there exists a positive and self-adjoint operatorH p such that, up to a subsequence,H p h G-converges toH p , whereH p has domain D(H p ) = X p where X p = ⊕ k∈N N k is a closed subspace of Y and where N k = {u ∈ X;H p u = λ k u}.
Moreover, by Theorem 6, the sequence of k th eigenvalues {λ k h } associated to the sequence {H p h } converges to the k th eigenvalue of λ k h ofH p and the corresponding sequence {u k h } converges to u k weakly in X. The limit shifted Dirac operator restricted to X p is explicitly given bỹ