Discrete approximations to Dirac operators and norm resolvent convergence

We consider continuous Dirac operators defined on 𝐑^d, d∈{1,2,3}, together with various discrete versions of them. Both forward-backward and symmetric finite differences are used as approximations to partial derivatives. We also allow a bounded, Hölder continuous, and self-adjoint matrix-valued potential, which in the discrete setting is evaluated on the mesh. Our main goal is to investigate whether the proposed discrete models converge in norm resolvent sense to their continuous counterparts, as the mesh size tends to zero and up to a natural embedding of the discrete space into the continuous one. In dimension one we show that forward-backward differences lead to norm resolvent convergence, while in dimension two and three they do not. The same negative result holds in all dimensions when symmetric differences are used. On the other hand, strong resolvent convergence holds in all these cases. Nevertheless, and quite remarkably, a rather simple but non-standard modification to the discrete models, involving the mass term, ensures norm resolvent convergence in general.

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1 Introduction

We study in detail in what sense continuous Dirac operators can be approximated by a family of discrete operators indexed by the mesh size. To investigate spectral properties based on the discrete models, it is essential to know whether we can obtain norm resolvent convergence or only strong resolvent convergence of the discrete models (suitably embedded into the continuum) to the continuous Dirac operators.

In this paper we present a remarkable new phenomenon. In dimensions two and three we cannot obtain norm resolvent convergence of the discrete operators (embedded into the continuum) as the mesh size tends to zero, if we use the natural discretizations based on either symmetric first order differences or a pair of forward-backward first order differences. The models require a simple modification to obtain norm resolvent convergence. In dimension one the discretization using a pair of forward-backward first order differences does lead to norm resolvent convergence, whereas the model based on symmetric first order differences does not.

These results are now described in some detail. To unify the notation we define and . The Hilbert spaces used for the continuous Dirac operators are

For mesh size the corresponding discrete spaces are denoted by

The norm on is given by

Here denotes the Euclidean norm on . We index by ; the dependence is in the subscript of .

To relate the spaces and we introduce embedding operators and discretization operators , constructed from a pair of biorthogonal Riesz sequences, as in [1, section 2]. We describe the construction briefly, with further details and assumptions given in section 2. Let and assume that and are a pair of biorthogonal Riesz sequences in . Define , and , , , . The embedding operator is then defined as

Note that here

is a scalar multiplying a vector

. To construct the discretization operator, let be defined as with replaced by . The discretization operator is then defined as . For , it can be written explicitly as

A similar formula holds for . We have , where is the identity in , and is a projection in onto .

Let be the free Dirac operator in , , and let be an approximation defined on . We compare the operators

acting on . The question of interest is in what sense will converge to as . We now summarize the results obtained. First we briefly define the operators considered.

Let , , denote the Pauli matrices

(1.1)

Let denote the mass. To simplify we do not indicate dependence on the mass in the notation for operators. In dimension the free Dirac operator is given by the operator matrix

on . We consider two discrete approximations based on replacing by finite difference operators. Let denote the identity operator on . We define

Here the forward and backward finite difference operators are defined as

(1.2)

and satisfies . The symmetric difference operator is the self-adjoint operator , i.e.

(1.3)

In dimension the free Dirac operator is defined as

on . As in the case, there are two natural discrete models given by

and

Here and are the corresponding finite differences in the ’th coordinate. It turns out that these two discrete models do not lead to norm resolvent convergence, so we also define two modified versions. Let denote the discrete Laplacian; see (2.4). Then the modified operators are given by

Here is understood to be the operator matrix

The details on the discretizations in dimension can be found in section 5.

Let and be two Hilbert spaces. The space of bounded operators from to is denoted by . If we write . In the following theorem we collect the positive results obtained on norm resolvent convergence in . We use the convention in the statements of results.

Theorem 1.1.

Let be equal to , , or equal to , , or equal to , . Let denote the free Dirac operator in the corresponding dimension. Then the following result holds.

Let be compact. Then there exists such that

(1.4)

for all and .

Theorem 1.1 can be generalized to also include a potential, by following the approach in [1]. Let be bounded and Hölder continuous. Assume is self-adjoint for each . Define the discretization as for . Then we can define self-adjoint operators on and on for all the discrete models. The results in Theorem 1.1 then generalize to and

, with an estimate

, where depends on the Hölder exponent for ; see section 7.

In the next theorem we summarize some negative results with non-convergence in the -operator norm in part (i), and in part (ii) a result using the Sobolev spaces is given. In particular, the estimate (1.5) implies strong resolvent convergence in .

Theorem 1.2.

Let be equal to , , or equal to , . Let denote the free Dirac operator in the corresponding dimension. Then the following results hold.

  1. Let . Then does not converge to in the operator norm on as .

  2. Let be compact. Then there exists such that

    (1.5)

    for all and .

The estimate (1.4) implies results on the spectra of the operators and and their relation, see [1, section 5]. Such results are not obtainable from the strong convergence implied by the estimate (1.5). Thus we are in the remarkable situation that in dimensions we need to modify the natural discretizations in order to obtain spectral information. Furthermore, in dimension to obtain spectral information we must use either the forward-backward discretizations or the modified symmetric discretizations.

Results of the type (1.4) were first obtained by Nakamura and Tadano [4] for on and on for a large class of real potentials , including unbounded . They used special cases of the and as defined here, i.e. the pair of biorthogonal Riesz sequences is replaced by a single orthonormal sequence. Recently their results have been applied to quantum graph Hamiltonians [2]. In [3] the continuum limit is studied for a number of different problems. Here strong resolvent convergence is proved up to the spectrum and scattering results are derived.

In [1] the authors proved results of the type (1.4) for a class of Fourier multipliers  and their discretizations , and obtained results of the type (1.4) for perturbations and with a bounded, real-valued, and Hölder continuous potential. Note that the results in [1] do not directly apply to Dirac operators, since the free Dirac operators do not satisfy an essential symmetry condition [1, Assumption 3.1(4)]. In [5] Schmidt and Umeda proved strong resolvent convergence for Dirac operators in dimension using the discretization . They allow a class of bounded non-self-adjoint potentials and also state corresponding results for dimensions .

The remainder of this paper is organized as follows. Section 2 introduces additional notation and operators used in the paper. Sections 3, 4, and 5 prove Theorem 1.1 and Theorem 1.2(i) in the one-, two-, and three-dimensional cases, respectively. Since some of the arguments are very similar in the different dimensions, we will give the full details in dimension two, and omit parts of the proofs in dimensions one and three that are essentially the same verbatim. Theorem 1.2(ii) is proved in section 6. Finally we show how a potential can be added to our results in section 7.

2 Preliminaries

In this section we collect a number of definitions and results used in the sequel.

2.1 Notation for identity operators

We use the following notation for identity operators on various spaces: on , on , on , on , on , and on . In section 5, in the definitions of the operator matrices for the free Dirac operator and its discretizations, denotes the identity on and denotes the identity on .

2.2 Finite differences

The forward, backward, and symmetric difference operators on are defined in (1.2) and (1.3). Let be the canonical basis in . The forward partial difference operators for mesh size are defined by

(2.1)

and backward partial difference operators by

(2.2)

The symmetric difference operators are given by

(2.3)

Note that and .

The discrete Laplacian acting on is given by

(2.4)

2.3 Fourier transforms

We use Fourier transforms extensively. They are normalized to be unitary. Write

and let be the Fourier transform given by

with adjoint . We suppress their dependence on in the notation, as it will be obvious in which dimension they are used.

Let , , and . The discrete Fourier transform and its adjoint are given by

for .

2.4 Embedding and discretization operators

We describe in some detail how the the embedding and discretization operators in [1, section 2] are adapted to the Dirac case.

Let be a Hilbert space. Let and be two sequences in . They are said to be biorthogonal if

where is Kronecker’s delta.

A sequence is called a Riesz sequence if there exist and such that

for all .

Assumption 2.1.

Let or . Let . Define

Assume that and are biorthogonal Riesz sequences in .

To simplify, we omit the dependence on in the notation for embedding and discretization operators. The embedding operators are defined by

(2.5)

For , and , the notation above means

with an obvious modification in case . As a consequence of the Riesz sequence assumption we get a uniform bound

The operators are defined as above by replacing by in (2.5). Then the discretization operators are defined as . Explicitly, for ,

with an obvious modification for . We have the uniform bound

Biorthogonality implies that

and that is a projection onto in . A further assumption on the functions and is needed.

Assumption 2.2 ([1, Assumption 2.8]).

Let be essentially bounded and satisfy Assumption 2.1. Assume further that there exists such that

and

For examples of and satisfying Assumption 2.2, see [1, subsection 2.1].

2.5 Two lemmas

We often use the following elementary result, where the identity matrix is denoted by

.

Lemma 2.3.

Let be a self-adjoint matrix. Then

(2.6)
(2.7)
Proof.

It suffices to prove (2.6). We use the -identity in to get

The following lemma will be used in the proofs related to the non-convergence results.

Lemma 2.4.

Let or . Assume that is a continuous matrix-valued function. Let denote the operator of multiplication by on . Then

(2.8)
Proof.

It is easy to see that

We need to prove the opposite inequality. Let be such that

We can find with such that . We have for any

Let be arbitrary. By the continuity assumption we can find an open set with , such that

Let with and . Then we have . Thus

Since is arbitrary, the result (2.8) follows. ∎

3 The 1D free Dirac operator

We state and prove results for the 1D Dirac operator. On the one-dimensional free Dirac operator with mass is given by the operator matrix

where denotes the identity operator on .

3.1 The 1D forward-backward difference model

Using (1.2) the forward-backward difference model of is defined as

where denotes the identity operator on . The operators and are given as multipliers in Fourier space by the functions and , respectively, where

(3.1)

and

(3.2)

We define

(3.3)

and

(3.4)

Then

(3.5)
Lemma 3.1.

Assume . Then we have

(3.6)

There exists such that for we have

(3.7)
Proof.

Using Lemma 2.3 together with (3.3) and (3.5) we get

proving (3.6).

To prove (3.7) we use Lemma 2.3, (3.4), and (3.5) to get

There exists such that for we have . For then (3.7) follows. ∎

Lemma 3.2.

There exists such that

for .

Proof.

We have

To estimate the and entries in we use Taylor’s formula:

It follows that the and entries are estimated by . Using Lemma 3.1 the result follows. ∎

Using Lemmas 3.1 and 3.2 we can adapt the arguments in [1] to obtain the following result. We omit the details here, and refer the reader to the proof of Theorem 4.4 where details of the adaptation are given.

Theorem 3.3.

Let be compact. Then there exists such that

for all and .

3.2 The 1D symmetric difference model

The discrete model based on the symmetric difference operator (1.3) is

(3.8)

In Fourier space it is a multiplier with symbol

(3.9)

We have

(3.10)
Lemma 3.4.

There exists such that

(3.11)

for all .

Proof.

We have

Using Lemma 2.3, (3.4), and (3.10) we get