Continuous limits of linear and nonlinear quantum walks

TL;DR

This paper rigorously demonstrates that nonlinear quantum walks on a lattice converge to solutions of nonlinear Dirac equations as the lattice spacing approaches zero, bridging discrete quantum models with continuous relativistic quantum equations.

Contribution

The paper provides a rigorous proof of the continuous limit of nonlinear quantum walks, connecting them to nonlinear Dirac equations using Shannon interpolation.

Findings

Nonlinear quantum walks converge to nonlinear Dirac solutions as lattice spacing decreases.

Uniform convergence in Sobolev space $H^s$ is established.

The approach bridges discrete quantum models with continuous relativistic quantum equations.

Abstract

In this paper, we consider the continuous limit of a nonlinear quantum walk (NLQW) that incorporates a linear quantum walk as a special case. In particular, we rigorously prove that the walker (solution) of the NLQW on a lattice $\delta \mathbb Z$ uniformly converges (in Sobolev space $H^s$) to the solution to a nonlinear Dirac equation (NLD) on a fixed time interval as $\delta\to 0$. Here, to compare the walker defined on $\delta\mathbb Z$ and the solution to the NLD defined on $\mathbb R$, we use Shannon interpolation.

Equations

$$\Vert u(m\delta )-v_{\delta }(m\delta ){\Vert }_{L^2}\to 0\text{as}\delta \to 0,\text{uniformly for }m\in \mathbb{Z},0\le m\le T/\delta ,$$

$$\delta \mathbb{Z}:=\{\delta n|n\in \mathbb{Z}\}.$$

$${\mathcal{H}}_{\delta }:=l^2(\delta \mathbb{Z},{\mathbb{C}}^2),{⟨u,v⟩}_{{\mathcal{H}}_{\delta }}:=\delta \sum _{x\in \delta \mathbb{Z}}{⟨u(x),v(x)⟩}_{{\mathbb{C}}^2}\text{ and }\Vert u{\Vert }_{{\mathcal{H}}_{\delta }}^2:={⟨u,u⟩}_{{\mathcal{H}}_{\delta }},$$

$${\sigma }_0:=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),{\sigma }_1:=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),{\sigma }_2:=\left(\begin{array}{cc}0& -\mathrm{i}\\ \mathrm{i}& 0\end{array}\right),{\sigma }_3:=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right).$$

$${\mathcal{S}}_{\delta }:=\left(\begin{array}{cc}{\mathcal{T}}_{+,\delta }& 0\\ 0& {\mathcal{T}}_{-,\delta }\end{array}\right),{\mathcal{T}}_{\pm ,\delta }f:=f(\cdot \mp \delta ).$$

$${\mathcal{S}}_{\delta }=\left(\begin{array}{cc}{e}^{-\delta {\partial }_x}& 0\\ 0& e^{\delta {\partial }_x}\end{array}\right)=e^{-\delta {\sigma }_3{\partial }_x}=e^{\mathrm{i}\delta A},A:=\mathrm{i}{\sigma }_3{\partial }_x.$$

$$\left({\mathcal{C}}_{\delta ,\mathbf{s}}u\right)(x):=e^{-\mathrm{i}\delta \mathbf{s}(x)\cdot \mathbf{\sigma }}u(x),x\in \delta \mathbb{Z},$$

$$\left({\mathcal{N}}_{\delta ,\gamma ,g}u\right)(x)=e^{-\mathrm{i}\delta g({⟨u(x),\gamma u(x)⟩}_{{\mathbb{C}}^2})\gamma }u(x),x\in \delta \mathbb{Z}.$$

$$\Vert {\mathcal{N}}_{\delta }u{\Vert }_{{\mathcal{H}}_{\delta }}=\Vert u{\Vert }_{{\mathcal{H}}_{\delta }}.$$

$${\mathcal{U}}_{\delta }(0)u_0=u_0,{\mathcal{U}}_{\delta }(m+1)u_0={\mathcal{S}}_{\delta }{\mathcal{C}}_{\delta }{\mathcal{N}}_{\delta }\left({\mathcal{U}}_{\delta }(m)u_0\right).$$

$${\mathcal{C}}_{\delta ,(0,-1,0,0)}=e^{\mathrm{i}\delta {\sigma }_1}=\left(\begin{array}{cc}\cos \delta & \mathrm{i}\sin \delta \\ \mathrm{i}\sin \delta & \cos \delta \end{array}\right).$$

$${\mathcal{C}}_{1,\frac{\pi }{4}(2,0,1,-2)}=e^{-\mathrm{i}\frac{\pi }{4}\left(2,0,1,-2\right)\cdot \mathbf{\sigma }}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right).$$

$${\mathcal{C}}_1=R(\theta )=\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$$

$${\mathcal{U}}_{\mathrm{NPR}}:={\mathcal{S}}_1{\mathcal{C}}_{1,\frac{\pi }{4}(2,0,1,-2)}{\mathcal{N}}_{1,{\gamma }_1,g}{\mathcal{N}}_{1,{\gamma }_2,g},$$

$${\mathcal{N}}_{1,{\gamma }_1,g}u(x)=\left(\begin{array}{cc}{e}^{\mathrm{i}\lambda |u_1(x){|}^2}& 0\\ 0& 1\end{array}\right)u(x),{\mathcal{N}}_{1,{\gamma }_2,g}u(x)=\left(\begin{array}{cc}1& 0\\ 0& e^{\mathrm{i}\lambda |u_2(x){|}^2}\end{array}\right)u(x).$$

$${\mathcal{U}}_{LKN}:={\mathcal{S}}_{\delta }{\mathcal{C}}_{\delta }{\mathcal{N}}_{\delta ,{\sigma }_j,g},$$

$${\mathcal{N}}_{\delta ,{\sigma }_3,g}u(x)=\left(\begin{array}{cc}{e}^{-\mathrm{i}\delta (|u_1(x){|}^2-|u_2(x){|}^2)}& 0\\ 0& e^{\mathrm{i}\delta (|u_1(x){|}^2-|u_2(x){|}^2)}\end{array}\right).$$

$${\mathcal{N}}_{\delta ,{\sigma }_3,g}u(x)=e^{-\mathrm{i}\delta (|u_1(x){|}^2+|u_2(x){|}^2)}u(x).$$

$$\mathrm{i}{\partial }_{t}u=-\mathrm{i}{\sigma }_3{\partial }_{x}u+\mathbf{s}\cdot \mathbf{\sigma }u+g({⟨u,\gamma u⟩}_{{\mathbb{C}}^2})\gamma u,(t,x)\in \mathbb{R}\times \mathbb{R},u:\mathbb{R}\to {\mathbb{C}}^2.$$

$$u(t)=U_{\mathrm{Dirac}}(t)u_0.$$

$$\mathrm{i}{\partial }_{t}u=-\mathrm{i}{\sigma }_{3}u+\mathbf{s}\cdot \mathbf{\sigma }u+2{⟨u,u⟩}_{{\mathbb{C}}^2}u-{⟨u,{\gamma }_{1}u⟩}_{{\mathbb{C}}^2}{\gamma }_{1}u-{⟨u,{\gamma }_{2}u⟩}_{{\mathbb{C}}^2}{\gamma }_{2}u.$$

$$\mathrm{i}{\partial }_{t}u=-\mathrm{i}{\sigma }_{3}u+\mathbf{s}\cdot \mathbf{\sigma }u+{⟨u,u⟩}_{{\mathbb{C}}^2}^{2}u-2{⟨u,{\gamma }_{1}u⟩}_{{\mathbb{C}}^2}^2{\gamma }_{1}u-2{⟨u,{\gamma }_{2}u⟩}_{{\mathbb{C}}^2}^2{\gamma }_{2}u.$$

$$⟨u,v⟩:={\int }_{\mathbb{R}}{⟨u(x),v(x)⟩}_{{\mathbb{C}}^2}dx.$$

$$\Vert u{\Vert }_{H^s}^2:=\sum _{j=0}^s\Vert {\partial }_x^{j}u{\Vert }_{L^2}^2.$$

$${⟨u,v⟩}_{H^s}:=\sum _{j=0}^s⟨{\partial }_x^{j}u,{\partial }_x^{j}v⟩.$$

$$\underset{t\in [0,T]}{\sup }\Vert U_{\mathrm{Dirac}}(t)u_1-U_{\mathrm{Dirac}}(t)u_2{\Vert }_{H^s}\le C\Vert u_1-u_2{\Vert }_{H^s},$$

$$\underset{0\le t\le T}{\sup }\Vert U_{\mathrm{Dirac}}(t)u_1-U_{\mathrm{Dirac}}(t)u_2{\Vert }_{H^s}\le C_{T,L}\Vert u_1-u_2{\Vert }_{H^s}.$$

$${⟨u,v⟩}_{{\hat{\mathcal{H}}}_{\delta }}:={\int }_{-\pi /\delta }^{\pi /\delta }{⟨u(\xi ),v(\xi )⟩}_{{\mathbb{C}}^2}d\xi ,\Vert u{\Vert }_{{\hat{\mathcal{H}}}_{\delta }}^2:={⟨u,u⟩}_{{\hat{\mathcal{H}}}_{\delta }}.$$

$${\mathcal{F}}_{\delta }u(\xi ):=\frac{\delta }{\sqrt{2\pi }}\sum _{x\in \delta \mathbb{Z}}{e}^{-\mathrm{i}x\xi }u(x),{\mathcal{F}}_{\delta }^{-1}v(x)=\frac{1}{\sqrt{2\pi }}{\int }_{-\pi /\delta }^{\pi /\delta }{e}^{\mathrm{i}x\xi }v(\xi )d\xi ,$$

$${⟨{\mathcal{F}}_{\delta }u,{\mathcal{F}}_{\delta }v⟩}_{{\hat{\mathcal{H}}}_{\delta }}={⟨u,v⟩}_{{\mathcal{H}}_{\delta }},{⟨{\mathcal{F}}_{\delta }^{-1}u,{\mathcal{F}}_{\delta }^{-1}v⟩}_{{\mathcal{H}}_{\delta }}={⟨u,v⟩}_{{\hat{\mathcal{H}}}_{\delta }}.$$

Full text

1 Introduction

Quantum walks (QWs), or more precisely discrete time QWs, are quantum counterparts of classical random walks [1, 2, 19]. We can find the early prototypes in the context of Feynman path integral [14, 38] and Quantum Lattice Boltzmann methods [6, 28, 43]. QWs are now attracting diverse interest because of its connection to various fields of mathematics and physics such as orthogonal polynomials on the unit circle [10], quantum search algorithms [3], topological insulators [21, 22]. Further, since the early works studying QWs [6, 14, 19, 38, 43] are all more or less motivated by the discretization of Dirac equation, the relation between QWs and Dirac equation and other wave equation have repeatedly discussed by many authors from various viewpoints [4, 7, 9, 23, 30, 31, 32, 41, 42].

Nonlinear QWs (NLQWs), which are nonlinear versions of the usual (linear) QWs with the nonlinearity coming into the dynamics through the state-dependence of the quantum coin, was first proposed in [35] as an “optical Galton board [8]” with Kerr effect. From then, several models of NLQWs have been proposed motivated by simulating nonlinear Dirac equations (NLD) [23, 33] and studying the nonlinear effect to the topological insulators [15]. See also [24, 25, 26] for the study of scattering phenomena, weak limit theorem and soliton-like behavior for NLQWs. We note that for continuous time QWs, which are substantially described by discrete Schrödinger equations, nonlinear models are also attracting interest because it can speed up the quantum search [29].

In this paper, motivated by the above works, we study the connection between NLQWs and nonlinear Dirac equations (NLD). In particular, we show that the walker (or the solution) of NLQWs converges to the solution to the NLD. Very roughly, we show that for fixed $T>0$,

[TABLE]

where $u$ is the solution to the NLD, $v_{\delta}$ is the walker of NLQW on $\delta{\mathbb{Z}}$(for the precise statement, see Theorem 1.14 below). Thus, we see that the walker converges to the solution to the NLD uniformly in a fixed time interval. We emphasize that our model incorporates a linear quantum walk as a particular case and hence Theorem 1.14 says that the walker of the linear QW also converges to the solution to a Dirac equation.

This paper is organized as follows. In Subsections 1.1 and 1.2 we introduce NLQW and NLD respectively. In Subsection 1.3, we state our main result Theorem 1.14. In Section 2 we recall some facts of the Shannon interpolation. In Section 3, we prove Theorem 1.14.

1.1 Nonlinear quantum walks

We now introduce NLQWs, which are space-time discretized dynamics conserving $l^{2}$ norm (in the linear case, it is a unitary dynamics). Let $\delta>0$ be a constant and set

[TABLE]

We set

[TABLE]

where $\left\langle\cdot,\cdot\right\rangle_{\mathbb{C}^{2}}$ is the inner-product of $\mathbb{C}^{2}$, i.e. for $u=(u_{1},u_{2})\in\mathbb{C}^{2}$ and $v=(v_{1},v_{2})\in\mathbb{C}^{2}$, $\left\langle u,v\right\rangle_{\mathbb{C}^{2}}=\sum_{j=1,2}u_{j}\bar{v}_{j}$, and we set $\|u\|_{\mathbb{C}^{2}}^{2}:=\left\langle u,u\right\rangle_{\mathbb{C}^{2}}$. We also use the standard Pauli matrices

[TABLE]

We define a shift operator ${\mathcal{S}_{\delta}}:{\mathcal{H}_{\delta}}\to{\mathcal{H}_{\delta}}$ by

[TABLE]

It is clear that ${\mathcal{S}_{\delta}}$ is a unitary operator on ${\mathcal{H}_{\delta}}$.

Remark 1.1*.*

Recall (formally) $\left(e^{t\partial_{x}}f\right)(x)=f(x+t)$. Thus, we can express ${\mathcal{S}_{\delta}}$ as

[TABLE]

In the following, we will use the expression of (1.6) for ${\mathcal{S}_{\delta}}$.

We next fix a smooth function $\mathbf{s}=(s_{0},s_{1},s_{2},s_{3}):{\mathbb{R}}\to{\mathbb{R}}^{4}$ and set a linear coin operator $\mathcal{C}_{\delta,\mathbf{s}}:{\mathcal{H}_{\delta}}\to{\mathcal{H}_{\delta}}$ by

[TABLE]

where $\mathbf{s}(x)\cdot{\boldsymbol{\sigma}}=\sum_{\alpha=0}^{3}s_{\alpha}(x)\sigma_{\alpha}$. For simplicity, we often suppress the explicit dependence on $\mathbf{s}$ and write ${\mathcal{C}_{\delta}}$ for $\mathcal{C}_{\delta,\mathbf{s}}$. Since for each $x\in{\delta\mathbb{Z}}$, $e^{-{\rm i}\delta\mathbf{s}(x)\cdot{\boldsymbol{\sigma}}}$ is a $2\times 2$ unitary matrix, it is clear that ${\mathcal{C}_{\delta}}$ is unitary operator on ${\mathcal{H}_{\delta}}$.

To define a nonlinear (or state-dependent) coin operator, we fix $\gamma$ to be a $2\times 2$ Hermitian matrix and smooth function $g\in C^{\infty}({\mathbb{R}},{\mathbb{R}})$. We now define the nonlinear coin $\mathcal{N}_{\delta,\gamma,g}:{\mathcal{H}_{\delta}}\to{\mathcal{H}_{\delta}}$ by

[TABLE]

When there is no ambiguity we drop the dependence on $\gamma$ and $g$ and write just ${\mathcal{N}_{\delta}}$ for $\mathcal{N}_{\delta,\gamma,g}$. Notice that since $\left\langle{\mathcal{N}_{\delta}}u(x),{\mathcal{N}_{\delta}}u(x)\right\rangle_{\mathbb{C}^{2}}=\left\langle u(x),u(x)\right\rangle_{\mathbb{C}^{2}},$ we have

[TABLE]

Definition 1.2**.**

For $u_{0}\in{\mathcal{H}_{\delta}}$ and $m\in{\mathbb{Z}}$, $m\geq 0$, we define ${\mathcal{U}_{\delta}}(m)u_{0}\in{\mathcal{H}_{\delta}}$ by the recurrence relation

[TABLE]

Remark 1.3*.*

If $g=0$, then $\mathcal{U}_{\delta}$ is a linear unitary operator. However, if $g\neq 0$, $\mathcal{U}_{\delta}$ becomes a nonlinear operator. This is the reason why we need to define $\mathcal{U}_{\delta}(t)u_{0}$ be the recurrence relation (1.9).

We give several examples of our model, which cover various QWs appeared in the literature.

Example 1.4* (Free QWs).*

When $g=0$ and $\mathbf{s}$ do not depend on $x\in{\mathbb{Z}}$, we will call the corresponding QW a free QW. This quantum walk is also called homogeneous since the coin operator is spatially homogeneous. A typical example is the case $\mathbf{s}=(0,-1,0,0)$, which appeared in the Feynman checkerboard model [14]. In particular, the coin operator in this case have the form

[TABLE]

Another important example is the Hadamard walk [2], which is usually considered for the case $\delta=1$, and the coin operator is given by the Hadamard matrix:

[TABLE]

Example 1.5* (Linear QWs).*

When $g=0$, we will call the corresponding QW a linear QW. A typical example will be the case $\mathbf{s}(x)=(0,0,\theta(x),0)$ where $\theta:{\mathbb{R}}\to{\mathbb{R}}$ is a function converging to some limit $\theta_{\pm}$ as $x\to\pm\infty$. In this case, the coin operator

[TABLE]

is spatially inhomogeneous and called a position-dependent coin. Such a model appears in the context of topological insulators [21] and the scattering theory for linear QWs are studied in [27, 34, 39, 40, 44].

Example 1.6* (NLQWs, I).*

When $g\neq 0$, we will call the corresponding QW a NLQW. NLQW first proposed in [35] is of the form

[TABLE]

where the linear coin is given by the Hadamard matrix (1.10) and the two nonlinear coins are defined by $g(s)=\lambda s$ ($\lambda\in{\mathbb{R}}$), $\gamma_{1}=\frac{1}{2}(\sigma_{0}+\sigma_{3})$ and $\gamma_{2}=\frac{1}{2}(\sigma_{0}-\sigma_{3})$. In particular, for $u=(u_{1},u_{2})$, we have

[TABLE]

Remark 1.7*.*

Even though our result in this paper is proved for the case of a single nonlinear coin, it can be extended to the case of two nonlinear coins as stated above without difficulty.

Example 1.8* (NLQWs, II).*

Another example of NLQW, which was proposed in [23] as a simulator of a nonlinear Dirac equation, is

[TABLE]

where $g(s)=s$, $\mathcal{C}_{1}=R(\theta)$ ($\theta\in{\mathbb{R}}$) and $j=0$ or $3$. The case $j=3$ is for simulating the Gross-Neveu model (scaler type interaction) and the nonlinear coin is of the form

[TABLE]

The case $j=0$ is for simulating the Thirring model (vector type interaction) and the nonlinear coin is of the form

[TABLE]

1.2 Nonlinear Dirac equations in $1+1$ space-time

The Dirac equation on ${\mathbb{R}}$ is given by

[TABLE]

Here, $\mathbf{s}:{\mathbb{R}}\to{\mathbb{R}}^{4}$, $\gamma$ is a $2\times 2$ Hermitian matrix and $g:{\mathbb{R}}\to{\mathbb{R}}$ corresponds to the ones given in the definition of the NLQW. Indeed, we will show that a solution to the NLQW converges to a solution to the NLD with the same $\mathbf{s}$, $\gamma$ and $g$. We will denote the solution $u=u(t)$ to the Dirac equation (1.11) with the initial condition $u(0)=u_{0}$ by

[TABLE]

Note that if the Dirac equation (1.11) is nonlinear (i.e. if $g$ is not a constant), then so is $U_{\mathrm{Dirac}}$. For a comprehensive introduction for the linear Dirac equation, see [45].

As the NLQW, we introduce several examples of the NLD.

Example 1.9* (NLD: Gross-Neveu model and Thirring model).*

For the case $\mathbf{s}=(0,0,m,0)$ ($m$ is a constant), $\gamma=\sigma_{3}$ (resp. $\gamma=\sigma_{0}$), $g(s)=s$, NLD (1.11) is called the Gross-Neveu model [18] (resp. Thirring model [46]). Further, a generalized Gross-Neveu model, which is the case of general $g\in C^{\infty}({\mathbb{R}},{\mathbb{R}})$, has been studied in [12].

Example 1.10* (Nonlinear coupled mode equations).*

Let $\mathbf{s}=(V,\kappa,0,0)$ and suppose that $V,\kappa\in C^{\infty}({\mathbb{R}},{\mathbb{R}})$ are bounded functions and $\gamma_{1}=\frac{1}{2}(\sigma_{0}+\sigma_{3})$, $\gamma_{2}=\frac{1}{2}(\sigma_{0}-\sigma_{3})$. Then the NLD becomes

[TABLE]

Such a model appears in the study of nonlinear propagation of light in an optical fiber waveguide [16, 17]. A similar model also appears in the study of Bose-Einstein condensation. In particular, in [37], the following model is studied:

[TABLE]

As we remarked in Example 1.6, our result in this paper can be generalized to the case of several nonlinear coins without difficulty.

We introduce some mathematical results about NLD. To do so, we prepare several notations.

We set $L^{2}=L^{2}({\mathbb{R}},\mathbb{C}^{2})$ and $H^{s}:=H^{s}({\mathbb{R}},\mathbb{C}^{2})$ ($s\in{\mathbb{N}}$), the $\mathbb{C}^{2}$-valued Sobolev spaces. The inner product of $L^{2}$ will be denoted by

[TABLE]

We set $\|u\|_{L^{2}}:=\left\langle u,u\right\rangle^{1/2}$. The norm of $H^{s}$ is defined by

[TABLE]

We further, define the innerproduct of $H^{s}$ by

[TABLE]

Since for $s\geq 1$, $H^{s}$ becomes an algebra, one can show the following result by standard fixed point argument.

Proposition 1.11**.**

Let $s\geq 1$ and suppose that $\|\mathbf{s}\|_{L^{\infty}}+\|\mathbf{s}^{\prime}\|_{H^{s-1}}<\infty$. Let $L>0$. Then there exists $T>0$ such that a unique solution $u(t)=U_{\mathrm{Dirac}}(t)u_{0}\in C([0,T],H^{s})$ of NLD (1.11) exists for any $u_{0}\in H^{s}$ with $\|u_{0}\|_{H^{s}}\leq L$. Further, for $u_{j}\in H^{s}$ with $\|u_{j}\|_{H^{s}}\leq L$ ($j=1,2$), $U_{\mathrm{Dirac}}(t)u_{j}$ satisfies

[TABLE]

where $C$ is a constant depends only on $L$.

Inspired by the above result for the solutions of nonlinear Dirac equations, we define the condition $(\mathrm{Lip})_{s}$ as follows.

Definition 1.12**.**

Let $s\geq 1$ and $T,L>0$. We say that the pair $(T,L)$ satisfies condition $(\mathrm{Lip})_{s}$ if there exists a constant $C_{T,L}>0$ such that for $u_{j}\in H^{s}$ with $\|u_{j}\|_{H^{s}}\leq L$ ($j=1,2$), ${U_{\mathrm{Dirac}}}(\cdot)u_{j}\in C([0,T],H^{s})$ and

[TABLE]

Remark 1.13*.*

By proposition 1.11, for any $L>0$, we can always find $T>0$ such that $(T,L)$ satisfies $(\mathrm{Lip})_{s}$. Moreover, if NLD (1.11) is globally wellposed, we can take $T=\infty$ in proposition 1.11 for arbitrary $L>0$ and hence $(\mathrm{Lip})_{s}$ holds for arbitrary $(T,L)\in{\mathbb{R}}_{+}\times{\mathbb{R}}_{+}$. It is known that if the nonlinearity comes only from $\sigma_{0}$, $\gamma_{1}$ and $\gamma_{2}$ (as Example 1.10), then the NLD is globally wellposed (see [36] for more information).

1.3 Main results

To state our result precisely, we introduce some notation. When there exists a constant $C>0$ such that $a\leq Cb$, we write $a\lesssim b$ or $b\gtrsim a$. If the implicit constant $C$ depends on some parameter $\alpha$ ($C=C_{\alpha}$), then we write $a\lesssim_{\alpha}b$. If $a\lesssim b$ and $b\lesssim a$, we write $a\sim b$.

For Banach spaces $X,Y$, we set $\mathcal{L}(X,Y)$ to be the Banach space of all bounded linear operators from $X$ to $Y$. We set $\mathcal{L}(X):=\mathcal{L}(X,X)$.

We set ${\hat{\mathcal{H}}_{\delta}}:=L^{2}({\mathbb{R}}/2\pi\delta^{-1}{\mathbb{Z}},\mathbb{C}^{2})$ and define the inner product and norm by

[TABLE]

We define the discrete Fourier transform $\mathcal{F}_{\delta}\in\mathcal{L}({\mathcal{H}_{\delta}},{\hat{\mathcal{H}}_{\delta}})$ and its inverse $\mathcal{F}_{\delta}^{-1}\in\mathcal{L}({\hat{\mathcal{H}}_{\delta}},{\mathcal{H}_{\delta}})$ as

[TABLE]

where ${\mathcal{H}_{\delta}}$ is defined in (1.3). Then, we have

[TABLE]

We denote the Fourier transform on $L^{2}$ by $F$. In particular, we set

[TABLE]

We define $H_{\delta}\subset L^{2}$ by

[TABLE]

Since the Fourier transform has compact support, we have $H_{\delta}\subset\cap_{s\geq 0}H^{s}$. We define the projection to $H_{\delta}$ by

[TABLE]

where $\chi_{A}$ is the characteristic function of $A$. Obviously, we have

[TABLE]

We next define the Shannon interpolation (see, e.g.  [5]), which is an isometry from ${\mathcal{H}_{\delta}}$ to $H_{\delta}\subset L^{2}$, by

[TABLE]

where ${\hat{\mathfrak{I}}_{\delta}}$ is the natural identification between $\hat{\mathcal{H}}_{\delta}$ and $H_{\delta}$ given by

[TABLE]

We are now in a position to state our main result precisely.

Theorem 1.14**.**

Let $s\geq 1$, $T>0$ and $L>0$. Assume $\|\mathbf{s}\|_{L^{\infty}}+\|\mathbf{s}^{\prime}\|_{H^{s}({\mathbb{R}},{\mathbb{R}}^{4})}<\infty$ and $(T,L)$ satisfies condition $(\mathrm{Lip})_{s}$. Assume $\|u_{0}\|_{H^{s+1}}\leq L$. Then, there exists $\delta_{0}>0$ such that for $\delta\in(0,\delta_{0}]$,

[TABLE]

where the implicit constant is independent of $\delta$.

Since $\|\cdot\|_{L^{2}}\leq\|\cdot\|_{H^{s}}$ for $s\geq 0$, we have the following continuous limit.

Corollary 1.15**.**

*Under the same assumptions as in Theorem 1.14, the walker of the NLQW converges to the solution to the NLD in the following sense: *

[TABLE]

Remark 1.16*.*

Theorem 1.14 calls for some explanation. For the solution $u(t)={U_{\mathrm{Dirac}}}(t)u_{0}$ to the NLD in ${\mathbb{R}}$ with the initial condition $u(0)=u_{0}\in H^{s+1}$ (on ${\mathbb{R}}$), we discretize the initial condition $u_{0}$ by ${\mathfrak{I}_{\delta}}^{-1}\circ j_{\delta}$, evolve it by the NLQW by (1.9). After m steps of the NLQW evolution, we can put it back to a function by the Shannon interpolation ${\mathfrak{I}_{\delta}}$ since it is unitary from ${\mathcal{H}_{\delta}}$ to $H_{\delta}$ (see Lemma 2.1). Theorem 1.14 ensures that the resulting function ${\mathfrak{I}_{\delta}}\circ{\mathcal{U}_{\delta}}(m)\circ{\mathfrak{I}_{\delta}}^{-1}\circ j_{\delta}u_{0}$ successively approximates the solution to the NLD. In this sense, we can say that the continuous limit of the NLQW is the NLD.

Although many works discuss the continuous limit of (linear and nonlinear) QWs, it seems that they just informally compare the equation of QWs and the Dirac equation by, for instance, expanding the equation or referring the Trotter-Kato formula. What is really needed is the estimate of the difference of the walker of the QW and solution to the Dirac equation as given in (1.18). The only result of such kind we are aware is [4], where the authors show (1.18) for $m=1$. In this sense, our result is new even in the linear QWs.

From the viewpoint of numerical analysis, the NLQW gives a splitting method of the NLD. Splitting methods are now popular numerical schemes for approximating semilinear Hamiltonian partial differential equations such as nonlinear Schrödinger equations [13]. In this point of view, it may be interesting to investigate higher-order methods such as Strang splitting for the NLD. However, to make our paper simple, we will not investigate them.

For the proof of Theorem 1.14, we employ the energy method of Holden-Karlsen-Risebro-Tao [20], which was originally applied to the KdV equation.

2 Preliminary

In this section, we collect technical tools which we use in the proof of Theorem 1.14. Recall the definitions of $\left\langle\cdot,\cdot\right\rangle_{\mathcal{H}_{\delta}}$ and ${\mathfrak{I}_{\delta}}$ given in (1.3) and (1.16) respectively.

Lemma 2.1** ([5]).**

${\mathfrak{I}_{\delta}}:{\mathcal{H}_{\delta}}\to H_{\delta}$* is unitary and*

[TABLE]

Proof.

By definition, ${\mathfrak{I}_{\delta}}$ is an isometry and the image of ${\mathfrak{I}_{\delta}}$ is $H_{\delta}$. Hence, ${\mathfrak{I}_{\delta}}$ is unitary. (2.1) is shown by

[TABLE]

∎

Recall $j_{\delta}$ given in (1.14).

Lemma 2.2**.**

For $\sigma\geq 1$,

[TABLE]

Proof.

By the definition of $j_{\delta}$ and the norm of $H^{s}$ given in (1.12), we have

[TABLE]

where we have used the fact that $(\delta\left\langle\xi\right\rangle)^{\sigma}\geq 1$ if $|\xi|>\pi/\delta$. Therefore, we have (2.2). ∎

Recall $\mathcal{T}_{-,\delta}u(x)=u(x+\delta)$. We set

[TABLE]

Formally, we can write $\mathcal{D}_{\delta}=\frac{e^{\delta\partial_{x}}-1}{\delta}$. Further, we set

[TABLE]

Remark 2.3*.*

The operators $\mathcal{D}_{\delta}$ and $D_{\delta}$ formally have the same definition. However, $\mathcal{D}_{\delta}$ is defined on $\mathcal{H}_{\delta}=l^{2}(\delta{\mathbb{Z}},\mathbb{C}^{2})$ and $D_{\delta}$ is defined on $L^{2}({\mathbb{R}},\mathbb{C}^{2})$.

Lemma 2.4**.**

Let $s\geq 0$. For $u\in H^{1}$, we have

[TABLE]

Proof.

By $e^{\delta\partial_{x}}=1+\delta\int_{0}^{1}e^{\delta t\partial_{x}}\,dt\partial_{x}$, we have

[TABLE]

Since

[TABLE]

we have $\|\int_{0}^{1}e^{\delta t\partial_{x}}\,dt\|_{\mathcal{L}(L^{2})}\leq 1$. Therefore, we have the conclusion. ∎

Lemma 2.5**.**

Let $s\geq 0$ and $u\in H^{s+1}$. Then, for $0\leq j\leq s$, we have

[TABLE]

Remark 2.6*.*

By Sobolev embedding, we have $u\in H^{1}({\mathbb{R}},\mathbb{C}^{2})\hookrightarrow C^{1}({\mathbb{R}},\mathbb{C}^{2})$. Therefore $u$ is defined pointwise and $D_{\delta}^{j}u(x)$ has a meaning.

Proof.

We first prove the case $j=0$. Fix $x\in\delta{\mathbb{Z}}$. Set $F_{x}(t)=\|u(x+t)\|_{\mathbb{C}^{2}}^{2}$. Then, since $\partial_{t}F_{x}=2\mathop{\rm Re}\nolimits\left\langle u(x+t),\partial_{x}u(x+t)\right\rangle_{\mathbb{C}^{2}}\in L^{1}({\mathbb{R}})$, we have

[TABLE]

By the Fubini Theorem,

[TABLE]

Therefore, by Schwartz, we have the conclusion.

Next, for $j\geq 1$, assume that we have (2.4) for $j-1$. Then, by Lemma 2.4, since $\partial_{x}$ and $D_{\delta}$ commute, we have

[TABLE]

Therefore, we have the conclusion. ∎

Lemma 2.7**.**

Let $u\in H_{\delta}$. Then, we have

[TABLE]

Here, the implicit constant is independent of $\delta$.

Proof.

First, for $|\eta|\leq\pi$, we have

[TABLE]

We have

[TABLE]

Therefore, we have the conclusion. ∎

Proposition 2.8**.**

Let $s\geq 0$, $\sigma\geq 1$. Let $u\in H_{\delta}$ and $v\in H^{s+\sigma}$ with $u(x)=v(x)$ for all $x\in\delta{\mathbb{Z}}\subset{\mathbb{R}}$. Then, we have

[TABLE]

Here, the implicit constant is independent of $u,v$ and $\delta$.

Remark 2.9*.*

Notice that the right hand side of (2.5) does not depend on $u$.

Proof.

First, by Lemma 2.2, we have

[TABLE]

For $0\leq j\leq s$, since $u-j_{\delta}v\in H_{\delta}$, by Lemma 2.7 we have

[TABLE]

By Lemmas 2.1, 2.2 and 2.5, we have

[TABLE]

Therefore, we have the conclusion. ∎

3 Proof of Theorem 1.14

In this section, we prove Theorem 1.14. In the following, as claimed in Theorem 1.14, we fix $T,L>0$ and $s\geq 1$ and assume

[TABLE]

and

[TABLE]

Since $\mathbf{s}$ is fixed, we will not denote the dependence of $\|\mathbf{s}\|_{L^{\infty}({\mathbb{R}},{\mathbb{R}}^{4})}+\|\mathbf{s}^{\prime}\|_{H^{s}({\mathbb{R}},{\mathbb{R}}^{4})}$ in the implicit constant in the inequalities below.

We start with decomposing $\|{\hat{\mathfrak{I}}_{\delta}}\circ{\mathcal{U}_{\delta}}(m)\circ{\hat{\mathfrak{I}}_{\delta}}^{-1}\circ j_{\delta}u_{0}-{U_{\mathrm{Dirac}}}(m\delta)u_{0}\|_{H^{s}}$ as

[TABLE]

For $m\delta\leq T$, one can estimate the second term of the right hand side of (3.3) by the assumptions (3.1) and (3.2). Indeed, by Lemma 2.2,

[TABLE]

We further decompose the first term of (3.3) as

[TABLE]

where

[TABLE]

and

[TABLE]

Remark 3.1*.*

$U_{\delta}$, $S_{\delta}$, $C_{\delta}$ and $N_{\delta}$ are the continuous counterparts of ${\mathcal{U}_{\delta}}$, ${\mathcal{S}_{\delta}}$, ${\mathcal{C}_{\delta}}$ and ${\mathcal{N}_{\delta}}$ respectively. That is, $U_{\delta}$, $S_{\delta}$, $C_{\delta}$ and $N_{\delta}$ is defined on $L^{2}({\mathbb{R}},\mathbb{C}^{2})$ instead of $\mathcal{H}_{\delta}$ with formally the same definition as ${\mathcal{U}_{\delta}}$, ${\mathcal{S}_{\delta}}$, ${\mathcal{C}_{\delta}}$ and ${\mathcal{N}_{\delta}}$.

We next bound the second term in (LABEL:pthm2:2) following Holden-Karlsen-Risebro-Tao [20]. To this end, we introduce $v_{\delta}(t_{1},t_{2},t_{3})$ as follows. Let

[TABLE]

where $\Omega_{\delta}^{m}:=[m\delta,(m+1)\delta]^{3}$. We define self-adjoint operators $A$ and $B$ as

[TABLE]

We define a nonlinear operator $G$ as

[TABLE]

Let $v_{\delta}(0,0,0)=j_{\delta}u_{0}\in L^{2}({\mathbb{R}},\mathbb{C}^{2})$ and define $v_{\delta}(t_{1},t_{2},t_{3})\in L^{2}({\mathbb{R}},\mathbb{C}^{2})$ for $(t_{1},t_{2},t_{3})\in\Omega_{\delta}$ by

[TABLE]

More precisely, given the value of $v_{\delta}(m\delta,m\delta,m\delta)$, we are defining $v_{\delta}(\tilde{t}_{1},\tilde{t}_{2},\tilde{t}_{3})$ for $(\tilde{t}_{1},\tilde{t}_{2},\tilde{t}_{3})\in\Omega_{\delta}^{m}$ by first solving the first equation of (3.9) in the $t_{1}$ direction up to $t_{1}=\tilde{t}_{1}$ and then solve the second equation of (3.9) in the $t_{2}$ direction up to $t_{2}=\tilde{t}_{2}$ and finally solve the third equation of (3.9) in the $t_{3}$ direction up to $t_{3}=\tilde{t}_{3}$. By this procedure we can define $v_{\delta}((m+1)\delta,(m+1)\delta,(m+1)\delta)$ and thus we can define the value of $v_{\delta}$ for all $(t_{1},t_{2},t_{3})\in\Omega_{\delta}$ by induction because $v_{\delta}(0,0,0)=j_{\delta}u_{0}$ is given.

Remark 3.2*.*

We note that $v_{\delta}=v_{\delta}(t_{1},t_{2},t_{3},x)$ is a $\mathbb{C}^{2}$-valued function defined on $\Omega_{\delta}\times{\mathbb{R}}$. However, since we want to view $v_{\delta}$ as an $L^{2}({\mathbb{R}},\mathbb{C}^{2})$-valued function on $\Omega_{\delta}$, we write $v_{\delta}=v_{\delta}(t_{1},t_{2},t_{3})$ and suppress the dependence on the spatial variable $x$. We further remark that the differential operator $A=-{\rm i}\sigma_{3}\partial_{x}$ acts on this spatial variable $x$.

Lemma 3.3**.**

Let $v_{\delta}$ be the solution to (3.9) with $v_{\delta}(0,0,0)=j_{\delta}u_{0}$. Then, $v_{\delta}$ correspond to $U_{\delta}(\cdot)j_{\delta}u_{0}$ at the diagonal lattice point. That is, we have

[TABLE]

Proof.

Recall (3.6) and (3.7). We prove (3.10) by induction. Thus, we can assume (3.10). Our goal will be to show (3.10) with $m$ replaced by $m+1$. We first show

[TABLE]

By the first equation of (3.9),

[TABLE]

Hence, $\left\langle v_{\delta}(t_{1},\delta m,\delta m),\gamma v_{\delta}(t_{1},\delta m,\delta m)\right\rangle$ conserves. By (3.7) and the first equation of (3.9) again, we obtain (3.11). Similarly, from the second and third equations of (3.9), we can prove

[TABLE]

and

[TABLE]

Therefore, we have the conclusion. ∎

Setting $v_{\delta}(t):=v_{\delta}(t,t,t)$, we show the following proposition.

Proposition 3.4**.**

For sufficiently small $\delta>0$, we have

[TABLE]

where $u_{\delta}(t):={U_{\mathrm{Dirac}}}(t)j_{\delta}u_{0}$.

Remark 3.5*.*

By Lemma 3.3 and Proposition 3.4, we obviously have

[TABLE]

where the implicit constant are independent of $m,\delta$. Thus, we obtain the bound for the second term of (LABEL:pthm2:2). It remains to obtain the bound for the first term of (LABEL:pthm2:2).

Before proving Proposition 3.4, we prepare several notations and lemmas. First, we set

[TABLE]

where $G^{\prime}(v)$ is the Fréchet derivative of $G$, $\mathop{\rm Re}\nolimits\left\langle w,\gamma v\right\rangle_{\mathbb{C}^{2}}$ is the real part of $\left\langle w,\gamma v\right\rangle_{\mathbb{C}^{2}}$ and

[TABLE]

Lemma 3.6**.**

We have

[TABLE]

Proof.

First, recall (3.1). Thus, by

[TABLE]

the bound for $\|[A,B]v\|_{H^{s}}$ is obvious since for $s\geq 1$, $H^{s}$ becomes an algebra. Next, we have

[TABLE]

Again, since $H^{s}$ is an algebra, we can bound each term by using the elementary inequality

[TABLE]

By a similar manner, we have the estimate for $\|[B,G]v\|_{H^{s}}$. ∎

Lemma 3.7**.**

Let $T^{\prime}>0$. Suppose there exists $\delta_{1}>0$ such that for $\delta\in(0,\delta_{1}]$,

[TABLE]

Then, there exists $\delta_{0}>0$ such that for $\delta\in(0,\delta_{0}]$,

[TABLE]

In particular, the implicit constant is independent of $\delta$.

Proof.

For $0\leq\tau\leq\delta$, we set

[TABLE]

Since we have

[TABLE]

by the 3rd line of (3.9), we see

[TABLE]

Similarly, by the 2nd line of (3.9), we have

[TABLE]

Thus,

[TABLE]

where we have used assumption (3.1). Therefore, we conclude

[TABLE]

Now, since $v_{\delta,1}(\delta m+\tau)$ is the solution to ${\rm i}\partial_{\tau}v_{\delta,1}=G(v_{\delta,1})$ with $v_{\delta,1}(\delta m)=v_{\delta}(\delta m).$ Therefore, since we can express $v_{1,\delta}=e^{{\rm i}g(\left\langle v_{\delta}(\delta m),\gamma v_{\delta}(\delta m)\right\rangle)\gamma}v_{\delta}(\delta)$, we have

[TABLE]

Further, since

[TABLE]

and by (3.18), we have

[TABLE]

Therefore, by comparison theorem of ordinarily differential equation (or Gronwall’s inequality), we have

[TABLE]

where $C_{M}>0$ is the implicit constant in (3.19). Combining (3.17) and (3.19), we have

[TABLE]

with $C_{0}=C+C_{M}$. Thus for $0\leq t\leq T^{\prime}$, we have

[TABLE]

This gives us the conclusion. ∎

Lemma 3.8**.**

Let $T^{\prime}>0$ and suppose

[TABLE]

Then, we have

[TABLE]

Proof.

Set $w_{\delta}(t):=v_{\delta}(t)-u_{\delta}(t)$. By (3.20) and Lemma 3.7, we have

[TABLE]

Next, by (3.9), we have

[TABLE]

Then, setting $F_{12}(t_{1},t_{2},t_{3}):={\rm i}\partial_{t_{1}}v_{\delta}-G(v_{\delta})+{\rm i}\partial_{t_{2}}v_{\delta}-Bv_{\delta}$, we have

[TABLE]

and thus

[TABLE]

Recall we have $\|Bw_{\delta}\|_{H^{s}}\lesssim\|w_{\delta}\|_{H^{s}}$.

By (3.9) we have $F_{12}(t,\delta m,\delta m)=0$. Further,

[TABLE]

By lemma 3.6, we have

[TABLE]

Therefore, we have

[TABLE]

where $C=C_{T^{\prime},L,M}>0$ is a constant. Now, we set

[TABLE]

By (3.9), we have $F_{1}(t,\delta m)=0$ and

[TABLE]

The estimate of $\|F_{1}(t,t)\|_{H^{s}}$ need a little care since $B$ do not commutate with the derivatives. Since, by Lemma 3.6, we have $\|[B,G]v_{\delta}\|_{H^{s}}\lesssim_{T^{\prime},L,M}1,$ we first get the estimate

[TABLE]

Suppose that for $s^{\prime}\leq s$, we have the estimate

[TABLE]

Since

[TABLE]

we have

[TABLE]

Therefore, we obtain (3.25) with $s^{\prime}-1$ replaced by $s^{\prime}$ and thus by induction we have (3.25) with $s^{\prime}-1$ replaced by $s$. Therefore, substituting (3.24) into (3.23), we have

[TABLE]

Next, since

[TABLE]

we have

[TABLE]

Therefore, by (3.22), (3.26) and (3.27), we have

[TABLE]

This gives us the conclusion. Indeed, if we have such inequality, setting $A(t)$ by

[TABLE]

we have $\|w_{\delta}(t)\|_{H^{s}}\leq A(t)$, where $\widetilde{C}=\widetilde{C}_{T^{\prime},L,M}>0$ is the implicit constant in (3.28). Moreover, since we have $A(t)=e^{\widetilde{C}t}(A(0)+\delta)-\delta$, we can conclude

[TABLE]

which is the desired estimate. ∎

Proof of Proposition 3.4.

By Lemma 3.8, it suffices to prove (3.20) for $T^{\prime}=T$. Let $\widetilde{C}_{T,L}$ be the constant given by the assumption that $(T,L)$ satisfies $(\mathrm{Lip})_{s}$. Without loss of generality, we can assume $\widetilde{C}_{T,L}L>\max(1,L)$. Set $M=M_{T,L}:=4\widetilde{C}_{T,L}L$. Let $C_{T,L,M}$ the implicit constant given in (3.21) in Lemma 3.8. Set $\delta_{T,L}:=C_{T,L,M_{T,L}}^{-1}$. For $\delta\in(0,\delta_{T,L})$, we set

[TABLE]

Then, it suffices to show $T\in\mathcal{T}_{\delta}$.

First, $0\in\mathcal{T}_{\delta}$ so $\mathcal{T}_{\delta}$ is not empty. Further, since $v_{\delta}$ is continuous in $H^{s}$, we see that $\mathcal{T}_{\delta}$ is an open interval in $[0,T]$ (i.e. there exists an open interval $\mathcal{O}\subset{\mathbb{R}}$ s.t. $\mathcal{T}_{\delta}=[0,T]\cap\mathcal{O}$). Now, suppose $T^{*}:=\sup\mathcal{T}_{\delta}<T$. Then, for any $T^{\prime}<T^{*}$, by Lemma 3.8, we have

[TABLE]

Therefore, by continuity, we have

[TABLE]

and thus for sufficiently small $\epsilon>0$, we have $T^{*}+\epsilon\in\mathcal{T}_{\delta}$, which contradicts the definition of $T^{*}$. Therefore, we have $T\in\mathcal{T}_{\delta}$. ∎

Proof of Theorem 1.14.

As mentioned in Remark 3.5, we need only bound the first term of (LABEL:pthm2:2). We note that from Proposition 3.4, Lemma 3.7 and the assumption (3.2) we have

[TABLE]

Further, notice that from the definition of $U_{\delta}$ and $\mathcal{U}_{\delta}$ and (2.1), we have

[TABLE]

Thus we can apply Proposition 2.8 and obtain

[TABLE]

This completes the proof of Theorem 1.14. ∎

Acknowledgments

M.M. was supported by the JSPS KAKENHI Grant Numbers 19K03579, JP17H02851 and

JP17H02853. A. S. was supported by JSPS KAKENHI Grant Number JP26800054 and JP18K03327.