On fractional semidiscrete Dirac operators of Lévy-Leblond type

The study of null solutions of Dirac-like operators is known as the heart of several function theories in the context of Clifford algebras. From the fact that Dirac-like operators factorize the Laplace operator and its analogues, it is of foremost importance to study them algebraically and analytically in order to obtain refinements of well known results in harmonic analysis and to develop applications in the fields of mathematical physics and applied mathematics as well.

In 1967 Lévy-Leblond investigates the factorization of the Schrödinger operator with the aim of obtaining non-relativistic analogues of the Dirac operator in the $(1+3)$–dimensional space carrying any spin (cf. [1]).However, we had to wait for contemporary times to realize how the $(1+n)-$dimensional generalization of the Lévy-Leblond type picture, coined as parabolic Dirac type operators, can be adopted to cover a wide range of applications on the crossroads of function theory and boundary value problems. Between the many notable results regarding this new class of operators, it is worth to stress the importance of the groundbreaking works of Cerejeiras, Kähler & Sommen [2] and Cerejeiras, Sommen & Vieira [3], who have proposed the matter of factorizing the heat operator and its relatives. Since then, their work have conducted many researchers to explore this type of approach to a wide variety of model problems, including higher-dimensional analogues of the nonlinear Schrödinger equation [4, 5].

Interestingly, the Lévy-Leblond picture had continue to receiving an increasing interest during the last decade. Mainly, on several physical-phase-space formulations of spinning particles through supersymmetric Lie algebraic representations of PDEs (see [6, 7]) as well as on operational models involving parabolic Dirac operators and their fractional analogues (see e.g. [8, 9, 10, 11] and the references therein), to mention a few. Therewith, it is reasonable to say that the Lévy-Leblond framework is no longer just an emerging topic in the mathematics and physics communities, so that nowadays one can say that its foundations and applications are well understood by many scholars.

In parallel, the study of discrete Dirac operators has experienced a rapid increase in hypercomplex analysis during the last two decades, strongly influenced by the pioneering papers [12, 13, 14, 15, 16]. The literature towards this topic, whose physical roots may be found e.g. on the research papers of Kogut & Susskind [17] and Rabin [18] (see also [19, 20] and the references therein) is very diverse. To the purpose of this paper we will depict only an abridged overview of it to motivate our approach.

For the construction of a faithful discretization for the discrete Dirac operator on the lattice $h{\mathbb{Z}}^{n}$, say $D_{h}$, whose precise definition will be introduced afterwards, the Clifford algebra of signature of $(n,n)$ may be embody on the ladder structure of $D_{h}$, with the aid of the so-called Witt basis $\left\{{\bf e}_{j}^{+},{\bf e}_{j}^{-}\leavevmode\nobreak\ :\leavevmode\nobreak\ j=1,2,\ldots,n\right\}$, to seamlessly encode the canonical structure of the exterior algebra (cf. [21, Chapter 1] and [22, Chapter 2]).

Originally highlighted in [14] and on the series of papers [15, 16], such fact was only duly clarified in author’s recent paper [23], through the canonical isomorphism $C\kern-1.00006pt\ell_{n,n}\cong\mbox{End}(C\kern-1.00006pt\ell_{0,n})$ between the Clifford algebra of signature $(n,n)$, $C\kern-1.00006pt\ell_{n,n}$, and the algebra of endomorphisms acting on the Clifford algebra of signature $(0,n)$, $C\kern-1.00006pt\ell_{0,n}$. This in turn allows us to establish a canonical correspondence between the discrete counterpart of the Dirac-Kähler operator, $d-d^{*}$, and the multivector approximation of Dirac operator, $D_{h}$, over $h{\mathbb{Z}}^{n}$ (see also [24]).

In a larger extend, starting from forward and backward discretizations of the Dirac operator $\displaystyle D=\sum_{j=1}^{n}{\bf e}_{j}\partial_{x_{j}}$, $D_{h}^{+}$ and $D_{h}^{-}$ respectively, formely introduced in [13], one can make use of the wedge ($\wedge$) and the dot $(\bullet)$ actions on $C\kern-1.00006pt\ell_{0,n}$ to establish the canonical one-to-one correspondences

$\displaystyle d\overset{\text{1-1}}{\longleftrightarrow}D_{h}^{-}\wedge(\cdot)$

and

$\displaystyle d^{*}\overset{\text{1-1}}{\longleftrightarrow}D_{h}^{+}\bullet(\cdot)$

in a way that the resulting discrete Dirac operator $D_{h}$ is nothing else than a $C\kern-1.00006pt\ell_{n,n}-$valued representation of the multivector counterpart of the Dirac-Kähler operator, $\displaystyle D_{h}^{-}\wedge(\cdot)-D_{h}^{+}\bullet(\cdot)$ (cf. [23, Subsection 2.3]).

In this paper, we will focus our attention on the time-fractional and space-fractional Dirac operators, carrying the parameters $0<\alpha\leq 1$, $\beta\geq 1$ and $|\theta|\leq\frac{\alpha\pi}{2}$. On the construction neatly summarized in Figure 1, the notation $\Delta_{h}$ stands for the discrete Laplacian on the lattice $h{\mathbb{Z}}^{n}$ considered in several author’s contributions such as [23, 24].

In addition, the family of fractional discrete operators $(-\Delta_{h})^{\sigma}$ ($0<\sigma\leq 1$) will be defined in terms of its Fourier symbol in the streamlines of [26, Section 6.] (see also [27, Section 21.4.3]). For our main purposes, we will adopt the notation $\mathbb{D}_{t}^{\beta}$ for the so-called right-sided Riemann-Liouville fractional derivative (cf. [28, Chapter 2]). The operator $\mathbb{D}_{t}^{\beta}$, defined as follows:

$\displaystyle\mathbb{D}_{t}^{\beta}\Psi(x,t)=\left\{\begin{array}[]{lll}\displaystyle(-\partial_{t})^{k}\int_{t}^{+\infty}g_{k-\beta}(s-t)\Psi(x,s)ds&\mbox{for}&k-1<\beta<k\\ \\ (-\partial_{t})^{k}\Psi(x,t)&\mbox{for}&\beta=k,\end{array}\right.$

(4)

where $k=\lfloor\beta\rfloor+1$ ($\lfloor\beta\rfloor$ denotes the integer part of $\beta$), is an integro-differential operator for values of $\beta\neq k$, involving the higher order time-derivative $(-\partial_{t})^{k}:=(-1)^{k}(\partial_{t})^{k}$ and an integral part, corresponding to the convolution between $\Psi(x,t)$ and the Gel’fand-Shilov function $g_{\nu}:\mathbb{R}\rightarrow[0,\infty)$, defined for $-\nu\not\in{\mathbb{N}}_{0}$ by

$\displaystyle g_{\nu}(p)=\left\{\begin{array}[]{lll}\displaystyle\frac{p^{\nu-1}}{\Gamma(\nu)}&\mbox{for}&p>0\\ \\ 0&\mbox{for}&p\leq 0.\end{array}\right.$

(8)

It should be noted that the Eulerian integral representation involving the Gamma function $\Gamma(\cdot)$ (cf. [29, formula 2.3.3. (1), p.322]):

$\displaystyle\int_{0}^{\infty}e^{-p\lambda}p^{\nu-1}dp=\Gamma(\nu)\lambda^{-\nu},$

$\displaystyle\Re(\nu)>0\leavevmode\nobreak\ \leavevmode\nobreak\ \&$

$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \Re(\lambda)>0$

(9)

assures that for every $k-1<\beta<k$ the function $p\mapsto g_{k-\beta}(p)$, appearing on the definition of $\mathbb{D}_{t}^{\beta}$, defines a probability density function converging to the Dirac delta function, that is

$$\lim_{\beta\rightarrow k^{-}}g_{k-\beta}(p)=\delta(p).$$

The later set of properties allows us to say that $\mathbb{D}_{t}^{\beta}$ ($k-1<\beta<k$) provides us a regularization of $(-\partial_{t})^{k}$ in the sense of the topology of the underlying space of tempered distributions.

The motivation to this paper comes from the series of contributions [30, 26, 31], focused on the study of Cauchy problems involving fractional discrete Laplacians, and from the interrelationship between space-fractional and time-fractional operators highlighted in [32] towards superdiffusion equations. Such link has played a crucial role in the theory of PDEs (cf. [33, 34]) and on stochastic calculus as well (cf. [35]). For a survey on both topics, we also refer to the monograph [36].

Thus, we are not only interested on fractional difference analogues for the parabolic operator of heat type ($\alpha=\beta=1$, $\theta=0$ & $h\rightarrow 0$) and of Schrödinger type ($\alpha=\beta=1$, $\theta=\pm\frac{\pi}{2}$ & $h\rightarrow 0$), considered in the series of papers [2, 4, 3, 6, 7, 10], neither fractional counterparts of semidiscrete models involving the semidiscrete heat operator $\partial_{t}-\Delta_{h}=(D_{h}+{\mathfrak{f}}\partial_{t}+{\mathfrak{f}}^{\dagger})^{2}$ (i.e when $\alpha=\beta=1$ & $\theta=0$), already considered in [37], but also on the interface between space-fractional and time-fractional semidiscrete operators in a way that the fractional discrete Laplacian $-(-\Delta_{h})^{\alpha}$, carrying the parameter $0<\alpha\leq 1$, turns out be treated as a temporal regularization of order $\beta:=\frac{1}{\alpha}\geq 1$, encoded on the time-fractional derivative $\mathbb{D}_{t}^{\beta}$.

Moreover, the parameter condition $|\theta|\leq\frac{\alpha\pi}{2}$ encoded on the [fractional semidiscrete] analytic semigroup $\displaystyle\left\{\exp(-te^{i\theta}(-\Delta_{h})^{\alpha})\right\}_{t\geq 0}$ is ubiquitous on space-fractional diffusion models (cf. [38]). Its incorporation on our model problem is seamlessly justified by the constraint $|\theta|=|\arg(te^{i\theta})|<\pi$, carrying the integral representation obtained in [27, p. 458, eq. (21.35)] for modified Bessel functions of the first kind (cf. [26, Subsection 2.1 & Subsection 3.1]). This will be the main novelty of this paper in comparison with the semidiscrete heat semigroup representations considered in the series of author’s contributions [27, 25, 39].

Up to author’s knowledge, the overlap between space-fractional Dirac operators – such as the ones introduced by Bernstein in [40] – and time-fractional Dirac operators of Riemann-Liouville type – such as the ones considered e.g. by Ferreira & Vieira [8] – was not yet addressed so that the idea of connecting fractional order in time and space on this paper shall be seen eventually as another step further to pursue the goal of studying the mild solutions for time-fractional Navier-Stokes equations in the superdiffusive case, from a hypercomplex analysis perspective (see e.g. [41, 42] and the references therein for an overview on the subdiffusive case).

This paper is organized as follows:

• 1.

  In Section 2 we briefly provide some background on Clifford algebras and on discrete Fourier analysis required throughout the paper. From the point of view of the theory of pseudo-differential operators, that will allows us to shift all the well-known constructions in the space of square-integrable functions to the space of Clifford-valued distributions over the lattice $h{\mathbb{Z}}^{n}$ (see, for instance, [27, Subsection 21.2] & [27, Subsection 2.2.]).

• 2.

  In Section 3 we introduce a time-fractional and a space-fractional variant of the semidiscrete Dirac operator $D_{h}+{\mathfrak{f}}\partial_{t}+{\mathfrak{f}}^{\dagger}e^{-i\theta}$. Guided by the approaches of Cerejeiras, Kähler & Sommen [2] and Cerejeiras, Sommen & Vieira [3], we introduce the time-fractional regularization of $D_{h}+{\mathfrak{f}}\partial_{t}+{\mathfrak{f}}^{\dagger}e^{-i\theta}$ by replacing the time-derivative $\partial_{t}$ by a time-fractional counterpart $-e^{i\theta(1-\beta)}\mathbb{D}_{t}^{\beta}$ ($\beta\geq 1$), mixing the Riemann-Liouville derivative and a unitary term lying on the unit circle $\mathbb{S}^{1}$. For the space-fractional regularization, we consider the space-fractional counterpart $(-\Delta_{h})^{\alpha-1}\partial_{t}$ of $\partial_{t}$, involving the fractional discrete operator $(-\Delta_{h})^{\alpha-1}$ ($0<\alpha\leq 1$) instead of $\partial_{t}$ (case of $\alpha=1$). With the proof of Theorem 3.1 & Theorem 3.2, we will show that the formulations highlighted on Figure 1 retain all of the salient features of the null solutions of the Lévy-Leblond type operator $D+{\mathfrak{f}}\partial_{t}+{\mathfrak{f}}^{\dagger}e^{-i\theta}$ (limit case $h\rightarrow 0$) considered by several authors (see also the papers of Bernstein [4], and Bao, Constales, De Bie & Mertens [10]).

• 3.

  In Section 4 we will show that the null solutions of both fractional semidiscrete operators are indeed interrelated. Starting from the pseudo-differential representation of the fractional semidiscrete analytic semigroup
  $\displaystyle\left\{\exp(-te^{i\theta}(-\Delta_{h})^{\alpha})\right\}_{t\geq 0}$ in terms of its Fourier symbol we will provide first, with the proof of Theorem 4.1, a bridge result that establishes the correspondence between Cauchy problems of space-fractional and time-fractional type, encoded by the set of fractional operators, $e^{-i\theta}\partial_{t}+(-\Delta_{h})^{\alpha}$ and $-e^{-i\theta\beta}\mathbb{D}_{t}^{\beta}-\Delta_{h}$ respectively. This theorem is essentially a wise reformulation of [32, Theorem 3] in terms of the right-sided Riemann-Liouville fractional derivative. Afterwards, we will prove Theorem 4.2 – the main contribution of this paper – by combining Theorem 3.1, Theorem 3.2 & Theorem 4.1.

• 4.

  In Section 5 we will briefly discuss an alternative formulation for the semidiscrete Dirac operator of space-fractional type that factorizes the space-fractional operator $e^{-i\theta}\partial_{t}+(-\Delta_{h})^{\alpha}$. In the end we will also comment on the key ingredients considered to obtain our main results and depict further directions of research on the crosssroads of function spaces and Helmholtz-Leray type decompositions.

Let ${\bf e}_{0},{\bf e}_{1},{\bf e}_{2},\ldots,{\bf e}_{n},{\bf e}_{n+1},{\bf e}_{n+2}\,\ldots,{\bf e}_{2n},{\bf e}_{2n+1}$ be the generators of the Clifford algebra of signature $(n+1,n+1)$, $C\kern-1.00006pt\ell_{n+1,n+1}$, satisfying

$\displaystyle\begin{array}[]{lll}{\bf e}_{j}{\bf e}_{k}+{\bf e}_{k}{\bf e}_{j}=-2\delta_{jk},&0\leq j,k\leq n\\ {\bf e}_{j}{\bf e}_{n+k}+{\bf e}_{n+k}{\bf e}_{j}=0,&0\leq j\leq n\leavevmode\nobreak\ \leavevmode\nobreak\ \&\leavevmode\nobreak\ \leavevmode\nobreak\ 1\leq k\leq n+1\\ {\bf e}_{n+j}{\bf e}_{n+k}+{\bf e}_{n+k}{\bf e}_{n+j}=2\delta_{jk},&1\leq j,k\leq n+1.\end{array}$

(13)

As it is well-known from the literature (cf. [22, Chapter 3]), $C\kern-1.00006pt\ell_{n+1,n+1}$ is a universal algebra of dimension $2^{2n+2}$ linear isomorphic to the exterior algebra $\Lambda^{*}\left({\mathbb{R}}^{n+1,n+1}\right)$, containing the field of real numbers ${\mathbb{R}}$, the $n-$dimensional Euclidean space ${\mathbb{R}}^{n}$ and the Minkowski space ${\mathbb{R}}^{n+1,n+1}$ of signature $(n+1,n+1)$ as proper subspaces.

In particular, the ladder structure of $C\kern-1.00006pt\ell_{n+1,n+1}$ allows us to represent the space-time tuple $(x_{1},x_{2},\ldots,x_{n},t)$ of ${\mathbb{R}}^{n+1}$ through the paravector representation $\displaystyle t+x=t+\sum_{j=1}^{n}x_{j}{\bf e}_{j}$ of ${\mathbb{R}}\oplus{\mathbb{R}}^{n}$.

Here and elsewhere, the $1-$vector representations $\displaystyle x=\sum_{j=1}^{n}x_{j}{\bf e}_{j}$ and $\displaystyle x\pm h{\bf e}_{j}$ of ${\mathbb{R}}^{n}$ will be adopted to describe the lattice point $(x_{1},x_{2},\ldots,x_{n})$ of $h{\mathbb{Z}}^{n}$ and the forward/backward shifts $(x_{1},x_{2},\ldots,x_{j}\pm h,\ldots,x_{n})$ over $h{\mathbb{Z}}^{n}$, respectively. For a sake of readibility, one will use throughout the manuscript the tuple notations $(x+h{\bf e}_{j},t)$ and $(x-h{\bf e}_{j},t)$ to define shifts over the semidiscrete space-time lattice

$$h{\mathbb{Z}}^{n}\times\left[0,\infty\right):=\left\{(x,t)\in{\mathbb{R}}^{n+1}\leavevmode\nobreak\ :\leavevmode\nobreak\ \frac{x}{h}\in{\mathbb{Z}}^{n}\leavevmode\nobreak\ \leavevmode\nobreak\ \wedge\leavevmode\nobreak\ \leavevmode\nobreak\ t\geq 0\right\}.$$

We notice also that the Clifford algebras $C\kern-1.00006pt\ell_{0,n}$, $C\kern-1.00006pt\ell_{1,1}$ and $C\kern-1.00006pt\ell_{1,n+1}$, considered e.g. on the series of papers [2, 4, 8, 9, 10], are subalgebras of $C\kern-1.00006pt\ell_{n+1,n+1}$. For our purposes, we assume that $C\kern-1.00006pt\ell_{1,1}$ is generated by the nilpotents

$\displaystyle{\mathfrak{f}}=\frac{1}{2}({\bf e}_{2n+1}+{\bf e}_{0})$

and

$\displaystyle{\mathfrak{f}}^{\dagger}=\frac{1}{2}({\bf e}_{2n+1}-{\bf e}_{0}).$

(14)

Thereby, the graded anti-commuting relations (13) are equivalent to

$\displaystyle\begin{array}[]{lll}{\bf e}_{j}{\mathfrak{f}}+{\mathfrak{f}}{\bf e}_{j}=0,&{\bf e}_{n+j}{\mathfrak{f}}+{\mathfrak{f}}{\bf e}_{n+j}=0,&(1\leq j\leq n)\\ \\ {\bf e}_{j}{\mathfrak{f}}^{\dagger}+{\mathfrak{f}}^{\dagger}{\bf e}_{j}=0,&{\bf e}_{n+j}{\mathfrak{f}}^{\dagger}+{\mathfrak{f}}^{\dagger}{\bf e}_{n+j}=0,&(1\leq j\leq n)\\ \\ ({\mathfrak{f}})^{2}=({\mathfrak{f}}^{\dagger})^{2}=0,&{\mathfrak{f}}{\mathfrak{f}}^{\dagger}+{\mathfrak{f}}^{\dagger}{\mathfrak{f}}=1.&\end{array}$

(20)

Furthermore, based on the decomposition $C\kern-1.00006pt\ell_{n+1,n+1}=C\kern-1.00006pt\ell_{1,1}\otimes C\kern-1.00006pt\ell_{n,n}$, we will represent the Clifford-vector-valued functions $(x,t)\mapsto\Psi(x,t)$ with membership in the complexified Clifford algebra ${\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n+1,n+1}$, through the ansatz

$\displaystyle\Psi(x,t)=\Psi^{[0]}(x,t)+{\mathfrak{f}}\Psi^{[1]}(x,t)+{\mathfrak{f}}^{\dagger}\Psi^{[2]}(x,t)+{\mathfrak{f}}{\mathfrak{f}}^{\dagger}\Psi^{[3]}(x,t),$

whereby $(x,t)\mapsto\Psi^{[m]}(x,t)$ ($m=0,1,2,3$) are Clifford-vector-valued functions with membership in ${\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}$.

To introduce in the sequel function spaces and operators underlying to the ${\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n+1,n+1}-$valued functions $(x,t)\mapsto\Psi(x,t)$ and to its ${\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}-$valued components $(x,t)\mapsto\Psi^{[m]}(x,t)$ ($m=0,1,2,3$), as well as ${\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}-$valued representations of the discrete Dirac operator $D_{h}$ in terms of its Fourier multipliers (cf. [27, Subsection 21.2.2] and [39, Subsection 2.3]), one has to consider the ${\dagger}-$conjugation operation ${\bf a}\mapsto{\bf a}^{\dagger}$ on the complexified Clifford algebra ${\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n+1,n+1}$, defined recursively as follows:

$\displaystyle\begin{array}[]{lll}(\textbf{a}\textbf{b})^{\dagger}=\textbf{b}^{\dagger}\textbf{a}^{\dagger}\\ (a_{J}{\bf e}_{J})^{\dagger}=\overline{a_{J}}\leavevmode\nobreak\ {\bf e}_{j_{r}}^{\dagger}\ldots{\bf e}_{j_{2}}^{\dagger}{\bf e}_{j_{1}}^{\dagger}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ (0\leq j_{1}<j_{2}<\ldots<j_{r}\leq 2n+1)\\ {\bf e}_{j}^{\dagger}=-{\bf e}_{j}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mbox{and}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ {\bf e}_{n+1+j}^{\dagger}={\bf e}_{n+1+j}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ (0\leq j\leq n).\end{array}.$

(24)

From (24), the $\dagger-$conjugation identities

$({\mathfrak{f}}^{\dagger})^{\dagger}={\mathfrak{f}}$ and $({\mathfrak{f}}{\mathfrak{f}}^{\dagger})^{\dagger}={\mathfrak{f}}{\mathfrak{f}}^{\dagger}$,

involving the nilpotents ${\mathfrak{f}}$ and ${\mathfrak{f}}^{\dagger}$ defined viz eq. (14), are then immediate. Also, from (24) one readily has that

$$\textbf{a}\mapsto\|\textbf{a}\|:=\sqrt{\textbf{a}^{\dagger}\textbf{a}}$$

defines a $\|\cdot\|-$norm endowed by the complexified Clifford algebra structure of ${\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n+1,n+1}$, since $\textbf{a}^{\dagger}\textbf{a}=\textbf{a}\textbf{a}^{\dagger}$ is a non-negative real number. In case where a belongs to ${\mathbb{C}}\otimes{\mathbb{R}}^{n+1,n+1}$, the quantity $\|\textbf{a}\|$ equals to the standard norm of a on ${\mathbb{C}}^{2n+2}$.

Let us define by $\mathcal{S}(h{\mathbb{Z}}^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}):=\mathcal{S}(h{\mathbb{Z}}^{n})\otimes\left({\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}\right)$ the Schwartz class of ${\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}-$valued functions on the lattice $h{\mathbb{Z}}^{n}$, consisting on rapidly decaying functions $x\mapsto\varPhi(x,t)$ ($t\in[0,\infty)$) defined for any $M\in[0,\infty)$ by the semi-norm condition

$$\displaystyle\sup_{x\in h{\mathbb{Z}}^{n}}(1+\|x\|^{2})^{M}\leavevmode\nobreak\ \|\varPhi(x,t)\|<\infty,$$

and by $\ell_{2}(h{\mathbb{Z}}^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}):=\ell_{2}(h{\mathbb{Z}}^{n})\otimes\left({\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}\right)$ the right Hilbert module endowed by the Clifford-valued sesquilinear form

$\displaystyle\langle\varPhi(\cdot,t),\varPsi(\cdot,t)\rangle_{h}=\sum_{x\in h{\mathbb{Z}}^{n}}h^{n}\leavevmode\nobreak\ \varPhi(x,t)^{\dagger}\varPsi(x,t).$

(25)

By exploiting [43, Exercise 3.1.7] to the Clifford-valued setting, it is easy to check that the seminorm condition

$$\displaystyle\sup_{x\in h{\mathbb{Z}}^{n}}(1+\|x\|^{2})^{-M}\leavevmode\nobreak\ \|\varPhi(x,t)\|<\infty$$

induces the set of all continuous linear functionals with membership in the Schwarz class $\mathcal{S}(h{\mathbb{Z}}^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n})$, induced by the mapping

$$\varPhi(\cdot,t)\mapsto\langle\varPhi(\cdot,t),\varPsi(\cdot,t)\rangle_{h},$$

whereby the family of functions $\varPsi(\cdot,t):h{\mathbb{Z}}^{n}\rightarrow{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}$ ($t\in[0,\infty)$) belong to the space of ${\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}-$valued tempered distributions on the lattice $h{\mathbb{Z}}^{n}$, denoted as

$$\mathcal{S}^{\prime}(h{\mathbb{Z}}^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}):=\mathcal{S}^{\prime}(h{\mathbb{Z}}^{n})\otimes\left({\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}\right).$$

In particular, the mapping property $\varPhi(\cdot,t)\mapsto\langle\varPhi(\cdot,t),\varPsi(\cdot,t)\rangle_{h}$ together with density arguments allows us to define, for every $x\mapsto\varPhi(x,t)$ with membership in $\ell_{2}(h{\mathbb{Z}}^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n})$, a distribution $\varPhi(\cdot,t)\mapsto\langle\varPhi(\cdot,t),\varPsi(\cdot,t)\rangle_{h}$ lying to $\mathcal{S}^{\prime}(h{\mathbb{Z}}^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n})$.

Next, we denote by $\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n}$ the $n-$dimensional Brioullin zone representation for the $n-$torus ${\mathbb{R}}^{n}/\frac{2\pi}{h}{\mathbb{Z}}^{n}$ (cf. [18, p.  324]), by

$$L_{2}\left(\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}\right):=L_{2}\left(\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n}\right)\otimes\left({\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}\right)$$

the ${\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}-$Hilbert module endowed by the sesquilinear form

$\displaystyle\displaystyle\langle{\bf f}(\cdot,t),\textbf{g}(\cdot,t)\rangle_{\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n}}=\int_{\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n}}{\bf f}(\xi,t)^{\dagger}\textbf{g}(\xi,t)d\xi,$

(26)

and by $C^{\infty}\left(\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}\right)$ the space of ${\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}-$valued test functions. The discrete Fourier transform, defined by

$\displaystyle(\mathcal{F}_{h}\varPhi)(\xi,t)=\left\{\begin{array}[]{lll}\displaystyle\frac{h^{n}}{\left(2\pi\right)^{\frac{n}{2}}}\displaystyle\sum_{x\in h{\mathbb{Z}}^{n}}\varPhi(x,t)e^{ix\cdot\xi},&\xi\in\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n}\\ \\ 0&\xi\in{\mathbb{R}}^{n}\setminus\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n}\end{array}\right.$

(30)

yields the isometric isomorphism

$$\mathcal{F}_{h}:\ell_{2}(h{\mathbb{Z}}^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n})\rightarrow L_{2}\left(\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}\right),$$

whose inverse $(\mathcal{F}_{h}^{-1}\textbf{g})(x,t)=\widehat{\textbf{g}}_{h}(x,t)$ is provided by the Fourier coefficients

$\displaystyle\widehat{\textbf{g}}_{h}(x,t)=\frac{1}{(2\pi)^{\frac{n}{2}}}\int_{\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n}}(\mathcal{F}_{h}\textbf{g})(\xi,t)e^{-ix\cdot\xi}d\xi.$

(31)

Moreover, by noticicing that the sesquilinear form (26) allows us to define a mapping that identifies $C^{\infty}\left(\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}\right)$ with the dual space $C^{\infty}\left(\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}\right)^{\prime}$, the so-called space of ${\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}-$valued distributions over $\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n}$ (cf. [43, Exercise 3.1.15.] & [43, Definition 3.1.25]), we immediately get that the Parseval type relation

$\displaystyle\langle\mathcal{F}_{h}\varPhi(\cdot,t),\textbf{g}(\cdot,t)\rangle_{\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n}}=\left\langle\varPhi(\cdot,t),\widehat{\textbf{g}}_{h}(\cdot,t)\right\rangle_{h},$

(32)

involving the sesquilinear forms (25) and (26) (cf. [43, Definition 3.1.27]), allows us to extend propertly $\mathcal{F}_{h}$ (see eq.  (30)) to the setting of distributions, through the mapping property (cf. [43, Definition 3.1.27 & 3.1.28])

$$\mathcal{F}_{h}:\mathcal{S}(h{\mathbb{Z}}^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n})\rightarrow C^{\infty}\left(\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}\right),$$

in a way that the Fourier coefficients $x\mapsto\widehat{\textbf{g}}_{h}(x,t)$, defined viz eq. (31), belong to $\mathcal{S}^{\prime}(h{\mathbb{Z}}^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n})$.

In the same order of ideas of [27, Subsection 21.1.3] & [39, Subsection 2.2] (see also [44, Section 6]), one can also define the discrete convolution operation $\star_{h}$ between the discrete distribution $\varPsi(\cdot,t)\in\mathcal{S}^{\prime}(h{\mathbb{Z}}^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n})$, and the discrete function $\Phi(x)\in\mathcal{S}(h{\mathbb{Z}}^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n})$ as follows:

$\displaystyle\left(\varPsi(\cdot,t)\star_{h}\Phi\right)(x)=\sum_{y\in h{\mathbb{Z}}^{n}}h^{n}\leavevmode\nobreak\ \Phi(y)\varPsi(x-y,t).$

(33)

Indeed, from the duality condition

$\displaystyle\left\langle\leavevmode\nobreak\ \varPsi(\cdot,t)\star_{h}\Phi,\textbf{g}(\cdot,t)\leavevmode\nobreak\ \right\rangle_{h}=\langle\leavevmode\nobreak\ \varPsi(\cdot,t),\widetilde{\Phi}\star_{h}\textbf{g}(\cdot,t)\leavevmode\nobreak\ \rangle_{h},$

with

$\displaystyle\widetilde{\Phi}(x)=[\Phi(-x)]^{\dagger}$

that yields straightforwardly from the following sequence of identities:

	$\displaystyle\langle\leavevmode\nobreak\ \mathcal{F}_{h}\left[\varPsi(\cdot,t)\star_{h}\Phi\right],\textbf{g}(\cdot,t)\leavevmode\nobreak\ \rangle_{\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n}}$	$\displaystyle=$	$\displaystyle\langle\leavevmode\nobreak\ \varPsi(\cdot,t)\star_{h}\Phi,\mathcal{F}_{h}^{-1}[\textbf{g}(\cdot,t)]\leavevmode\nobreak\ \rangle_{h}$
		$\displaystyle=$	$\displaystyle\langle\leavevmode\nobreak\ \varPsi(\cdot,t),\widetilde{\Phi}\star_{h}\mathcal{F}_{h}^{-1}[\textbf{g}(\cdot,t)]\leavevmode\nobreak\ \rangle_{h}$
		$\displaystyle=$	$\displaystyle\left\langle\leavevmode\nobreak\ \varPsi(\cdot,t),\mathcal{F}_{h}^{-1}\left(\mathcal{F}_{h}\widetilde{\Phi}\leavevmode\nobreak\ \textbf{g}(\cdot,t)\right)\leavevmode\nobreak\ \right\rangle_{h}$
		$\displaystyle=$	$\displaystyle\left\langle\leavevmode\nobreak\ \mathcal{F}_{h}\varPsi(\cdot,t),\mathcal{F}_{h}\widetilde{\Phi}\leavevmode\nobreak\ \textbf{g}(\cdot,t)\leavevmode\nobreak\ \right\rangle_{h}$
		$\displaystyle=$	$\displaystyle\langle\leavevmode\nobreak\ \left(\mathcal{F}_{h}\varPsi(\cdot,t)\right)\left(\mathcal{F}_{h}\Phi\right),\textbf{g}(\cdot,t)\leavevmode\nobreak\ \rangle_{\left(-\frac{\pi}{h},\frac{\pi}{h}\right]^{n}},$

one can say the discrete convolution operation (33) is well-defined.

Let us now recall the basic setup and results from the series of papers [23, 24, 27, 25] to discuss further aspects of our construction. On the sequel, we will use the nilpotents ${\mathfrak{f}}$ and ${\mathfrak{f}}^{\dagger}$ of $C\kern-1.00006pt\ell_{1,1}$ to introduce firstly on ${\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n+1,n+1}$, with $C\kern-1.00006pt\ell_{n+1,n+1}=C\kern-1.00006pt\ell_{1,1}\otimes C\kern-1.00006pt\ell_{n,n}$, the semidiscrete Dirac type operator

$\displaystyle{\leavevmode\nobreak\ }_{\theta}\mathcal{D}_{h,t}:=D_{h}+{\mathfrak{f}}\partial_{t}+{\mathfrak{f}}^{\dagger}e^{-i\theta},$

(34)

carrying the discrete Dirac operator

$\displaystyle\begin{array}[]{lll}D_{h}\Psi(x,t)&=&\displaystyle\sum_{j=1}^{n}{\bf e}_{j}\frac{\Psi(x+h{\bf e}_{j},t)-\Psi(x-h{\bf e}_{j},t)}{2h}+\\ &+&\displaystyle\sum_{j=1}^{n}{\bf e}_{n+j}\frac{2\Psi(x,t)-\Psi(x+h{\bf e}_{j},t)-\Psi(x-h{\bf e}_{j},t)}{2h},\end{array}$

(37)

the time derivative $\partial_{t}$ and the unitary group parameter $\theta\mapsto e^{-i\theta}$.

From the factorization property $\left(D_{h}\right)^{2}=-\Delta_{h}$ involving the discrete Laplacian (cf. [23, Proposition 2.1])

$\displaystyle\displaystyle\Delta_{h}\varPsi(x,t)=\sum_{j=1}^{n}\frac{\varPsi(x+h{\bf e}_{j},t)+\varPsi(x-h{\bf e}_{j},t)-2\varPsi(x,t)}{h^{2}},$

(38)

and from the graded anti-commuting relations (20) we can easily prove that

$\displaystyle\left({\leavevmode\nobreak\ }_{\theta}\mathcal{D}_{h,t}\right)^{2}=e^{-i\theta}\partial_{t}-\Delta_{h}.$

Here, we notice that $\left({\leavevmode\nobreak\ }_{\theta}\mathcal{D}_{h,t}\right)^{2}$ is, up to the unitary term $e^{-i\theta}$, the semidiscrete heat operator considered in [37].

Accordingly to [8, 9, 11], a time-fractional counterpart of (34) can be straightforwardly obtained by replacing the time-derivative by a fractional analogue. In particular, based on the Riemann-Liouville derivative (4) we introduce the time-fractional regularization of ${\leavevmode\nobreak\ }_{\theta}\mathcal{D}_{h,t}$, defined for $\beta\geq 1$ by the time-fractional semidiscrete operator

$\displaystyle{\leavevmode\nobreak\ }_{\theta}\mathbb{D}_{h,t}^{\beta}:=D_{h}-{\mathfrak{f}}e^{i\theta(1-\beta)}\mathbb{D}_{t}^{\beta}+{\mathfrak{f}}^{\dagger}e^{-i\theta}.$

(39)

In addition, from the set of graded anti-commuting relations (20) one can also obtain the factorization property

$\displaystyle\left({\leavevmode\nobreak\ }_{\theta}\mathbb{D}_{h,t}^{\beta}\right)^{2}=-e^{-i\theta\beta}\mathbb{D}_{t}^{\beta}-\Delta_{h}$

for (39) in terms of the time-fractional semidiscrete operator $-e^{-i\theta\beta}\mathbb{D}_{t}^{\beta}-\Delta_{h}$.

By taking into account the discrete Fourier transform introduced viz eq. (30), one can obtain an equivalent formulation of ${\leavevmode\nobreak\ }_{\theta}\mathbb{D}_{h,t}^{\beta}$ resp. $-e^{-i\theta\beta}\partial_{t}-\Delta_{h}$ in terms of the Fourier multipliers (cf. [25, Section 3.3])

$\displaystyle\begin{array}[]{lll}d_{h}(\xi)^{2}&=&\displaystyle\sum_{j=1}^{n}\frac{4}{h^{2}}\sin^{2}\left(\frac{h\xi_{j}}{2}\right)\\ \\ \textbf{z}_{h}(\xi)&=&\displaystyle\sum_{j=1}^{n}-i{\bf e}_{j}\dfrac{\sin(h\xi_{j})}{h}+\sum_{j=1}^{n}{\bf e}_{n+j}\dfrac{1-\cos(h\xi_{j})}{h}\end{array}$

(43)

of $\mathcal{F}_{h}\circ(-\Delta_{h})\circ\mathcal{F}_{h}^{-1}$ and $\mathcal{F}_{h}\circ D_{h}\circ\mathcal{F}_{h}^{-1}$. Here, we emphasize that the Fourier multiplier formulation lead to the following mapping properties (cf. [27, Subsection 21.2.2])

	$\displaystyle D_{h}:\mathcal{S}(h{\mathbb{Z}}^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n})\rightarrow\mathcal{S}(h{\mathbb{Z}}^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n}),$
	$\displaystyle-\Delta_{h}:\mathcal{S}(h{\mathbb{Z}}^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n})\rightarrow\mathcal{S}(h{\mathbb{Z}}^{n};{\mathbb{C}}\otimes C\kern-1.00006pt\ell_{n,n})$

in a way that

	$\displaystyle\mathcal{F}_{h}\left[\left(-e^{-i\theta\beta}\mathbb{D}_{t}^{\beta}-\Delta_{h}\right)\Psi(x,t)\right]$	$\displaystyle=$	$\displaystyle\left(-e^{-i\theta\beta}\mathbb{D}_{t}^{\beta}+d_{h}(\xi)^{2}\right)\mathcal{F}_{h}\Psi(\xi,t),$
	$\displaystyle\mathcal{F}_{h}\left[{\leavevmode\nobreak\ }_{\theta}\mathbb{D}_{h,t}^{\beta}\Psi(x,t)\right]$	$\displaystyle=$	$\displaystyle\left(\textbf{z}_{h}(\xi)-{\mathfrak{f}}e^{i\theta(1-\beta)}\mathbb{D}_{t}^{\beta}+{\mathfrak{f}}^{\dagger}e^{-i\theta}\right)\mathcal{F}_{h}\Psi(\xi,t).$

Thereby, by mimicking [3, Theorem 4.1 & Corollary 4.2] we obtain the following theorem.