A series representation of the discrete fractional Laplace operator of arbitrary order

Abstract

Although fractional powers of non-negative operators have received much attention in recent years, there is still little known about their behavior if real-valued exponents are greater than one. In this article, we define and study the discrete fractional Laplace operator of arbitrary real-valued positive order. A series representation of the discrete fractional Laplace operator for positive non-integer powers is developed. Its convergence to a series representation of a known case of positive integer powers is proven as the power tends to the integer value. Furthermore, we show that the new representation for arbitrary real-valued positive powers of the discrete Laplace operator is consistent with existing theoretical results.

Introduction

Due to its wide array of applications in multi-physical sciences, the construction and approximation of fractional powers of the Laplace operator have been of great interest for nearly a century (cf., e.g., [8], [40], [44], [55], [63] and references therein). Conventionally, only powers of the order $s\in (0,1)$ are considered, and in this case, one may define the fractional Laplace operator applied to a smooth enough function in a natural way. Specifically, for $d\in \mathbb{N}=\{1,2,3,\dots \}$, $s\in (0,1)$ let $u:{\mathbb{R}}^d\to \mathbb{R}$ be a smooth function, and for every $\epsilon \in (0,\infty )$, $x\in {\mathbb{R}}^d$ let $B_{\epsilon }(x)$ be the d-dimensional ball of radius ε centered at x (with respect to the typical topology of ${\mathbb{R}}^d$). Then for every $x\in {\mathbb{R}}^d$ we can define the s-order fractional Laplace operator applied to u at x as

$$\left({(-\mathrm{\Delta })}^{s}u\right)(x)=c_{d,s}\underset{\epsilon \to 0^{+}}{\lim }\left[\underset{{\mathbb{R}}^d﹨B_{\epsilon }(x)}{\int }\frac{u(x)-u(y)}{|x-y{|}^{d+2s}}dy\right],$$

where $c_{d,s}\in [0,\infty )$ is a known normalization constant.

It is worth noting that the recent rapid increase in interest in the fractional Laplace operator is also due to the seminal work of Caffarelli and Silvestre [9]. In their work, it was shown that one may study the non-local operator given by (1.1) via the Dirichlet-to-Neumann operator associated with a particular extension problem posed in ${\mathbb{R}}^d\times [0,\infty )$ (albeit, one trades the non-locality for a problem posed in a higher dimension which is either singular or degenerate depending upon the value of $s\in (0,1)$). The employed Dirichlet-to-Neumann operator is a particular example of the Poincaré-Stecklov operator (cf., e.g., [38]). For a fixed domain, the Poincaré-Stecklov operator is known to map the boundary values of a harmonic function to the normal derivative values of the same harmonic function on the same boundary. We can summarize the results of Caffarelli and Silvestre (cf., e.g., [9, Eq. (3.1)]) as follows. Let $d\in \mathbb{N}$, $s\in (0,1)$, let $u:{\mathbb{R}}^d\to \mathbb{R}$ be a smooth function, and let $v:[0,\infty )\times {\mathbb{R}}^d\to \mathbb{R}$ satisfy for all $x\in {\mathbb{R}}^d$ that $v(0,x)=u(x)$ and for all $t\in (0,\infty )$, $x\in {\mathbb{R}}^d$ that

$$\left(\frac{{\partial }^2}{\partial t^2}v\right)(t,x)+\frac{1-2s}{t}\left(\frac{\partial }{\partial t}v\right)(t,x)+({\mathrm{\Delta }}_{x}v)(t,x)=0.$$

Then there exists $c\in [0,\infty )$ such that for all $x\in {\mathbb{R}}^d$ it holds that

$$\left({(-\mathrm{\Delta })}^{s}u\right)(x)=c[\underset{t\to 0^{+}}{\lim }{t}^{1-2s}\left(\frac{\partial }{\partial t}v\right)(t,x)].$$

Interestingly, the constant $c\in [0,\infty )$ in (1.3) depends only upon the parameter $s\in (0,1)$ and not upon $d\in \mathbb{N}$. More importantly, this demonstrates that one may trade out the highly non-local problem given by (1.1) for the local problem given by (1.2) and (1.3). This technique has also been recently further generalized to cases of arbitrary non-negative operators defined on Banach spaces [5], [21], [45], [46], [52].

While the above formulations (i.e., (1.1), (1.2), and (1.3)) may be utilized to provide insights into a continuous fractional Laplace operator with order $s\in (0,1)$, they cannot be generalized to provide any insight into the discrete case or the case where $s\in (0,\infty )$. The discrete case is a natural consideration as it arises in the study of numerous physically relevant phenomena (cf., e.g., [31], [32], [53] and references therein) and also in an attempt to numerically approximate (1.1). The consideration of a truly discrete case—that is, the case which is the fractional power of the discrete Laplace operator rather than a direct approximation of (1.1)—was originally studied by Ciaurri et al. [13]. By employing the basic language of semigroups (e.g., a special case of Ciaurri et al. [12, Eq. (1)] combined with, e.g., Padgett [52, Theorem 2.1]) Ciaurri et al. were able to develop the first series representation for the discrete fractional Laplace operator of order $s\in (0,1)$ (cf. Definition 4.10, for clarity). Moreover, it was shown that this formulation did converge to the continuous case via adaptive mesh refinements (cf. Ciaurri et al. [12, Theorems 1.7 and 1.8]). However, it is important to note that while this aforementioned convergence was observed, it is the case that the series representation developed by Ciaurri et al. is an exact representation and not a numerical approximation.

The consideration of higher-order fractional Laplace operators has recently received increased attention in continuous cases (cf., e.g., [11], [19], [22], [56], [64]). But to the authors' knowledge, the only study in discrete cases has been carried out by Padgett et al. [53]. Rectifying this aforementioned gap in theory is the primary goal of this article (although the applicability of such derivations in the study of localization will be outlined in Section 2 below). To this end, a series representation of the discrete fractional Laplace operator of order $s\in (0,\infty )$ is implemented. This development is illustrated through Theorem 1.1, which is also a partial description of the main result of this article focused on the case of positive non-integer powers of the discrete Laplace operator.

Theorem 1.1

Let $m\in \mathbb{N}$, $s\in (m-1,m)$, let $\mathbb{Z}=\{\dots ,-2,-1,0,1,2,\dots \}$, let $\mathbb{R}$ be the real number field, let ${\ell }^2(\mathbb{Z})$ be the set of all $w:\mathbb{Z}\to \mathbb{R}$ which satisfy that ${\sum }_{k\in \mathbb{Z}}|w(k){|}^2<\infty$, let $-\mathit{\Delta }:{\ell }^2(\mathbb{Z})\to {\ell }^2(\mathbb{Z})$ satisfy for all $w\in {\ell }^2(\mathbb{Z})$, $n\in \mathbb{Z}$ that $(-\mathit{\Delta }w)(n)=2w(n)-w(n-1)-w(n+1)$, let $u\in {\ell }^2(\mathbb{Z})$, and let¹ $v:[0,\infty )\times \mathbb{Z}\to \mathbb{R}$ satisfy for all $t\in (0,\infty )$, $n\in \mathbb{Z}$ that $v(0,n)=({(-\mathit{\Delta })}^{m-1}u)(n)$ and

$$\left(\frac{{\partial }^2}{\partial t^2}v\right)(t,x)+\frac{1-2(s-m+1)}{t}\left(\frac{\partial }{\partial t}v\right)(t,x)+\left(\mathit{\Delta }v\right)(t,x)=0.$$

Then

• there exists $c\in [0,\infty )$ such that for all $n\in \mathbb{Z}$ it holds that

  $$\left({(-\mathit{\Delta })}^{s-m+1}{(-\mathit{\Delta })}^{m-1}u\right)(n)=\left({(-\mathit{\Delta })}^{s}u\right)(n)=c[\underset{t\to 0^{+}}{\lim }{t}^{1-2(s-m+1)}\left(\frac{\partial }{\partial t}v\right)(t,n)]$$

  and

• there exists $K:\mathbb{Z}\to \mathbb{R}$, $C\in [0,\infty )$ such that for all $n\in \mathbb{Z}﹨\{0\}$ it holds that $|K(n)|\le C|n{|}^{-(1+2s)}$ and for all $n\in \mathbb{Z}$ it holds that $K(-n)=K(n)$ and

  $$\left({(-\mathit{\Delta })}^{s}u\right)(n)=\sum _{k\in \mathbb{Z}}K(k)(u(n)-u(n-k)).$$

We now provide some clarifying remarks regarding the objects in Theorem 1.1. In Theorem 1.1 we intend to construct an exact series representation of the so-called co-normal derivative of the function $v(0,\cdot ):\mathbb{Z}\to \mathbb{R}$. The positive real number $s\in (0,\infty )$ describes the fractional power of the discrete Laplace operator, the positive integer m describes the smallest positive integer that is greater than or equal to $s\in (0,\infty )$, and the set ${\ell }^2(\mathbb{Z})$ is the standard Hilbert space of square-summable sequences defined on the integers. The operator $-\mathrm{\Delta }:{\ell }^2(\mathbb{Z})\to {\ell }^2(\mathbb{Z})$ is the standard one-dimensional discrete Laplace operator and is the primary object used in the construction of the desired series representation. The function $v:[0,\infty )\times \mathbb{Z}\to \mathbb{R}$ is the solution to the extension problem in (1.4) and the trace of this function coincides with the given square-summable function $u:\mathbb{Z}\to \mathbb{R}$ to which we are applying the discrete fractional Laplace operator of order $s\in (0,\infty )$.

Let us now provide some clarifying remarks regarding the results in Theorem 1.1. Item (i) of Theorem 1.1 above is a direct consequence of combining Definition 4.11 and Padgett [52, Theorem 2.1] (applied for every $n\in \mathbb{Z}$ with $s\curvearrowleft s-m+1$, $A\curvearrowleft \mathrm{\Delta }$, $u_0\curvearrowleft ({(-\mathrm{\Delta })}^{m-1}u)(n)$, ${(u(t))}_{t\in [0,\infty )}\curvearrowleft {(v(t,n))}_{t\in [0,\infty )}$ in the notation of Padgett [52, Theorem 2.1]). See the beginning of Section 3 for an explanation of this “applied with” notation (i.e., the symbol “↶”). The right-hand side of (1.5) is not considered in detail, herein, as it is an elementary consequence of, e.g., Padgett [52, Theorem 2.1]. Item (ii) of Theorem 1.1 follows directly from Lemma 5.4 and Lemma 6.1.

The main result of this article is Theorem 6.4 in Section 6 below. This result provides a complete description of the series representation of the discrete fractional Laplace operator of order $s\in (0,\infty )$. The most surprising implication of Theorem 6.4 is that the formula for the function $K:\mathbb{Z}\to \mathbb{R}$ in Theorem 1.1 depends only on the parameter $s\in (0,\infty )$ (cf. Definition 5.1 below). In fact, this function is continuous with respect to the parameter s for all $s\in (0,\infty )﹨\mathbb{N}$ with the points $s\in \mathbb{N}$ all being removable singularities of the function $K:\mathbb{Z}\to \mathbb{R}$. Hence, we may extend the definition of $K:\mathbb{Z}\to \mathbb{R}$ to that of an analytic function (cf. (6.20) of Theorem 6.4).

The remainder of this article is organized as follows. In Section 2 we briefly motivate our interest in the development of a series representation for the discrete fractional Laplace operator of arbitrary order. In particular, we focus on its application to the study of the Anderson localization problem in materials science and its application to transport problems in plasma physics. Next, in Section 3 we recall several basic definitions and properties of sequence spaces and introduce the so-called logarithmic norm. Afterwards, in Section 4 we define the discrete Laplace operator of arbitrary real-valued positive order. We do so by introducing the heat semigroup generated by the discrete Laplace operator and then defining higher-order powers via induction. In Section 5 we define a discrete fractional kernel function and provide a detailed investigation of its various quantitative and qualitative properties. Thereafter, in Section 6 we construct a series representation for real-valued positive powers of the discrete fractional Laplace operator by employing the results from Sections 4 and 5. Finally, in Section 7, a number of useful concluding remarks are provided. Continuing avenues of research based on the results developed in this article are outlined.