Discrete approximations to Dirac operators and norm resolvent convergence

We consider continuous Dirac operators defined on $\mathbf{R}^d$, $d\in\{1,2,3\}$, 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, H\"older 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.


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.
For mesh size h > 0 the corresponding discrete spaces are denoted by H d h = ℓ 2 (hZ d ) ⊗ C ν(d) , d = 1, 2, 3. The norm on H d h is given by Here | · | denotes the Euclidean norm on C ν(d) . We index u h by k ∈ Z d ; the h dependence is in the subscript of u h .  [2, section 2]. We describe the construction briefly, with further details and assumptions given in section 2. Let ϕ 0 , ψ 0 ∈ L 2 (R d ) and assume that {ϕ 0 ( · −k)} k∈Z d and {ψ 0 ( · −k)} k∈Z d are a pair of biorthogonal Riesz sequences in L 2 (R d ). Define ϕ h,k (x) = ϕ 0 ((x − hk)/h), and ψ h,k (x) = ψ 0 ((x − hk)/h), x ∈ R d , k ∈ Z d , h > 0. The embedding operator J h is then defined as Note that here ϕ h,k (x) is a scalar multiplying a vector u h (k) ∈ C ν(d) . To construct the discretization operator, let J h be defined as J h with ϕ 0 replaced by ψ 0 . The discretization operator is then defined as K h = ( J h ) * . For d = 1, 2, it can be written explicitly as acting on H d . The question of interest is in what sense will J h (H 0,h − zI h ) −1 K h converge to (H 0 − zI) −1 as h → 0. We now summarize the results obtained. First we briefly define the operators considered. Let σ j , j = 1, 2, 3, denote the Pauli matrices Let m ≥ 0 denote the mass. To simplify we do not indicate dependence on the mass in the notation for operators. In dimension d = 1 the free Dirac operator is given by the operator matrix on H 1 . We consider two discrete approximations based on replacing −i d dx by finite difference operators. Let I h denote the identity operator on ℓ 2 (hZ). We define Here the forward and backward finite difference operators are defined as In dimension d = 2 the free Dirac operator is defined as on H 2 . As in the d = 1 case, there are two natural discrete models given by Here D ± h;j and D s h;j are the corresponding finite differences in the j'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 −∆ h denote the discrete Laplacian; see (2.4). Then the modified operators are given by The details on the discretizations in dimension d = 3 can be found in section 5. Let K 1 and K 2 be two Hilbert spaces. The space of bounded operators from K 1 to K 2 is denoted by B(K 1 , K 2 ). If K 1 = K 2 = K we write B(K) = B(K, K). In the following theorem we collect the positive results obtained on norm resolvent convergence in B(H d ). We use the convention (−0, 0) = ∅ in the statements of results.
for all z ∈ K and h ∈ (0, 1]. Theorem 1.1 can be generalized to also include a potential, by following the approach in [2]. Let V : R d → B(C ν(d) ) be bounded and Hölder continuous. Assume V (x) is selfadjoint for each x ∈ R d . Define the discretization as V h (k) = V (hk) for k ∈ Z d . Then we can define self-adjoint operators H = H 0 + V on H d and H h = H 0,h + V h on H d h for all the discrete models. The results in Theorem 1.1 then generalize to H and H h , with an estimate Ch θ ′ , where 0 < θ ′ < 1 depends on the Hölder exponent for V ; see section 7.
In the next theorem we summarize some negative results with non-convergence in the B(H d )-operator norm in part (i), and in part (ii) a result using the Sobolev spaces for all z ∈ K and h ∈ (0, 1]. The estimate (1.4) implies results on the spectra of the operators H 0,h and H 0 and their relation, see [2, 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 d = 2, 3 we need to modify the natural discretizations in order to obtain spectral information. Furthermore, in dimension d = 1 to obtain spectral information we must use either the forward-backward discretizations or the modified symmetric discretizations. Moreover, this is relevant for resolving the unwanted fermion doubling phenomenon that is present in some discretizations of Dirac operators [1].
Results of the type (1.4) were first obtained by Nakamura and Tadano [5] for H = −∆ + V on L 2 (R d ) and H h = −∆ h + V h on ℓ 2 (hZ d ) for a large class of real potentials V , including unbounded V . They used special cases of the J h and K h 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 [3]. In [4] 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 [2] the authors proved results of the type (1.4) for a class of Fourier multipliers H 0 and their discretizations H 0,h , and obtained results of the type (1.4) for perturbations H = H 0 + V and H h = H 0,h + V h with a bounded, real-valued, and Hölder continuous potential. Note that the results in [2] do not directly apply to Dirac operators, since the free Dirac operators do not satisfy an essential symmetry condition [2, Assumption 3.1(4)]. In [7] Schmidt and Umeda proved strong resolvent convergence for Dirac operators in dimension d = 2 using the discretization H fb 0,h . They allow a class of bounded non-selfadjoint potentials and also state corresponding results for dimensions d = 1, 3.
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 V can be added to our results in section 7.

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

Notation for identity operators
We use the following notation for identity operators on various spaces: , 1 on C 2 , and 1 on C 4 . In section 5, in the definitions of the operator matrices for the free Dirac operator and its discretizations, 1 denotes the identity on L 2 (R 3 ) ⊗ C 2 and 1 h denotes the identity on ℓ 2 (hZ 3 ) ⊗ C 2 .

Finite differences
The forward, backward, and symmetric difference operators on H 1 h are defined in (1.2) and (1.3). Let {e 1 , e 2 , e 3 } be the canonical basis in Z 3 . The forward partial difference operators for mesh size h are defined by and backward partial difference operators by The symmetric difference operators are given by Note that (D + h;j ) * = D − h;j and (D s h;j ) * = D s h;j . The discrete Laplacian acting on ℓ 2 (hZ d ) is given by with adjoint F * : H d → H d . We suppress their dependence on d in the notation, as it will be obvious in which dimension they are used.

Embedding and discretization operators
We describe in some detail how the the embedding and discretization operators in [2, section 2] are adapted to the Dirac case.
Let K be a Hilbert space. Let {u k } k∈Z d and {v k } k∈Z d be two sequences in K. They are said to be biorthogonal if Assume that {ϕ 1,k } k∈Z d and {ψ 1,k } k∈Z d are biorthogonal Riesz sequences in L 2 (R d ).
To simplify, we omit the dependence on d in the notation for embedding and discretization operators. The embedding operators J h : , the notation above means with an obvious modification in case d = 3. As a consequence of the Riesz sequence assumption we get a uniform bound The operators J h are defined as above by replacing ϕ h,k by ψ h,k in (2.5). Then the discretization operators are defined as K h = ( J h ) * . Explicitly, for d = 1, 2, with an obvious modification for d = 3. We have the uniform bound

Biorthogonality implies that
A further assumption on the functions ϕ 0 and ψ 0 is needed.
. Let ϕ 0 , ψ 0 ∈ L 2 (R d ) be essentially bounded and satisfy Assumption 2.1. Assume further that there exists c 0 > 0 such that

Two lemmas
We often use the following elementary result, where the identity matrix is denoted by I.
Proof. It suffices to prove (2.6). We use the C * -identity in B(C n ) to get The following lemma will be used in the proofs related to the non-convergence results; see e.g. [6, Theorem XIII.83]. Then

The 1D free Dirac operator
We state and prove results for the 1D Dirac operator. On H 1 the one-dimensional free Dirac operator with mass m ≥ 0 is given by the operator matrix where I denotes the identity operator on L 2 (R).

The 1D forward-backward difference model
where I h denotes the identity operator on ℓ 2 (hZ). The operators H 0 and H fb 0,h are given as multipliers in Fourier space by the functions G 0 and G fb 0,h , respectively, where and and 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 c > 0 such that for |θ| ≤ 3π To estimate the 12 and 21 entries in G fb 0,h (ξ) − G 0 (ξ) we use Taylor's formula: It follows that the 12 and 21 entries are estimated by Ch|ξ| 2 . Using Lemma 3.1 the result follows.
Using Lemmas 3.1 and 3.2 we can adapt the arguments in [2] 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.

The 1D symmetric difference model
The discrete model based on the symmetric difference operator (1.3) is In Fourier space it is a multiplier with symbol for all h ∈ (0, 1].
Using Lemmas 2.4 and 3.4 together with Theorem 3.3 and properties of J h and K h , we get the following result.
We can introduce a modified operator H s 0,h given by where −∆ h is the 1D discrete Laplacian; see (2.4). We obtain norm resolvent convergence for the modified symmetric difference model, similar to the results in dimensions two and three; see Theorems 4.4 and 5.1. The proof is omitted as it is nearly identical to the proof of Theorem 4.4.

The 2D free Dirac operator
In two dimensions the free Dirac operator on H 2 with mass m ≥ 0 is given by where the Pauli matrices are given in (1.1). In H 2 it is a Fourier multiplier with symbol The corresponding discrete Dirac operator can be obtained by replacing the derivatives in (4.1) by finite differences.

The 2D symmetric difference model
We first consider the model obtained by using the symmetric difference operators; see (2.3) for the definition.
In H 2 h it acts as a Fourier multiplier with symbol The 2D discrete Laplacian is defined in (2.4). We introduce the modified symmetric difference model as h the operator H s 0,h acts as a Fourier multiplier with symbol Related to the symbols G 0 , G s 0,h , and G s 0,h , we define We have G 0 (ξ) 2 = g 0 (ξ)1, G s 0,h (ξ) 2 = g s 0,h (ξ)1, and G s 0,h (ξ) 2 =g s 0,h (ξ)1.
To prove (4.12) we first use Lemma 2.3 and (4.10) to get Then note that there exists c > 0 such that Combining these estimates we get The estimate (4.12) follows.
Lemma 4.2. There exists C > 0 such that The 11 entry in G s 0,h (ξ) − G 0 (ξ) is estimated using |sin(θ)| ≤ |θ|. We get This result implies the estimates that are used to estimate the 12 and 21 entries in G s 0,h (ξ) − G 0 (ξ). Combining these results with the estimates from Lemma 4.1 we get Lemma 3.3] in a form adapted to the Dirac operators and outline its proof.
Proof. We assume d = 2. It suffices to consider K = {i}, since (H 0 − iI)(H 0 − zI) −1 is bounded uniformly in norm for z ∈ K. Let u ∈ S(R 2 ) ⊗ C 2 , the Schwartz space. Going through the computations in [2, section 2] using that ϕ 0 and ψ 0 are scalar functions, we get the result Here G 0 is given by (4.2). If hξ ∈ [− π 2 , π 2 ] 2 then the j = 0 term is the only non-zero term in the sum. Using [2, Lemma 2.7] we conclude that this term and the last term cancel. For hξ / ∈ [− π 2 , π 2 ] 2 we use Lemma 4.1 to get (G 0 (ξ) − i1) −1 B(C 2 ) ≤ Ch, 0 < h ≤ 1. Since ϕ 0 and ψ 0 are assumed essentially bounded, we conclude that the j = 0 term in the sum and the last term are bounded by Ch u H 2 .
Due to the support assumptions on ϕ 0 and ψ 0 , only the terms in the sum with |j| ≤ 1 contribute. Assume |j| = 1 and hξ ∈ supp( ϕ 0 ) ∩ supp( ψ 0 (· + 2πj)). Then for some c 0 > 0 we have |ξ + 2π h j| ≥ c 0 h , which by Lemma 4.1 implies Again using the boundedness of ϕ 0 and ψ 0 we conclude that Squaring and integrating the result gives an estimate of the form Ch u H 2 . By density, adding up the finite number of terms corresponding to |j| ≤ 1 gives the final result.
We have now established the estimates necessary to repeat the arguments from [2]. Using the embedding operators J h and discretization operators K h defined in section 2, we state the result and then show in some detail how the arguments in [2] are adapted to the Dirac case.
Proof. We start by proving the result for K = {i}. We have The last term is estimated using Lemma 4.3.
To estimate the remaining terms we go to Fourier space. We have Let u ∈ S(R 2 ) ⊗ C 2 . We now use a modified version of the computation leading to [2, equation (2.11)]. For the first term we get For the second term we get We need to rewrite (4.16). First we note that Next we can rewrite part of (4.16) as follows, since ψ 0 is a scalar-valued function: We now insert (4.17) and the rewritten (4.16) into (4.15) to get Due to the support conditions on ϕ 0 and ψ 0 in Assumption 2.2, only terms with |j| ≤ 1 contribute. First consider j = 0. We have assumed supp( ϕ 0 ), supp( ψ 0 ) ⊆ [− 3π 2 , 3π 2 ] 2 . Using Lemma 4.2 and Assumption 2.2 we get From the supports of ϕ 0 and ψ 0 we have Assume hξ ∈ M, then Lemma 4.1 implies Since we have a finite number of j with |j| ≤ 1 and since u is in a dense set, the estimate in Theorem 4.4 follows in the K = {i} case. For the general case we use the estimates This is the crucial estimate used above. Further details are omitted.
Next we show that, without modification to the symmetric difference model, the norm convergence stated in the theorem fails.
for all h ∈ (0, 1]. Proof. Using the notation from (4.5) we have From the same reasoning as in the proof of Lemma 3.4, we obtain Here g s 0,h (ξ) is given by (4.8),g s 0,h (ξ) by (4.9), and f h (ξ) by (4.6). Take hξ 1 = π and hξ 2 = π, and insert them in the last term in (4.19). We get The result (

The 2D forward-backward difference model
We now consider the model for the discrete Dirac operator obtained by using the forward and backward difference operators; see (2.1) and (2.2) for definitions. The discretized operator is given by In H 2 h it is a Fourier multiplier with the symbol We also consider the modified model, where the modification is the same as in the symmetric case, i.e.
The corresponding Fourier multiplier is where f h (ξ) is given by (4.6). We recall the expression Straightforward computations show that We now prove the analogue of (4.12) for G fb 0,h (ξ).
Lemma 4.8. There exists C > 0 such that Proof. We have The 11 and 22 entries in G fb 0,h (ξ) − G 0 (ξ) are estimated by Ch|ξ| 2 ; see (4.13). To estimate the 12 and 21 entries we use Taylor's formula: (4.25) It follows that the 12 and 21 entries also are estimated by Ch|ξ| 2 . Using Lemmas 4.1 and 4.7 the result follows.
We can now state the analogue of Theorem 4.4. The proof is omitted, since it is almost identical to the proof of Theorem 4.4; indeed the key ingredients are the estimates in Lemmas 4.7 and 4.8, that correspond to the results from Lemmas 4.1 and 4.2 with the modified symmetric difference model.
The negative result in Theorem 4.6 for the symmetric model holds also in the forwardbackward case.
Proof. As in the proof of Lemma 4.5 we get f h (ξ) (1 + g fb 0,h (ξ)) 1/2 (1 +g fb 0,h (ξ)) 1/2 . Using (4.6), (4.21), and (4.22) we get It follows that we have a lower bound  This result implies that the strong convergence result in [7] cannot be improved to a norm convergence result, without modifying the discretization.
For U, W ∈ C 3 there is the following identity related to the Pauli matrices, where the "dot" does not involve complex conjugation: The Dirac matrices α = (α 1 , α 2 , α 3 ) and β satisfy We can choose The free Dirac operator with mass m ≥ 0 in H 3 is given by where 1 in the context of (5.2) denotes the identity operator on L 2 (R 3 ) ⊗ C 2 . In Fourier space H 3 it is a multiplier with symbol then G 0 (ξ) 2 = g 0 (ξ)1. (5.5) As in dimension two there are two natural discretizations of (5.2), using either the pair of forward-backward partial difference operators or the symmetric partial difference operators.

The 3D symmetric difference model
The symmetric partial difference operators are defined in (2.3). We use the notation for the discrete symmetric gradient. The symmetric discretization of the 3D Dirac operator is defined as where 1 h is the identity operator on ℓ 2 (hZ 3 ) ⊗ C 2 . In Fourier space this operator is a multiplier with symbol . As in the two-dimensional case we also define a modified discretization. Let −∆ h denote the 3D discrete Laplacian; see (2.4). Let −∆ h 1 denote the 2 × 2 diagonal operator matrix with the discrete Laplacian on the diagonal elements. Then define Its symbol is

The 3D forward-backward difference model
Using the definitions (2.1) and (2.2) we introduce the discrete forward and backward gradients as . The forward-backward difference model is then given by The symbols of D ± h;j in Fourier space are ± 1 ih (e ±ihξ j − 1), j = 1, 2, 3.
The arguments for norm resolvent convergence of the 3D modified forward-backward difference model do not follow as straightforwardly as in the symmetric difference case, since in particular G fb 0,h (ξ) 2 is not a diagonal matrix. A computation reveals that .

(5.11)
We proceed to show the required estimates related to G fb 0,h (ξ) in detail.
Lemma 5.4. There exists C > 0 such that We estimate the entries in G fb 0,h (ξ) − G 0 (ξ) as in the proof of Lemma 4.8. Thus the entries are estimated by Ch|ξ| 2 . Using Lemma 5.3 the result follows.
Using Lemmas 5.3 and 5.4 we can adapt the arguments in [2] 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.
As in dimension two, the unmodified forward-backward difference model does not lead to norm resolvent convergence.

Sobolev space estimates and strong convergence
In sections 3-5 we have shown that J h (H 0,h − zI h ) −1 K h converges in the B(H d )-operator norm to (H 0 − zI) −1 for several choices of discrete model H 0,h , and we have also shown that in other cases this norm convergence does not hold. This section is dedicated to the cases where J h (H 0,h − zI h ) −1 K h does not converge to (H 0 − zI) −1 in the B(H d )-operator norm, and instead we will prove that convergence holds in the B(H 1 (R d ) ⊗ C ν(d) , H d )operator norm. These latter results obviously imply strong convergence. In particular, we recover the result in [7] for d = 2 with the discretization H fb 0,h .

The 1D model
The 1D symmetric model H s 0,h is defined in (3.8) and its symbol G s 0,h (ξ) in (3.9). The symbol for the continuous Dirac operator G 0 (ξ) is defined in (3.1). Lemma 6.1. There exists C > 0 such that . Proof. Note that Lemma 2.3 and (3.10) imply the estimate (G s 0,h (ξ) − i1) −1 B(C 2 ) ≤ (1 + m 2 ) −1 and that this estimate cannot be improved for hξ ∈ [− 3π 2 , 3π 2 ]. We have Proof. The result follows if we prove the estimate The proof is very similar to the proof of Theorem 4.4. In the arguments one replaces F * u by F * (H 0 − iI) −1 u and uses Lemma 6.1. Further details are omitted.

Perturbed Dirac operators
In this section we state results on perturbed Dirac operators and their discretizations, with respect to norm resolvent convergence. We use the following condition on the perturbation.
We require another assumption on ψ 0 in addition to Assumption 2.2. We emphasize that concrete examples of ψ 0 satisfying these assumptions are given in [2, subsection 2.1].
Assumption 7.2. Assume there exists τ > d such that Define a discretization of V by V h (k) = V (hk), k ∈ Z d .
Proof. The proof in [2] can be directly adapted to the current framework. We omit the details. Note that ψ 0 (x) is a scalar, such that we have ψ 0 (x)V (x)f (x) = V (x)ψ 0 (x)f (x), f ∈ H d .
We can then state our main result on the perturbed Dirac operators, which follows from Lemma 7.3 and a direct adaptation of the proof of [ where H 0 is the free Dirac operator in the relevant dimension. Assume V ≡ 0 and let θ ′ be given by (7.2). Then the following result holds.
Let K ⊂ C \ R be compact. Then there exist C > 0 and h 0 > 0 such that for all z ∈ K and h ∈ (0, h 0 ].