^{1}

^{1}

^{1}

We consider the linear Dirac operator with a (

In the present work we study the asymptotic behavior of Dirac operators

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 [

We will consider periodic perturbations, that is, we will assume that the potential

and the corresponding eigenvalue problem

The paper is arranged as follows: in Section

Let

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

Let

Here

Note that a separation of variables in (

It acts as a multiplication operator in

The operator

where

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

Let

See [

We will as a motivating example consider perturbations which are locally periodic and of the form

The evolution equation associated with the Dirac operator

In the sequel we will consider a shifted family of Dirac operators denoted by

For more detailed information on G-convergence we refer to, for example, [

Let

Let

The following result provides a useful criterion for G-convergence of self-adjoint operators. See [

Given

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

For more details see [

Completeness;

Countable additivity; if

If

Let

and

By the above notations

Let

Clearly

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

For a self-adjoint operator

If

See for example, [

Consider the family

The following theorem gives a bound for the inverse of operators of the class

Let

B is injective. Moreover, for every

See Propositions 12.1 and 12.3 in [

The connection between the operator and its G-limit of the class

Given a family of operators

Since

Consider

Let

The proof is straight forward by assuming

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

Let

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

Let

By the definition of supremum norm

Consider now the right-hand side of (

Let

See Theorem 1.4 and Lemma

We can now complete the proof of Theorem

Let us now return to the shifted and perturbed Dirac operator

We also assume that the entries of

in

We are now interested in the asymptotic behavior of the operator and the spectrum of the perturbed Dirac operator

where there exists a discrete set of eigenvalues

Let

We recall that

that is, the eigenspace of

where

It is clear that for

Let us now consider the restriction

where the spectral measure

is a closed subspace of

Moreover, by Theorem

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

We denote by

where

where the spectral measure

Let us consider the evolution equation

where

Finally, by considering the operator