Research Article | Open Access
Hasan Almanasreh, Nils Svanstedt, "G-Convergence of Dirac Operators", Journal of Function Spaces, vol. 2012, Article ID 789875, 13 pages, 2012. https://doi.org/10.1155/2012/789875
G-Convergence of Dirac Operators
We consider the linear Dirac operator with a ()-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.
In the present work we study the asymptotic behavior of Dirac operators with respect to a parameter as . We consider Dirac operators on , where ( denoting the identity matrix) is a shifted Dirac operator. The operators , , and are the free Dirac operator, the Coulomb potential (, where is the electric charge number), and a perturbation to , 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 [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 . In the present work we will base a lot of our framework on the results in Chapters 12 and 13 in . 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 is a periodic function with respect to some regular lattice in . 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
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.
Let be a linear operator on a Hilbert space. By , , and we mean the range, domain, and null-space of , respectively.
2.1. Dirac Operator
Let and denote the Hilbert spaces and , respectively. The free Dirac evolution equation reads where is the free Dirac operator with domain , which acts on the four-component vector . It is a first-order linear hyperbolic partial differential equation. The free Dirac operator has the form Here , is the Planck constant divided by , the constant is the speed of light, is the particle rest mass and and are the Dirac matrices given by
Here and 0 are the unity and zero matrices, respectively, and the ’s are the Pauli matrices
Note that a separation of variables in (2.1) yields the Dirac eigenvalue problem where is the spatial part of the wave function and is the total energy of the particle. The free Dirac operator is essentially self-adjoint on and self-adjoint on . Moreover, its spectrum, , is purely absolutely continuous (i.e., its spectral measure is absolutely continuous with respect to the Lebesgue measure) and given by
describes the motion of an electron that moves freely without external force. Let us now introduce an external field given by a matrix-valued function ,
It acts as a multiplication operator in , thus the free Dirac operator with additional field is of the form
The operator is essentially self-adjoint on and self-adjoint on the Sobolev space provided that is Hermitian and satisfies the following estimate (see for e.g., ): the constant is the speed of light, , and . From now on we let be the Coulomb potential , (without ambiguity, is usually dropped from the Coulomb term for simplicity). The spectrum of the Dirac operator with Coulomb potential is given by
where is a discrete sequence of eigenvalues in the “gap” and the remaining part of the spectrum is the continuous part .
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 matrix-valued function and define the operator as We recall that a function is called homogeneous of degree if for any nonzero scalar , . The next theorem is of profound importance for the present work.
Theorem 2.1. Let be Hermitian and satisfy the bound (2.9) above. Further, for any fixed , let be a measurable -homogeneous Hermitian matrix-valued function with entries in , . Then is essentially self-adjoint on and self-adjoint on . Moreover where is a discrete sequence of parameter dependent eigenvalues corresponding to the Dirac eigenvalue problem .Proof. See [9, 10].
We will as a motivating example consider perturbations which are locally periodic and of the form . The entries of are assumed to be -homogeneous. The entries of are assumed to be periodic with respect to a regular lattice in . This can also be phrased that they are defined on the unit torus .
The evolution equation associated with the Dirac operator reads By the Stone theorem, since is self-adjoint on , there exists a unique solution to (2.13) given by where is the strongly continuous unitary operator generated by the infinitesimal operator on , see for example,  or .
In the sequel we will consider a shifted family of Dirac operators denoted by and defined as , where . Also without loss of generality we will in the sequel put . By Theorem 2.1, for any , we then get
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  for the application to general self-adjoint operators. Here we recall some basic facts about G-convergence for self-adjoint operators in .
Let , by we denote the class of self-adjoint operators on a closed linear subspace of such that .
Definition 2.2. Let , and let then we say that G-converges to , denoted in if in , for all , where and refer to strong and weak topologies respectively, and and are the orthogonal projections onto and respectively. Also we say converges to in the strong resolvent sense (SRS) if in .
The following result provides a useful criterion for G-convergence of self-adjoint operators. See  for a proof.
Lemma 2.3. Given , and the orthogonal projection onto . Suppose that for every , converges strongly (resp. weakly) in , then there exists an operator such that (resp. ) in .
From now on we will just use the word “converge” instead of saying “strongly converge,” hence instead of .
2.3. Some Basic Results in Spectral Theory
For more details see [9, 11, 14]. Given a Hilbert space , let be a measurable space for and let be a -algebra on . Let be the set of orthogonal projections on , then is called a spectral measure if it satisfies the following:(i) (this condition is superfluous given the next properties);(ii)Completeness; ;(iii)Countable additivity; if is a finite or a countable set of disjoint elements and , then .
If is spectral measure then , also is modular, that is, . For an increasing sequence of sets , , while if is a decreasing sequence then . Because of the idempotence property of the spectral measure we have , which means that the weak convergence and the strong convergence of a sequence of spectral measures are equivalent.
Let be the finite scalar measure on generated by ,
and be the complex measure
By the above notations .
Let . The spectral measure on the real line corresponding to an operator is denoted by (where the superscript indicates that the spectral measure corresponds to a specific operator )
Clearly is monotonic (nondecreasing), that is, for . Also and . are self-adjoint, idempotent, positive, bounded, right continuous operator (), and discontinuous at each eigenvalue of the spectrum. If is an eigenvalue, then we define to be the point projection onto the eigenspace of . For being in the continuous spectrum .
Now we state the spectral theorem for self-adjoint operators.
Theorem 2.4. For a self-adjoint operator defined on a Hilbert space there exists a unique spectral measure on such that (i);(ii) if ;(iii)If is open and , then .Proof. See for example, .
3. The Main Results
Consider the family of Dirac operators with domain . We will state and prove some useful theorems for operators of the class for , where and 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 for .
Theorem 3.1. Let be a positive and self-adjoint operator on and put . Then for ,(i)B is injective. Moreover, for every and .(ii).
Proof. See Propositions 12.1 and 12.3 in .
The connection between the operator and its G-limit of the class for to their corresponding eigenvalue problems is addressed in the next two theorems. Here we prove the critical case when , where for the proof is analogous and even simpler.
Theorem 3.2. Given a family of operators of the class G-converging to in the strong resolvent sense. Let be the solution of , where is converging to in . If converges to in , then solves the G-limit problem .
Proof. Since G-converges to in the strong resolvent sense
where and are and , respectively. Note that by Theorem 3.1, , so the projections and are unnecessary.
Consider which is equivalent to ; by the definition of we have . Define which is clearly convergent to in ; by the assumptions. Therefore , this is because The convergence to zero follows with help of (3.1) and the boundedness of the inverse operator . Thus, for all Hence for every , which implies ; therefore .
Theorem 3.3. Let be a sequence in which G-converges to in the strong resolvent sense, and let be the solution of the eigenvalue problem . If in , then the limit couple is the solution of the eigenvalue problem .
Proof. The proof is straight forward by assuming (which converges to in ) 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 of operators which converges uniformly to a limit operator nice results can be drawn for the spectrum. Exactly speaking converges to including the isolated eigenvalues. The same conclusion holds if the uniform convergence is replaced by the uniform resolvent convergence, see for example, . In the case of strong convergence (the same for strong resolvent convergence), if the sequence is strongly convergent to , then every is the limit of a sequence where , but not the limit of every such sequence lies in the spectrum of , (see the below example taken from ). For weakly convergent sequences of operators no spectral implications can be extracted. In the present work we frequently write converges to when we mean that the sequence is converging to .
Example 3.4. Let be an operator in defined by where The operator converges to in the strong resolvent sense as for both . One can compute the spectrum for the three operators and obtain for being a simple eigenvalue in and , whereas for the unperturbed limit operator the spectrum consists of just the continuous spectrum, that is, .
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 (and hence the G-convergence in the strong resolvent sense of operators of the class ) implies the convergence of the corresponding eigenvalues.
Theorem 3.5. Let be a family of operators in , , with domain . If G-converges to , then converges in the norm of is the set of bounded linear operators on ) to . Moreover the eigenvalue of converges to the eigenvalue of and the associated eigenvector converges to weakly in , .
Proof. By the definition of supremum norm
Also, by the definition of supremum norm there exists a sequence with such that
It is well known that and are compact self-adjoint operators on . Both are bounded operators, by Theorem 3.1, with compact range of .
Consider now the right-hand side of (3.7). We write this asThe first and the third terms converge to zero by the compactness of and on and the second term converges to zero by the G-convergence of to . Consequently Consider the eigenvalue problems associated to and Since and are compact and self-adjoint operators it is well known that there exists infinite sequences of eigenvalues and , accumulating at the origin, respectively. Define and for all . Consider now the spectral problems associated to and There exists infinite sequences of eigenvalues and , respectively. By the compactness of and the sets and are bounded in , thus the proof is complete by virtue of the following lemma.
Lemma 3.6. Let , , , , and be as in Theorem 3.5, and let , . There is a sequence converging to zero with such that where is a constant independent of , and is the eigenspace of corresponding to .Proof. See Theorem 1.4 and Lemma in  Chapter 3.
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 matrix-valued function is of the form where is (−1)-homogeneous and where the entries of are 1-periodic in , that is,
We also assume that the entries of belong to . It is then well known that in weakly*, where is the unit torus in . It easily also follows from this mean-value property that
in weakly for , 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,
where there exists a discrete set of eigenvalues , and a corresponding set of mutually orthogonal eigenfunctions . We know, by Theorem 2.1, that the eigenvalues (or point spectrum) . We also know that has a continuous spectrum . 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 be a fixed -algebra of subsets of , and let be a measurable space. Consider the spectral measures and of the families of Dirac operators and respectively, each one of these measures maps onto , where is the set of orthogonal projections on . By the spectral theorem By the spectral theorem we can also write since is a multiplication operator.
We recall that , let now
that is, the eigenspace of corresponding to the eigenvalue . Further define the sum of mutual orthogonal eigenspaces
where is a closed subspace of invariant with respect to .
It is clear that for we have
Let us now consider the restriction of to which can be written as
where the spectral measure is the point measure, that is, the orthogonal projection onto . With this construction is a positive and self-adjoint operator on with compact inverse . By Lemma 2.3, see also Proposition 13.4 in , we conclude that there exists a positive and self-adjoint operator such that, up to a subsequence, G-converges to , where has domain where
is a closed subspace of and where
Moreover, by Theorem 3.5, the sequence of eigenvalues associated to the sequence converges to the eigenvalue of of and the corresponding sequence converges to weakly in . The limit shifted Dirac operator restricted to is explicitly given by
This follows by standard arguments in homogenization theory, see for example, .
We continue now to study the asymptotic analysis of the projection to the closed subspace of corresponding to the positive part of the continuous spectrum of .
We denote by the orthogonal complement in to the eigenspace . Thus, is the closed subspace invariant with respect to corresponding to the absolutely continuous spectrum . We now define the two mutually orthogonal subspaces and with
where corresponds to the positive part and corresponds to the negative part , respectively. Next we define the restriction of to which can be written as
where the spectral measure is the continuous spectral measure corresponding to . By construction is a positive and self-adjoint operator on . Therefore by Proposition 13.4 in , there exists a sequence which G-converges to a positive and self-adjoint operator . Moreover, since is not an eigenvalue, the corresponding sequence of spectral measures converges to the spectral measure corresponding to .
Let us consider the evolution equation By the Stone theorem, there exists a unique solution to (3.28) given by
where is the strongly continuous unitary group of transformations generated by the infinitesimal operator on . By the G-convergence of the sequence it follows that the associated sequence of unitary groups of transformations converges to a unitary group of transformations which for every defines the solution to the limit evolution equation
Finally, by considering the operator where is the restriction to , that is, the closed subspace corresponding to the negative part of the continuous spectrum we can repeat all the arguments from the positive part of the continuous spectrum.
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