Discrete approximations to Dirac operators and norm resolvent convergence

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 $ν (1) = ν (2) = 2$ and $ν (3) = 4$ . The Hilbert spaces used for the continuous Dirac operators are

H^{d} = L^{2} (R^{d}) \otimes C^{ν (d)}, d = 1, 2, 3.

For mesh size $h > 0$ the corresponding discrete spaces are denoted by

H_{h}^{d} = ℓ^{2} (h Z^{d}) \otimes C^{ν (d)}, d = 1, 2, 3.

The norm on $H_{h}^{d}$ is given by

∥ u_{h} ∥_{H_{h}^{d}}^{2} = h^{d} \sum k \in Z^{d} | u_{h} (k) |^{2}, u_{h} \in H_{h}^{d} .

Here $| \cdot |$ denotes the Euclidean norm on $C^{ν (d)}$ . We index $u_{h}$ by $k \in Z^{d}$ ; the $h$ dependence is in the subscript of $u_{h}$ .

To relate the spaces $H_{h}^{d}$ and $H^{d}$ we introduce embedding operators $J_{h} : H_{h}^{d} \to H^{d}$ and discretization operators $K_{h} : H^{d} \to H_{h}^{d}$ , 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 $φ_{0}, ψ_{0} \in L^{2} (R^{d})$ and assume that ${φ_{0} (\cdot - k)}_{k \in Z^{d}}$ and ${ψ_{0} (\cdot - k)}_{k \in Z^{d}}$ are a pair of biorthogonal Riesz sequences in $L^{2} (R^{d})$ . Define $φ_{h, k} (x) = φ_{0} ((x - h k) / h)$ , and $ψ_{h, k} (x) = ψ_{0} ((x - h k) / h)$ , $x \in R^{d}$ , $k \in Z^{d}$ , $h > 0$ . The embedding operator $J_{h}$ is then defined as

(J_{h} u_{h}) (x) = \sum k \in Z^{d} φ_{h, k} (x) u_{h} (k) .

Note that here $φ_{h, k} (x)$

is a scalar multiplying a vector

u_{h} (k) \in 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

(K_{h} f) (k) = \frac{1}{h^{d}} [\begin{matrix} ⟨ ψ_{h, k}, f^{1} ⟩ ⟨ ψ_{h, k}, f^{2} ⟩ \end{matrix}], f = [\begin{matrix} f^{1} f^{2} \end{matrix}] \in H^{d} .

A similar formula holds for $d = 3$ . We have $K_{h} J_{h} = I_{h}$ , where $I_{h}$ is the identity in $H_{h}^{d}$ , and $J_{h} K_{h}$ is a projection in $H^{d}$ onto $J_{h} H_{h}^{d}$ .

Let $H_{0}$ be the free Dirac operator in $H^{d}$ , $d = 1, 2, 3$ , and let $H_{0, h}$ be an approximation defined on $H_{h}^{d}$ . We compare the operators

J_{h} (H_{0, h} - z I_{h})^{- 1} K_{h} and (H_{0} - z I)^{- 1}

acting on $H^{d}$ . The question of interest is in what sense will $J_{h} (H_{0, h} - z I_{h})^{- 1} K_{h}$ converge to $(H_{0} - z I)^{- 1}$ as $h \to 0$ . We now summarize the results obtained. First we briefly define the operators considered.

Let $σ_{j}$ , $j = 1, 2, 3$ , denote the Pauli matrices

σ_{1} = [\begin{matrix} 0 & 1 1 & 0 \end{matrix}], σ_{2} = [\begin{matrix} 0 & - i i & 0 \end{matrix}], σ_{3} = [\begin{matrix} 1 & 0 0 & - 1 \end{matrix}] .

(1.1)

Let $m \geq 0$ 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

H_{0} = - i \frac{d}{d x} σ_{1} + m σ_{3}

on $H^{1}$ . We consider two discrete approximations based on replacing $- i \frac{d}{d x}$ by finite difference operators. Let $I_{h}$ denote the identity operator on $ℓ^{2} (h Z)$ . We define

H_{0, h}^{f b} = [\begin{matrix} m I_{h} & D_{h}^{-} D_{h}^{+} & - m I_{h} \end{matrix}] and H_{0, h}^{s} = [\begin{matrix} m I_{h} & D_{h}^{s} D_{h}^{s} & - m I_{h} \end{matrix}] .

Here the forward and backward finite difference operators are defined as

(D_{h}^{+} u_{h}) (k) = \frac{1}{i h} (u_{n} (k + 1) - u_{h} (k)), (D_{h}^{-} u_{h}) (k) = \frac{1}{i h} (u_{n} (k) - u_{h} (k - 1)),

(1.2)

and satisfies $(D_{h}^{+})^{*} = D_{h}^{-}$ . The symmetric difference operator is the self-adjoint operator $D_{h}^{s} = \frac{1}{2} (D_{h}^{+} + D_{h}^{-})$ , i.e.

(D_{h}^{s} u_{h}) (k) = \frac{1}{2 i h} (u_{h} (k + 1) - u_{h} (k - 1)) .

(1.3)

In dimension $d = 2$ the free Dirac operator is defined as

H_{0} = - i \frac{\partial}{\partial x_{1}} σ_{1} - i \frac{\partial}{} \partial x_{2} σ_{2} + m σ_{3}

on $H^{2}$ . As in the $d = 1$ case, there are two natural discrete models given by

H_{0, h}^{f b} = [\begin{matrix} m I_{h} & D_{h; 1}^{-} - i D_{h; 2}^{-} D_{h; 1}^{+} + i D_{h; 2}^{+} & - m I_{h} \end{matrix}]

and

H_{0, h}^{s} = [\begin{matrix} m I_{h} & D_{h; 1}^{s} - i D_{h; 2}^{s} D_{h; 1}^{s} + i D_{h; 2}^{s} & - m I_{h} \end{matrix}] .

Here $D_{h; j}^{\pm}$ and $D_{h; j}^{s}$ 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 $- Δ_{h}$ denote the discrete Laplacian; see (2.4). Then the modified operators are given by

{˜ H}_{0, h}^{f b} = H_{0, h}^{f b} - h Δ_{h} σ_{3} % and {˜ H}_{0, h}^{s} = H_{0, h}^{s} - h Δ_{h} σ_{3} .

Here $- h Δ_{h} σ_{3}$ is understood to be the operator matrix

[\begin{matrix} - h Δ_{h} & 0 0 & h Δ_{h} \end{matrix}] .

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 $(- 0, 0) = \emptyset$ in the statements of results.

Theorem 1.1.

Let $H_{0, h}$ be equal to $H_{0, h}^{f b}$ , $d = 1$ , or equal to ${˜ H}_{0, h}^{f b}$ , $d = 2, 3$ , or equal to ${˜ H}_{0, h}^{s}$ , $d = 1, 2, 3$ . Let $H_{0}$ denote the free Dirac operator in the corresponding dimension. Then the following result holds.

Let $K \subset (C ∖ R) \cup (- m, m)$ be compact. Then there exists $C > 0$ such that

∥ J_{h} (H_{0, h} - z I_{h})^{- 1} K_{h} - (H_{0} - z I)^{- 1} ∥_{B (H^{d})} \leq C h

(1.4)

for all $z \in K$ and $h \in (0, 1]$ .

Theorem 1.1 can be generalized to also include a potential, by following the approach in [1]. Let $V : R^{d} \to B (C^{ν (d)})$ be bounded and Hölder continuous. Assume $V (x)$ is self-adjoint for each $x \in R^{d}$ . Define the discretization as $V_{h} (k) = V (h k)$ for $k \in Z^{d}$ . Then we can define self-adjoint operators $H = H_{0} + V$ on $H^{d}$ and $H_{h} = H_{0, h} + V_{h}$ on $H_{h}^{d}$ for all the discrete models. The results in Theorem 1.1 then generalize to $H$ and $H_{h}$

, with an estimate

C h^{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 $H^{1} (R^{d}) \otimes C^{ν (d)}$ is given. In particular, the estimate (1.5) implies strong resolvent convergence in $B (H^{d})$ .

Theorem 1.2.

Let $H_{0, h}$ be equal to $H_{0, h}^{f b}$ , $d = 2, 3$ , or equal to $H_{0, h}^{s}$ , $d = 1, 2, 3$ . Let $H_{0}$ denote the free Dirac operator in the corresponding dimension. Then the following results hold.

Let $z \in (C ∖ R) \cup (- m, m)$ . Then $J_{h} (H_{0, h} - z I_{h})^{- 1} K_{h}$ does not converge to $(H_{0} - z I)^{- 1}$ in the operator norm on $B (H^{d})$ as $h \to 0$ .
Let $K \subset (C ∖ R) \cup (- m, m)$ be compact. Then there exists $C > 0$ such that

$∥ J_{h} (H_{0, h} - z I_{h})^{- 1} K_{h} - (H_{0} - z I)^{- 1} ∥_{B (H^{1} (R^{d}) \otimes C^{ν (d)}, H^{d})} \leq C h$ (1.5)

for all $z \in K$ and $h \in (0, 1]$ .

The estimate (1.4) implies results on the spectra of the operators $H_{0, h}$ and $H_{0}$ 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 $d = 2, 3$ 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.

Results of the type (1.4) were first obtained by Nakamura and Tadano [4] for $H = - Δ + V$ on $L^{2} (R^{d})$ and $H_{h} = - Δ_{h} + V_{h}$ on $ℓ^{2} (h Z^{d})$ 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 [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 $H_{0}$ 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 [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 $d = 2$ using the discretization $H_{0, h}^{f b}$ . They allow a class of bounded non-self-adjoint 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.

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: $I$ on $H^{d}$ , $I_{h}$ on $H_{h}^{d}$ , $I$ on $L^{2} (R^{d})$ , $I_{h}$ on $ℓ^{2} (h Z^{d})$ , $1$ 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 $L^{2} (R^{3}) \otimes C^{2}$ and $1_{h}$ denotes the identity on $ℓ^{2} (h Z^{3}) \otimes C^{2}$ .

2.2 Finite differences

The forward, backward, and symmetric difference operators on $H_{h}^{1}$ 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

(2.1)

and backward partial difference operators by

(2.2)

The symmetric difference operators are given by

(D_{h; j}^{s} u_{h}) (k) = \frac{1}{2 i h} (u_{h} (k + e_{j}) - u_{h} (k - e_{j})), j = 1, 2, 3.

(2.3)

Note that $(D_{h; j}^{+})^{*} = D_{h; j}^{-}$ and $(D_{h; j}^{s})^{*} = D_{h; j}^{s}$ .

The discrete Laplacian acting on $ℓ^{2} (h Z^{d})$ is given by

(- Δ_{h} v_{h}) (k) = \frac{1}{h^{2}} d \sum j = 1 (2 v_{h} (k) - v_{h} (k + e_{j}) - v_{h} (k - e_{j})) .

(2.4)

2.3 Fourier transforms

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

{ˆ H}^{d} = L^{2} (R^{d}) \otimes C^{ν (d)}

and let

F : H^{d} \to {ˆ H}^{d}

be the Fourier transform given by

(F f) (ξ) = \frac{1}{(2 π)^{d / 2}} \int_{R^{d}} e^{- i x \cdot ξ} f (x) d x, ξ \in R^{d},

with adjoint $F^{*} : {ˆ H}^{d} \to H^{d}$ . We suppress their dependence on $d$ in the notation, as it will be obvious in which dimension they are used.

Let $T_{h}^{d} = [- \frac{π}{h}, \frac{π}{h}]^{d}$ , $d = 1, 2, 3$ , and ${ˆ H}_{h}^{d} = L^{2} (T_{h}^{d}) \otimes C^{ν (d)}$ . The discrete Fourier transform $F_{h} : H_{h}^{d} \to {ˆ H}_{h}^{d}$ and its adjoint $F_{h}^{*} : {ˆ H}_{h}^{d} \to H_{h}^{d}$ are given by

	$(F_{h} u_{h}) (ξ)$	$= \frac{h^{d}}{(2 π)^{d / 2}} \sum k \in Z^{d} u_{h} (k) e^{- i h k \cdot ξ}, ξ \in T_{h}^{d},$
	$(F_{h}^{*} g) (k)$	$= \frac{1}{(2 π)^{d / 2}} \int_{T_{h}^{d}} e^{i h k \cdot ξ} g (ξ) d ξ, k \in Z^{d},$

for $d = 1, 2, 3$ .

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 $K$ be a Hilbert space. Let ${u_{k}}_{k \in Z^{d}}$ and ${v_{k}}_{k \in Z^{d}}$ be two sequences in $K$ . They are said to be biorthogonal if

⟨ u_{k}, v_{n} ⟩ = δ_{k, n}, k, n \in Z^{d},

where $δ_{k, n}$ is Kronecker’s delta.

A sequence ${u_{k}}_{k \in Z^{d}}$ is called a Riesz sequence if there exist $A > 0$ and $B > 0$ such that

A \sum k \in Z^{d} | c_{k} |^{2} \leq ∥ \sum k \in Z^{d} c_{k} u_{k} ∥^{2} \leq B \sum k \in Z^{d} | c_{k} |^{2}

for all ${c_{k}}_{k \in Z^{d}} \in ℓ^{2} (Z^{d})$ .

Assumption 2.1.

Let $d = 1, 2,$ or $3$ . Let $φ_{0}, ψ_{0} \in L^{2} (R^{d})$ . Define

φ_{h, k} (x) = φ_{0} ((x - h k) / h), ψ_{h, k} (x) = ψ_{0} ((x - h k) / h), h > 0, k \in Z^{d} .

Assume that ${φ_{1, k}}_{k \in Z^{d}}$ and ${ψ_{1, k}}_{k \in Z^{d}}$ 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 $J_{h} : H_{h}^{d} \to H^{d}$ are defined by

J_{h} u_{h} = \sum k \in Z^{d} φ_{h, k} u_{h} (k), u_{h} \in H_{h}^{d} .

(2.5)

For $d = 1, 2$ , and $u_{h} (k) = [\begin{matrix} u_{h}^{1} (k) u_{h}^{2} (k) \end{matrix}]$ , the notation above means

φ_{h, k} u_{h} (k) = [\begin{matrix} u_{h}^{1} (k) φ_{h, k} u_{h}^{2} (k) φ_{h, k} \end{matrix}],

with an obvious modification in case $d = 3$ . As a consequence of the Riesz sequence assumption we get a uniform bound

sup h > 0 ∥ J_{h} ∥_{B (H_{h}^{d}, H^{d})} < \infty .

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 $d = 1, 2$ ,

(K_{h} f) (k) = \frac{1}{h^{d}} [\begin{matrix} ⟨ ψ_{h, k}, f^{1} ⟩ ⟨ ψ_{h, k}, f^{2} ⟩ \end{matrix}], k \in Z^{d},

with an obvious modification for $d = 3$ . We have the uniform bound

sup h > 0 ∥ K_{h} ∥_{B (H^{d}, H_{h}^{d})} < \infty .

Biorthogonality implies that

K_{h} J_{h} = I_{h}

and that $J_{h} K_{h}$ is a projection onto $J_{h} H_{h}^{d}$ in $H^{d}$ . A further assumption on the functions $φ_{0}$ and $ψ_{0}$ is needed.

Assumption 2.2 ([1, Assumption 2.8]).

Let ${ˆ φ}_{0}, {ˆ ψ}_{0} \in L^{2} (R^{d})$ be essentially bounded and satisfy Assumption 2.1. Assume further that there exists $c_{0} > 0$ such that

supp ({ˆ φ}_{0}) \subseteq [- \frac{3 π}{2}, \frac{3 π}{2}]^{d} and | {ˆ φ}_{0} (ξ) | \geq c_{0}, ξ \in [- \frac{π}{2}, \frac{π}{2}]^{d},

and

supp ({ˆ ψ}_{0}) \subseteq [- \frac{3 π}{2}, \frac{3 π}{} 2]^{d} and | {ˆ ψ}_{0} (ξ) | \geq c_{0}, ξ \in [- \frac{π}{2}, \frac{π}{2}]^{d} .

For examples of $φ_{0}$ and $ψ_{0}$ 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

I

Lemma 2.3.

Let $G \in B (C^{n})$ be a self-adjoint $n \times n$ matrix. Then

	$∥ G - i I ∥_{B (C^{n})}$	$= ∥ G^{2} + I ∥_{B (C^{n})}^{1 / 2},$		(2.6)
	$∥ (G - i I)^{- 1} ∥_{B (C^{n})}$	$= ∥ (G^{2} + I)^{- 1} ∥_{B (C^{n})}^{1 / 2} .$		(2.7)

Proof.

It suffices to prove (2.6). We use the $C^{*}$ -identity in $B (C^{n})$ to get

∥G−iI∥2B(Cn)=∥(G+iI)(G−iI)∥B(Cn)=∥G2+I∥B(Cn).\qed

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

Lemma 2.4.

Let $d = 1, 2,$ or $3$ . Assume that $M_{h} : T_{h}^{d} \to B (C^{ν (d)})$ is a continuous matrix-valued function. Let $T_{M_{h}}$ denote the operator of multiplication by $M_{h}$ on ${ˆ H}_{h}^{d}$ . Then

∥ T_{M_{h}} ∥_{B ({ˆ H}_{h}^{d})} = max ξ \in T_{h}^{d} ∥ M_{h} (ξ) ∥_{B (C^{ν (d)})} .

(2.8)

Proof.

It is easy to see that

∥ T_{M_{h}} ∥_{B ({ˆ H}_{h}^{d})} \leq max ξ \in T_{h}^{d} ∥ M_{h} (ξ) ∥_{B (C^{ν (d)})} .

We need to prove the opposite inequality. Let $ξ_{0} \in T_{h}^{d}$ be such that

∥ M_{h} (ξ_{0}) ∥_{B (C^{ν (d)})} = max ξ \in T_{h}^{d} ∥ M_{h} (ξ) ∥_{B (C^{ν (d)})} .

We can find $w_{0} \in C^{ν (d)}$ with $| w_{0} | = 1$ such that $∥ M_{h} (ξ_{0}) ∥_{B (C^{ν (d)})} = | M_{h} (ξ_{0}) w_{0} |$ . We have for any $ξ \in T_{h}^{d}$

| M_{h} (ξ_{0}) w_{0} | \geq ∥ M_{h} (ξ) ∥_{B (C^{ν (d)})} \geq | M_{h} (ξ) w_{0} | .

Let $ε > 0$ be arbitrary. By the continuity assumption we can find an open set $O \subset T_{h}^{d}$ with $ξ_{0} \in O$ , such that

| M_{h} (ξ) w_{0} |^{2} \geq | M_{h} (ξ_{0}) w_{0} |^{2} - ε, ξ \in O .

Let $ϕ : T_{h}^{d} \to C$ with $supp ϕ \subset O$ and $\int_{T_{h}^{d}} | ϕ (ξ) |^{2} d ξ = 1$ . Then we have $∥ ϕ (\cdot) w_{0} ∥_{{ˆ H}_{h}^{d}} = 1$ . Thus

∥ T_{M_{h}} ∥_{B ({ˆ H}_{h}^{d})}^{2} \geq \int_{T_{h}^{d}} | M_{h} (ξ) ϕ (ξ) w_{0} |^{2} d ξ \geq | M_{h} (ξ_{0}) w_{0} |^{2} - ε = max ξ \in T_{h}^{d} ∥ M_{h} (ξ) ∥_{B (C^{ν (d)})}^{2} - ε .

Since $ε > 0$ is arbitrary, the result (2.8) follows. ∎

3 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 $m \geq 0$ is given by the operator matrix

where $I$ denotes the identity operator on $L^{2} (R)$ .

3.1 The 1D forward-backward difference model

Using (1.2) the forward-backward difference model of $H_{0}$ is defined as

H_{0, h}^{f b} = [\begin{matrix} m I_{h} & D_{h}^{-} D_{h}^{+} & - m I_{h} \end{matrix}],

where $I_{h}$ denotes the identity operator on $ℓ^{2} (h Z)$ . The operators $H_{0}$ and $H_{0, h}^{f b}$ are given as multipliers in Fourier space by the functions $G_{0}$ and $G_{0, h}^{f b}$ , respectively, where

G_{0} (ξ) = [\begin{matrix} m & ξ ξ & - m \end{matrix}]

(3.1)

and

G_{0, h}^{f b} (ξ) = ⎡ ⎣ \begin{matrix} m & - \frac{1}{i h} (e^{- i h ξ} - 1) \frac{1}{i h} (e^{i h ξ} - 1) & - m \end{matrix} ⎤ ⎦ .

(3.2)

We define

g_{0} (ξ) = m^{2} + ξ^{2},

(3.3)

and

g_{0, h}^{f b} (ξ) = m^{2} + \frac{4}{h^{2}} {sin}^{2} (\frac{h}{2} ξ) .

(3.4)

Then

G_{0} (ξ)^{2} = g_{0} (ξ) 1 and G_{0, h}^{f b} (ξ)^{2} = g_{0, h}^{f b} (ξ) 1 .

(3.5)

Lemma 3.1.

Assume $ξ \neq 0$ . Then we have

∥ (G_{0} (ξ) - i 1)^{- 1} ∥_{B (C^{2})} \leq \frac{1}{| ξ |} .

(3.6)

There exists $C > 0$ such that for $h ξ \in [- \frac{3 π}{2}, \frac{3 π}{2}]$ we have

∥ (G_{0, h}^{f b} (ξ) - i 1)^{- 1} ∥_{B (C^{2})} \leq \frac{C}{| ξ |} .

(3.7)

Proof.

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

∥ (G_{0} (ξ) - i 1)^{- 1} ∥_{B (C^{2})}

= ∥ (G_{0} (ξ)^{2} + 1)^{- 1} ∥_{B (C^{2})}^{1 / 2} = \frac{1}{(1 + m^{2} + ξ^{2})^{1 / 2}} \leq \frac{1}{| ξ |},

proving (3.6).

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

	$∥ (G_{0}^{f b} (ξ) - i 1)^{- 1} ∥_{B (C^{2})}$	$= ∥ (G_{0}^{f b} (ξ)^{2} + 1)^{- 1} ∥_{B (C^{2})}^{1 / 2} = (1 + m^{2} + \frac{4}{h^{2}} {sin}^{2} (\frac{h}{2} ξ))^{- 1 / 2}$
		$\leq (\frac{4}{h^{2}} {sin}^{2} (\frac{h}{2} ξ))^{- 1 / 2} .$

There exists $c > 0$ such that for $| θ | \leq \frac{3 π}{4}$ we have $| sin (θ) | \geq c | θ |$ . For $h ξ \in [- \frac{3 π}{2}, \frac{3 π}{2}]$ then (3.7) follows. ∎

Lemma 3.2.

There exists $C > 0$ such that

∥ (G_{0} (ξ) - i 1)^{- 1} - (G_{0, h}^{f b} (ξ) - i 1)^{- 1} ∥_{B (C^{2})} \leq C h

for $h ξ \in [- \frac{3 π}{2}, \frac{3 π}{2}]$ .

Proof.

We have

	$(G_{0} (ξ) - i 1)^{- 1} -$	$(G_{0, h}^{f b} (ξ) - i 1)^{- 1}$
		$= (G_{0} (ξ) - i 1)^{- 1} (G_{0, h}^{f b} (ξ) - G_{0} (ξ)) (G_{0, h}^{f b} (ξ) - i 1)^{- 1} .$

To estimate the $12$ and $21$ entries in $G_{0, h}^{f b} (ξ) - G_{0} (ξ)$ we use Taylor’s formula:

e^{i h ξ} = 1 + i h ξ + (i h ξ)^{2} \int_{0}^{1} e^{i h t ξ} (1 - t) d t .

It follows that the $12$ and $21$ entries are estimated by $C h | ξ |^{2}$ . 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 $K \subset (C ∖ R) \cup (- m, m)$ be compact. Then there exists $C > 0$ such that

∥ J_{h} (H_{0, h}^{f b} - z I_{h})^{- 1} K_{h} - (H_{0} - z I)^{- 1} ∥_{B (H^{1})} \leq C h

for all $z \in K$ and $h \in (0, 1]$ .

3.2 The 1D symmetric difference model

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

H_{0, h}^{s} = [\begin{matrix} m I_{h} & D_{h}^{s} D_{h}^{s} & - m I_{h} \end{matrix}] .

(3.8)

In Fourier space it is a multiplier with symbol

G_{0, h}^{s} (ξ) = ⎡ ⎣ \begin{matrix} m & \frac{1}{h} sin (h ξ) \frac{1}{h} sin (h ξ) & - m \end{matrix} ⎤ ⎦ .

(3.9)

We have

G_{0, h}^{s} (ξ)^{2} = g_{0, h}^{s} (ξ) 1 where g_{0, h}^{s} (ξ) = m^{2} + \frac{1}{h^{2}} {sin}^{2} (h ξ) .

(3.10)

Lemma 3.4.

There exists $c > 0$ such that

max ξ \in T_{h}^{1} ∥ (G_{0, h}^{f b} (ξ) - i 1)^{- 1} - (G_{0, h}^{s} (ξ) - i 1)^{- 1} ∥_{B (C^{2})} \geq c

(3.11)

for all $h \in (0, 1]$ .

Proof.

We have

	$(G_{0, h}^{f b} (ξ) - i 1) [(G_{0, h}^{s} (ξ) - i 1)^{- 1}$	$- (G_{0, h}^{f b} (ξ) - i 1)^{- 1}] (G_{0, h}^{s} (ξ) - i 1)$
		$= G_{0, h}^{f b} (ξ) - G_{0, h}^{s} (ξ) = \frac{1}{h} (1 - cos (h ξ)) σ_{2} .$

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

∥ (G_{0, h}^{f b} (ξ) - i 1)^{- 1} - (G_{0, h}^{s} (ξ) - i 1)^{- 1} ∥_{B (C^{2})} \geq \frac{1 - cos (h ξ)}{h (1 + g^{f b}}

Discrete approximations to Dirac operators and norm resolvent convergence

Authors

Continuum limits for discrete Dirac operators on 2D square lattices

Derivatives of partial eigendecomposition of a real symmetric matrix for degenerate cases

Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure

On some nonlocal models with radially symmetric interaction domains and the effect of domain truncation

Algebraic function based Banach space valued ordinary and fractional neural network approximations

On the numerical approximations of the periodic Schrödinger equation

Analysis of a Backward Euler-type Scheme for Maxwell's Equations in a Havriliak-Negami Dispersive Medium

1 Introduction

Theorem 1.1.

Theorem 1.2.

2 Preliminaries

2.1 Notation for identity operators

2.2 Finite differences

2.3 Fourier transforms

2.4 Embedding and discretization operators

Assumption 2.1.

Assumption 2.2 ([1, Assumption 2.8]).

2.5 Two lemmas

Lemma 2.3.

Proof.

Lemma 2.4.

Proof.

3 The 1D free Dirac operator

3.1 The 1D forward-backward difference model

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Theorem 3.3.

3.2 The 1D symmetric difference model

Lemma 3.4.

Proof.

Discrete approximations to Dirac operators and norm resolvent convergence

Authors

Related Research

Continuum limits for discrete Dirac operators on 2D square lattices

Derivatives of partial eigendecomposition of a real symmetric matrix for degenerate cases

Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure

On some nonlocal models with radially symmetric interaction domains and the effect of domain truncation

Algebraic function based Banach space valued ordinary and fractional neural network approximations

On the numerical approximations of the periodic Schrödinger equation

Analysis of a Backward Euler-type Scheme for Maxwell's Equations in a Havriliak-Negami Dispersive Medium

This week in AI

1 Introduction

Theorem 1.1.

Theorem 1.2.

2 Preliminaries

2.1 Notation for identity operators

2.2 Finite differences

2.3 Fourier transforms

2.4 Embedding and discretization operators

Assumption 2.1.

Assumption 2.2 ([1, Assumption 2.8]).

2.5 Two lemmas

Lemma 2.3.

Proof.

Lemma 2.4.

Proof.

3 The 1D free Dirac operator

3.1 The 1D forward-backward difference model

Lemma 3.1.

Proof.

Lemma 3.2.

Proof.

Theorem 3.3.

3.2 The 1D symmetric difference model

Lemma 3.4.

Proof.