Spectral Theory and Continuum Norm Resolvent Convergence of a One-Sided Discrete Dirac-Type Operator

Introduction and preliminaries

A classical one-dimensional Dirac operator for free electrons is the differential operator $D\colon(L^2(\mathbb{R};\mathbb{C}))^2 \supset(W^{1,2}(\mathbb{R};\mathbb{C}))^2 \to(L^2(\mathbb{R};\mathbb{C}))^2$, given by

where $c>0$ denotes the speed of light and $m\ge0$ denotes the particle rest mass (see [7]). Defined on the direct sum of Sobolev spaces $\mathcal{D}(D)=(W^{1,2}(\mathbb{R};\mathbb{C}))^2$, the operator $D$ is an unbounded, closed, self-adjoint operator with purely absolutely continuous spectrum

One can define a natural discrete analogue of the operator (1) on a one-dimensional lattice and obtain a discrete one-dimensional Dirac-type operator $D_d:(\ell^2(\mathbb{Z};\mathbb{C}))^2\to(\ell^2(\mathbb{Z};\mathbb{C}))^2$, given by the formula

where $S_1:\ell^2(\mathbb{Z};\mathbb{C})\to\ell^2(\mathbb{Z};\mathbb{C})$ is the right shift operator, that is, for any sequence $x=(x_n)_{n\in\mathbb{Z}}\in\ell^2(\mathbb{Z};\mathbb{C})$, we have $(S_1x)_n=x_{n-1}$. Clearly, $S_1$ is a unitary operator on $\ell^2(\mathbb{Z};\mathbb{C})$ whose adjoint (and inverse) is the left shift $(S_1^*x)_n=x_{n+1}$.

In the mathematical physics literature, lattice discretizations of the Dirac operator (1) have been extensively studied in the context of spectral theory, supersymmetric quantum mechanics, and continuum limits [1, 2, 6]. When replacing the continuum derivative $\frac{d}{dx}$ on a lattice, one encounters a fundamental structural dichotomy:

1. If operator $\frac{d}{dx}$ is replaced by the skew-adjoint symmetric central difference $\frac{1}{2}(S_1^*-S_1)$, the resulting operator $D_{\mathrm{sym}}$ is self-adjoint, but its Fourier symbol $i\sin k$ vanishes both at the origin $k=0$ and at the boundary of the Brillouin zone $k=\pm\pi$. This produces a spurious second Dirac cone at $k=\pm\pi$, known as fermion doubling [4, 6]. Because of the high-frequency doubler mode, $D_{\mathrm{sym}}$ fails to converge to $D$ in the full-space norm resolvent topology as the lattice spacing $h\to0^+$ unless an auxiliary projection cutoff onto low frequencies is imposed [1].

2. If one instead places $-ic(I-S_1)$ in the lower-left off-diagonal entry and its adjoint $ic(I-S_1^*)$ in the upper-right entry, one obtains a self-adjoint supersymmetric discretization [2], which implicitly modifies the Pauli matrix structure of (1) by introducing a Wilson-type mass term proportional to $\sigma_2(2I-S_1-S_1^*)$.

In contrast, the operator $D_d$ defined in (2) preserves the exact Pauli matrix structure of (1) while employing the one-sided difference operator $\nabla_-:=I-S_1$ in both off-diagonal components. As a consequence, $D_d$ exhibits a distinctive combination of operator-theoretic features:

(1) For $m>0$, $D_d$ is neither self-adjoint nor normal, yet its square $D_d^2$ is a normal scalar-block operator (that is, $D_d$ is quadratically normal).

(2) The Fourier symbols of the one-sided difference $I-S_1$ are $1-e^{-ik}=2ie^{-ik/2}\sin(k/2)$, which vanishes on the torus $\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$ only at $k=0. Thus $D_d$ is completely free of fermion doubling at $k=\pm\pi$, at the cost of its spectrum becoming a two-component curve in the complex plane $\mathbb{C}$.

(3) When scaled by a lattice spacing $h>0$ and embedded isometrically into the continuum Hilbert space $(L^2(\mathbb{R};\mathbb{C}))^2$ via Whittaker–Shannon interpolation, the absence of fermion doubling provides global quadratic coercivity across the entire Brillouin zone $[-\pi/h,\pi/h]$. This allows us to prove unconditional $O(h)$ norm resolvent convergence of $D_{d,h}$ to the continuous self-adjoint Dirac operator $D$ on the full Hilbert space.

The paper is organized as follows. In Section 2, we analyze the basic operator-theoretic properties of $D_d$, including its Cartesian decomposition, self-commutator, quadratic normality, exact operator norm, and exact spectral radius. In Section 3, we determine the exact complex spectrum of $D_d$ and classify its geometric regimes. In Section 4, we construct the Whittaker–Shannon isometric embedding and prove $O(h)$ norm resolvent convergence in the continuum limit $h\to0^+$.

Notation. Throughout the paper, we denote by $\mathcal{H}:=(\ell^2(\mathbb{Z};\mathbb{C}))^2\cong\ell^2(\mathbb{Z};\mathbb{C}^2)$ the discrete Hilbert space equipped with the standard inner product

and by $\mathcal{B}(\mathcal{H})$ the $\mathrm{C}^*$-algebra of bounded linear operators on $\mathcal{H}$. We write the standard Pauli matrices as

In block-operator form on $\mathcal{H}=\ell^2(\mathbb{Z};\mathbb{C})\oplus\ell^2(\mathbb{Z};\mathbb{C})$, the operator $D_d$ in (1.2) acts as

Basic operator-theoretic properties of $D_d$

We begin by computing the adjoint and the Cartesian decomposition of $D_d$ into its real and imaginary self-adjoint parts. Recall that the standard one-dimensional discrete Laplacian on $\ell^2(\mathbb{Z};\mathbb{C})$ is the negative semidefinite self-adjoint operator $\Delta_{\mathrm{disc}}:=S_1+S_1^\dagger-2I$, so that

Proof. Since operator $S_1$ is unitary on the space $\ell^2(\mathbb{Z};\mathbb{C})$, it clearly follows that $\lVert I-S_1\rVert_{\mathcal{B}(\ell^2(\mathbb{Z};\mathbb{C}))}\le\lVert I\rVert_{\mathcal{B}(\ell^2(\mathbb{Z};\mathbb{C}))}+\lVert S_1\rVert_{\mathcal{B}(\ell^2(\mathbb{Z};\mathbb{C}))}=2$. So $I-S_1\in\mathcal{B}(\ell^2(\mathbb{Z};\mathbb{C}))$, and hence $D_d\in\mathcal{B}(\mathcal{H})$. Using the self-adjointness of the Pauli matrices $\sigma_1,\sigma_3$ and $(-ic)^\dagger=ic$, we obtain

which coincides with (5). Adding and subtracting $D_d$ and $D_d^\dagger$ yields

and

Since $2I-S_1-S_1^\dagger=-\Delta_{\mathrm{disc}}\ne0$ and $c>0$, we have $\operatorname{Im}(D_d)\ne0$, hence $D_d\ne D_d^\dagger$. $\square$

Remark 2.2. Identity (6) shows that the Hermitian part $\operatorname{Re}(D_d)$ is precisely the naive symmetric central-difference discrete Dirac operator $D_{\mathrm{sym}}$, while (7) reveals that the skew-Hermitian part $i\operatorname{Im}(D_d)$ acts as a dissipative term proportional to the discrete Laplacian $-\Delta_{\mathrm{disc}}$ coupled to $\sigma_1$.

We now examine the normality of $D_d$ and its square $D_d^2$.

Proof. Using the Cartesian decomposition $D_d=\operatorname{Re}(D_d)+i\operatorname{Im}(D_d)$ from Lemma 2.1, we have $[D_d,D_d^\dagger]=-2i[\operatorname{Re}(D_d),\operatorname{Im}(D_d)]$. Because $S_1$ and $S_1^\dagger$ commute on $\ell^2(\mathbb{Z};\mathbb{C})$, the difference operators $\frac{1}{2}(S_1^\dagger-S_1)$ and $-\Delta_{\mathrm{disc}}=2I-S_1-S_1^\dagger$ commute with each other and with the identity $I$. Since $[\sigma_1,\sigma_1]=0$ and $[\sigma_3,\sigma_1]=2i\sigma_2$, we compute

Multiplying by $-2i$ establishes equation (8). Since $\sigma_2$ is a unitary operator on $\mathbb{C}^2$ and we have $\|-\Delta_{\mathrm{disc}}\|_{\mathcal{B}(\ell^2(\mathbb{Z};\mathbb{C}))}=4$, (see [3], section 3.1) it follows that the norm $\|[D_d,D_d^\dagger]\|_{\mathcal{B}(\mathcal{H})}=2mc^3\|-\Delta_{\mathrm{disc}}\|_{\mathcal{B}(\ell^2(\mathbb{Z};\mathbb{C}))}=8mc^3$, which vanishes if and only if $m=0$. □

Despite the failure of normality for $m>0$, the anticommutation of the Pauli matrices $\sigma_1$ and $\sigma_3$ endows $D_d$ with quadratic normality.

(iii) (Charge conjugation) Let $K:\mathcal{H}\to\mathcal{H}$ denote componentwise complex conjugation, $(Ku)_n=\overline{u_n}$. Then the antiunitary involution $C:=\sigma_1K$ anticommutes with $D_d$:

Proof. (i) Since $\sigma_1^2=\sigma_3^2=\sigma_0$, $\sigma_1\sigma_3+\sigma_3\sigma_1=0$, and $I-S_1$ commutes with $mc^2I$, squaring (1.3) yields

Since $S_1$ is unitary, $I-S_1$ is normal, and any polynomial in a normal operator is normal; hence $D_d^2$ is normal.

(ii) For any $x\in\ell^2(\mathbb{Z};\mathbb{C})$ and $n\in\mathbb{Z}$, we have $(PS_1Px)_n=(S_1Px)_{-n}=(Px)_{-n-1}=x_{n+1}=(S_1^\dagger x)_n$. Since $\sigma_3\sigma_1\sigma_3=-\sigma_1$ and $\sigma_3^3=\sigma_3$, conjugating $D_d$ by $J=\sigma_3(P\oplus P)$ gives

(iii) Since $S_1$ and $mc^2I$ have real matrix entries in the canonical basis of $\ell^2(\mathbb{Z};\mathbb{C})$, they commute with $K$. Moreover, $K(-ic\sigma_1)K^{-1}=ic\sigma_1=ic\sigma_1$. Because $\sigma_1^3=\sigma_1$ and $\sigma_1\sigma_3\sigma_1=-\sigma_3$, we obtain $CD_dC^{-1}=-D_d$. □

We conclude this section by computing the exact operator norm $\|D_d\|_{\mathcal{B}(\mathcal{H})}$ and the exact spectral radius $r(D_d)$, demonstrating a strict norm–spectral-radius gap for all $m,c>0$.

Proof. Multiplying (2.2) and (1.3) and using (2.1), we compute the positive self-adjoint operator $D_d^\dagger D_d$:

Let $\mathcal{F}: \ell^2(\mathbb{Z};\mathbb{C}) \to L^2([-\pi,\pi];\mathbb{C})$ be the unitary discrete Fourier transform defined by $(\mathcal{F}x)(k) = \frac{1}{\sqrt{2\pi}}\sum_{n\in\mathbb{Z}}x_ne^{-ikn}$ for $k\in[-\pi,\pi]$. Under $\mathcal{F}$, the shift $S_1$ is unitarily equivalent to multiplication by $e^{-ik}$, and $-\Delta_{\mathrm{disc}}=2I-S_1-S_1^\dagger$ is unitarily equivalent to multiplication by $2-2\cos k=4\sin^2(k/2)$. Consequently, $\widehat{D}_d^\dagger\widehat{D}_d$ is unitarily equivalent to multiplication on $(L^2([-\pi,\pi];\mathbb{C}))^2$ by the $2\times2$ Hermitian matrix symbols

Since $\sigma_2$ has eigenvalues $\pm1$, and $\sigma_0$ and $\sigma_2$ commute, the eigenvalues of $\widehat{D}_d(k)^\dagger\widehat{D}_d(k)$ for each $k\in[-\pi,\pi]$ are

Because $m\geq0$, $c>0$, and $\sin^2(k/2)\in[0,1]$ attains its maximum value $1$ at $k=\pm\pi$, we obtain

which proves (12).

To establish (13), we use Gelfand’s spectral radius formula together with Theorem 2.4(i). Since $D_d^2$ is a normal operator, $\|D_d^2\|_{\mathcal{B}(\mathcal{H})}=r(D_d^2)=r(D_d)^2$. Under the Fourier transform $\mathcal{F}\oplus\mathcal{F}$, $D_d^2$ is unitarily equivalent to multiplication by $(m^2c^4-c^2(1-e^{-ik})^2)\sigma_0$. Writing $1-e^{-ik}=2ie^{-ik/2}\sin(k/2)$ and setting $M:=mc^2$ and $s:=\sin^2(k/2)\in[0,1]$, we have $\cos k=1-2s$ and therefore

Thus $r(D_d)=\max_{s\in[0,1]}P(s)^{1/4}$. Differentiating with respect to $s\in[0,1]$ gives

If $0\leq M\leq\frac{2}{\sqrt{3}}c$, then $M^2+4(c^2-M^2)s\geq M^2(1-4s)+4c^2s\geq0$ on $[0,1]$, so $P$ is non-decreasing on $[0,1]$ and attains its maximum at $s=1$, where $P(1)=(4c^2-M^2)^2$; hence $r(D_d)=\sqrt{4c^2-M^2}$. If $M>\frac{2}{\sqrt{3}}c$, then $c^2-M^2<0$ and $P$ is strictly concave on $[0,1]$ with interior maximum at $s_*=\frac{M^2}{4(M^2-c^2)}\in(0,1)$. Evaluating $P(s_*)$ yields

so $r(D_d)=P(s_*)^{1/4}=M^{3/2}(M^2-c^2)^{-1/4}$, proving (13). Finally, for all $M,c>0$ and $s\in[0,1]$, the triangle inequality gives us $\lvert M^2+4c^2se^{-ik}\rvert\leq M^2+4c^2s\leq M^2+4c^2$ (where equality cannot hold simultaneously with $s=1$ and $e^{-ik}=1$), yielding (14).

Fourier–Laurent symbol and exact complex spectrum

By means of the discrete Fourier transform $U := \mathcal{F} \oplus\mathcal{F} : \mathcal{H} \to\left(L^2([-\pi,\pi];\mathbb{C})\right)^2$, using techniques similar as in [7], the translation-invariant operator $D_d$ is unitarily equivalent to the multiplication operator $(U D_d U^{-1}\widehat{u})(k) = \widehat{D}_d(k)\widehat{u}(k)$, where the $2 \times2$ matrix-valued symbol $\widehat{D}_d : [-\pi,\pi] \to M_2(\mathbb{C})$ is given by

Proof. For any $\lambda\in\mathbb{C}$ and $k \in[-\pi,\pi]$, the characteristic polynomial of $\widehat{D}_d(k)$ is

Since $k \mapsto\widehat{D}_d(k)$ is continuous and $2\pi$-periodic on $[-\pi,\pi]$, the multiplication operator $U D_d U^{-1}$ is invertible on $\left(L^2([-\pi,\pi];\mathbb{C})\right)^2$ if and only if $\det(\widehat{D}_d(k)-\lambda\sigma_0) \ne0$ for all $k \in[-\pi,\pi]$, which proves (17).

To see that $\sigma_p(D_d) = \varnothing$, suppose $(\widehat{D}_d(k)-\lambda\sigma_0)\widehat{u}(k) = 0$ for Lebesgue-almost every $k \in[-\pi,\pi]$. Then $\det(\widehat{D}_d(k)-\lambda\sigma_0) = 0$ on the support of $\widehat{u}$. However, the Laurent polynomial $z \mapsto\lambda^2-m^2c^4+c^2(1-z^{-1})^2$ is non-constant for $c>0$ and therefore has at most two roots on the unit circle $|z|=1$. Thus $\det(\widehat{D}_d(k)-\lambda\sigma_0)=0$ holds for at most two values of $k\in[-\pi,\pi)$ (a set of Lebesgue measure zero), forcing $\widehat{u}=0$ in $(L^2([-\pi,\pi];\mathbb{C}))^2$ and $\sigma_p(D_d)=\varnothing$. By Theorem 2.4(ii), $D_d^{\dagger}$ is unitarily equivalent to $D_d$ via $J$, so $\sigma_p(D_d^{\dagger})=\varnothing$ as well. Since $\sigma_r(D_d)\subset\sigma_p(D_d^{\dagger})=\varnothing$, we conclude that $\sigma(D_d)=\sigma_c(D_d)$.

Finally, writing $\mathcal{N}(k):=m^2c^4-c^2(1-e^{-ik})^2$, we observe that $\mathcal{N}(-k)=\overline{\mathcal{N}(k)}$, so $\sigma(D_d)$ is symmetric across both axes. Moreover, we have $\operatorname{Im}\mathcal{N}(k)=-2c^2\sin k(1-\cos k)$ vanishes on $[-\pi,\pi]$ if and only if $k=0$ (where $\mathcal{N}(0)=m^2c^4$) or $k=\pm\pi$ (where $\mathcal{N}(\pm\pi)=m^2c^4-4c^2$). Evaluating $\pm\sqrt{\mathcal{N}(k)}$ at $k=0$ and $k=\pm\pi$ according to the sign of $m^2c^4-4c^2$ immediately yields regimes (i)–(iii).

The qualitative geometric structure of the complex spectrum $\sigma(D_d)$ across its three mass regimes is illustrated in Figures 1, 2, and 3 below.

The numerical routines and plotting scripts used to generate these spectral curves are openly available in the GitHub repository [5].

Continuum limit and norm resolvent convergence

We now turn to the central question of how the bounded, non-normal discrete operator $D_d$ approximates the unbounded, self-adjoint continuous Dirac operator $D$ in (1) when the lattice spacing tends to zero.

Lattice scaling and Whittaker–Shannon isometric embedding

For each $h>0$, consider the scaled lattice Hilbert space $\mathcal{H}_h := \left(\ell^2(h\mathbb{Z};\mathbb{C})\right)^2$ endowed with the inner product and norm

and define the scaled one-sided discrete Dirac-type operator $D_{d,h}:\mathcal{H}_h\to\mathcal{H}_h$ by replacing $I-S_1$ in (2) with the backward difference quotient $\frac{1}{h}(I-S_1)$:

Let $B_h:=[-\pi/h,\pi/h]$ denote the scaled Brillouin zone. The scaled discrete Fourier transform $\mathcal{F}_h:\ell^2(h\mathbb{Z};\mathbb{C})\to L^2(B_h;\mathbb{C})$, defined by

is unitary with inverse $(\mathcal{F}_h^{-1}\phi)_n=\frac{1}{\sqrt{2\pi}}\int_{-\pi/h}^{\pi/h}\phi(p)e^{\mathrm{i}pnh}\,dp$. Likewise, on the Hilbert space $\mathcal{H}_{\mathrm{cont}}:=\left(L^2(\mathbb{R};\mathbb{C})\right)^2$, the continuous Fourier–Plancherel transform $\mathcal{F}_{\mathrm{cont}}:L^2(\mathbb{R};\mathbb{C})\to L^2(\mathbb{R};\mathbb{C})$, defined on intersection $L^1(\mathbb{R};\mathbb{C})\cap L^2(\mathbb{R};\mathbb{C})$ by $(\mathcal{F}_{\mathrm{cont}}\psi)(p)=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\psi(x)e^{-\mathrm{i}px}\,dx$, is unitary.

$(\operatorname{sinc}(t) := \frac{\sin(\pi t)}{\pi t}$ for $t \ne0$ and $\operatorname{sinc}(0) := 1)$:

Because $\mathcal{F}_h$, $\mathcal{F}_{\mathrm{cont}}$, and $\iota_h$ are isometries, $T_h:\mathcal{H}_h\to\mathcal{H}_{\mathrm{cont}}$ is an isometry ($T_h^\dagger T_h=I_{\mathcal{H}_h}$), while $P_{B_h}:=T_hT_h^\dagger$ is the orthogonal projection of $\mathcal{H}_{\mathrm{cont}}$ onto the subspace of $\mathbb{C}^2$-valued functions, band-limited to $B_h$.

Global symbol coercivity and uniform resolvent bounds

Under the Fourier transforms $\mathcal{F}_h\oplus\mathcal{F}_h$ and $\mathcal{F}_{\mathrm{cont}}\oplus\mathcal{F}_{\mathrm{cont}}$, the discrete operator $D_{d,h}$ and the continuous operator $D$ act as multiplication by the $2\times2$ matrix symbols

where $\xi_h:B_h\to\mathbb{C}$ is the scaled one-sided difference symbol

The key analytical advantage of the one-sided discretization (4.1) over the symmetric central-difference discretization is that $|\xi_h(p)|$ is globally coercive on the entire Brillouin zone $B_h=[-\pi/h,\pi/h]$.

Proof. For $p\in[-\pi/h,\pi/h]$, the angle $\theta:=ph/2$ belongs to $[-\pi/2,\pi/2]$. Applying Jordan’s inequality $\frac{2}{\pi}|\theta|\leq|\sin\theta|\leq|\theta|$ to $|\xi_h(p)|=\frac{2}{h}|\sin(ph/2)|$ immediately yields (4.6).

Next, by Taylor’s formula with integral remainder applied to $t\mapsto e^{-ipt}$ on $[0,h]$, we have

Dividing by $h$ and taking absolute values gives

which proves (4.7). Finally, since $|\xi_h(p)+ip| \leq|\xi_h(p)|+|p| \leq2|p|$ by (4.6), multiplying (4.7) by $|\xi_h(p)+ip|$ gives (4.8).

We next establish that for any fixed spectral parameter $z \in\mathbb{C} \setminus\sigma(D)$, the determinant of $\widetilde{D}_{d,h}(p)-z\sigma_0$ satisfies a uniform quadratic lower bound across $B_h$ for all sufficiently small $h>0$. Let us set

so that $\det(\widetilde{D}_{d,h}(p)-z\sigma_0)=z^2-N_h(p)$ and $\det(\widehat{D}(p)-z\sigma_0)=z^2-E(p)^2$.

Therefore,

which completes the proof of (28).

To deduce inequalities (29), observe that $(\widehat{D}_{d,h}(p)-z\sigma_0)(\widehat{D}_{d,h}(p)+z\sigma_0)=(\mathcal{N}_h(p)-z^2)\sigma_0$. Since $\|\sigma_1\|_{\mathcal{B}(\mathbb{C}^2)}=\|\sigma_3\|_{\mathcal{B}(\mathbb{C}^2)}=1$ and $|\xi_h(p)|\le|p|$, the triangle and Cauchy–Schwarz inequalities give

Dividing by $|\mathcal{N}_h(p)-z^2|\ge\kappa(z)(1+p^2)$ yields the first bound in (29), and taking $h\to0^+$ at fixed $p$ yields the second bound.

Optimal $O(h)$ norm resolvent convergence

We are now ready to prove our main continuum limit theorem.

Proof. Conjugating the operator difference $T_h(D_{d,h}-zI)^{-1}T_h^\dagger-(D-zI)^{-1}$ by the unitary continuous Fourier transform $\mathcal{F}_{\mathrm{cont}}\oplus\mathcal{F}_{\mathrm{cont}}$ on $\mathcal{H}_{\mathrm{cont}}$ and using Definition 4.1, we obtain a multiplication operator on $(L^2(\mathbb{R};\mathbb{C}))^2$ whose $2\times2$ matrix multiplier splits orthogonally across $B_h=[-\pi/h,\pi/h]$ and $\mathbb{R}\setminus B_h$:

To bound the first supremum over $p\in B_h$, we apply the second resolvent identity:

By (4.3), (4.4), and Lemma 4.2, we have

Combining this estimate with the resolvent bounds (29) from Lemma 4.3 yields, for all $p \in B_h$,

Notice that the quadratic growth $p^2$ of the Taylor remainder is completely absorbed by the product $(1+p^2)^{-1}$ of the two resolvent decay factors across the entire Brillouin zone $B_h$.

To bound the second supremum over the high-frequency tail $p \in\mathbb{R} \setminus B_h$, we note that $|p|>\pi/h$ implies $\sqrt{1+p^2}>|p|>\pi/h$. Applying the second inequality in (29) gives

Substituting (33) and (34) into (32) establishes (30) with $C(z)$ given by (31).

As a direct consequence, we obtain the precise asymptotic flattening of the complex spectral curve $\sigma(D_{d,h})$ onto the real absolutely continuous spectrum $\sigma(D)=(-\infty,-mc^2]\cup[mc^2,\infty)$ as $h\to0^+$.

Proof. (i) For fixed $p\in\mathbb{R}$ and $h\to0^+$, the Taylor expansion $\xi_h(p)=ip+\frac{1}{2}p^2h+O(h^2)$ implies

Taking the square root via $\sqrt{1+w}=1+\frac{1}{2}w+O(w^{2})$ yields (35).

(ii) Consider $K \subset\mathbb{C}$ to be a compact set and choose $R>0$ such that $K \subset\{z \in\mathbb{C}: |z| \le R\}$. If $z \in\sigma(D_{d,h}) \cap K$, then $z=\lambda_{\pm,h}(p)$ for some $p \in B_h$. By the lower bound in (23),

so $|p|\le P_R:=\frac{\pi}{2c}\sqrt{R^{2}+m^{2}c^{4}}$. For all $|p|\le P_R$, (25) gives $|\mathcal{N}_h(p)-E(p)^{2}|\le c^{2}P_R^{3}h$. Since $\pm E(p)\in\sigma(D)$ and $|\sqrt{\mathcal{N}_h(p)}+E(p)|\ge mc^{2}>0$ for sufficiently small $h>0$, we obtain

which completes the proof.

Acknowledgments

The author would like to thank Professor Petru Cojuhari, for introducing him to this problem, which inspired this research.

Mariusz Niwiński

[email protected]

Jagiellonian University Doctoral School of Exact and Natural Sciences Faculty of Mathematics and Computer Science ul. Łojasiewicza 6, 30-348 Kraków, Poland