Fractional Discrete Laplacians

• Fractional discrete Laplacians are nonlocal operators that extend classical Laplacians to discrete frameworks such as lattices and graphs, exhibiting scale-free behavior.
• They are modeled using spectral calculus and explicit kernel formulas that capture algebraically decaying tails and self-adjoint properties.
• Applications span harmonic analysis, quantum mechanics, and numerical approximations where discretizations converge to continuous Riesz fractional Laplacians.

A fractional discrete Laplacian is a nonlocal operator extending the concept of fractional Laplacians from continuous domains to discrete settings such as lattices, periodic chains, or general graphs. These operators form the cornerstone for diverse research fields including analysis on graphs, nonlocal PDEs, harmonic analysis, fractional quantum mechanics, and discrete modeling of anomalous diffusion. Fractional discrete Laplacians interpolate between the identity and local (integer-order) Laplacians and capture the signature nonlocal, scale-free behavior characteristic of their continuous counterparts.

1. Algebraic and Spectral Definitions

On the uniform lattice $\mathbb{Z}_h = h\mathbb{Z}$, the standard discrete Laplacian, $\Delta_h u_j = (u_{j+1} - 2u_j + u_{j-1})/h^2$, admits fractional powers defined spectrally via the semigroup: $$(-\Delta_h)^s u_j = \frac{1}{\Gamma(-s)} \int_0^\infty (e^{t \Delta_h} u_j - u_j)\, t^{-1-s}\, dt, \quad 0 < s < 1.$$ On finite or infinite graphs, the operator is analogously defined by spectral calculus in terms of the Laplacian's (generalized) eigenpairs; for a finite graph $(V,E)$ with Laplacian $-\Delta$, the spectral definition is

$$(-\Delta)^s f = \sum_{k=0}^{n-1} \lambda_k^s \langle f, \phi_k\rangle \phi_k,$$

where $\{\phi_k\}$ and $\{\lambda_k\}$ are the eigenfunctions and eigenvalues of $-\Delta$ (Zhang et al., 2024). The heat semigroup formula generalizes to weighted graphs and locally finite graphs: $$(-\Delta)^s u(x) = \frac{1}{\Gamma(-s)} \int_0^\infty (e^{t\Delta}u(x) - u(x)) t^{-1-s} dt,$$ where $\Delta_h u_j = (u_{j+1} - 2u_j + u_{j-1})/h^2$0 is the graph heat semigroup (Zhang et al., 2024).

For translation-invariant cases, the Fourier representation yields the symbol $\Delta_h u_j = (u_{j+1} - 2u_j + u_{j-1})/h^2$1 (Ciaurri et al., 2015). In $\Delta_h u_j = (u_{j+1} - 2u_j + u_{j-1})/h^2$2 with anisotropic orders $\Delta_h u_j = (u_{j+1} - 2u_j + u_{j-1})/h^2$3, the operator is most naturally viewed as

$\Delta_h u_j = (u_{j+1} - 2u_j + u_{j-1})/h^2$4

where each $\Delta_h u_j = (u_{j+1} - 2u_j + u_{j-1})/h^2$5 is defined by the spectral calculus for the shift-invariant 1D Laplacian acting along $\Delta_h u_j = (u_{j+1} - 2u_j + u_{j-1})/h^2$6 (Athmouni, 12 Oct 2025, Athmouni, 7 Sep 2025).

2. Explicit Kernel Formulas and Nonlocal Series Representations

Fractional discrete Laplacians admit pointwise, global (nonlocal) series representations. In one dimension,

$\Delta_h u_j = (u_{j+1} - 2u_j + u_{j-1})/h^2$7

where the kernel $\Delta_h u_j = (u_{j+1} - 2u_j + u_{j-1})/h^2$8 has explicit expressions: $\Delta_h u_j = (u_{j+1} - 2u_j + u_{j-1})/h^2$9 This kernel decays as $$(-\Delta_h)^s u_j = \frac{1}{\Gamma(-s)} \int_0^\infty (e^{t \Delta_h} u_j - u_j)\, t^{-1-s}\, dt, \quad 0 < s < 1.$$0, so the operator is genuinely nonlocal but with algebraically decaying tails (Ciaurri et al., 2016, Ciaurri et al., 2015, Jones et al., 2021).

On general graphs or weighted graphs, the kernel $$(-\Delta_h)^s u_j = \frac{1}{\Gamma(-s)} \int_0^\infty (e^{t \Delta_h} u_j - u_j)\, t^{-1-s}\, dt, \quad 0 < s < 1.$$1 is built from the heat kernel $$(-\Delta_h)^s u_j = \frac{1}{\Gamma(-s)} \int_0^\infty (e^{t \Delta_h} u_j - u_j)\, t^{-1-s}\, dt, \quad 0 < s < 1.$$2 via

$$(-\Delta_h)^s u_j = \frac{1}{\Gamma(-s)} \int_0^\infty (e^{t \Delta_h} u_j - u_j)\, t^{-1-s}\, dt, \quad 0 < s < 1.$$3

and the operator writes as a discrete “Riesz-type” integral: $$(-\Delta_h)^s u_j = \frac{1}{\Gamma(-s)} \int_0^\infty (e^{t \Delta_h} u_j - u_j)\, t^{-1-s}\, dt, \quad 0 < s < 1.$$4 Such symmetric, positive, rapidly decaying kernels underpin the nonlocal character of discrete fractional Laplacians on arbitrary (possibly infinite) graphs (Zhang et al., 2024).

For the periodic chain of length $$(-\Delta_h)^s u_j = \frac{1}{\Gamma(-s)} \int_0^\infty (e^{t \Delta_h} u_j - u_j)\, t^{-1-s}\, dt, \quad 0 < s < 1.$$5, the operator is a matrix function of the Born–von Karman Laplacian, with explicit entries using generalized binomial coefficients: $$(-\Delta_h)^s u_j = \frac{1}{\Gamma(-s)} \int_0^\infty (e^{t \Delta_h} u_j - u_j)\, t^{-1-s}\, dt, \quad 0 < s < 1.$$6

3. Analytical Properties and Limiting Cases

Fractional discrete Laplacians interpolate continuously between the identity and the standard Laplacian:

• $$(-\Delta_h)^s u_j = \frac{1}{\Gamma(-s)} \int_0^\infty (e^{t \Delta_h} u_j - u_j)\, t^{-1-s}\, dt, \quad 0 < s < 1.$$7
• $$(-\Delta_h)^s u_j = \frac{1}{\Gamma(-s)} \int_0^\infty (e^{t \Delta_h} u_j - u_j)\, t^{-1-s}\, dt, \quad 0 < s < 1.$$8 on zero-mean functions, for finite graphs (Zhang et al., 2024)
• For even-integer $$(-\Delta_h)^s u_j = \frac{1}{\Gamma(-s)} \int_0^\infty (e^{t \Delta_h} u_j - u_j)\, t^{-1-s}\, dt, \quad 0 < s < 1.$$9 in chains, the kernel recovers the $(V,E)$0-point central difference formula (Michelitsch et al., 2014, Michelitsch et al., 2015).
• As $(V,E)$1 (continuum limit), $(V,E)$2 converges to the continuous space Riesz fractional Laplacian, both in the pointwise and spectral sense (Ciaurri et al., 2016, Michelitsch et al., 2014).

Spectral decompositions and kernel representations reveal that discrete fractional Laplacians are self-adjoint on the $(V,E)$3 space, have absolutely continuous spectrum in the infinite lattice setting, and inherit the nonlocality and scaling properties of their continuous counterparts.

4. Boundary Conditions, Compression, and Operator Extension

For bounded domains or compressed settings, forms of the fractional discrete Laplacian are defined by restriction/projection:

• The half-lattice discrete fractional Laplacian is given by

$(V,E)$4

where $(V,E)$5 is the projection to the half-lattice and $(V,E)$6 is a compact "boundary correction," ensuring that the essential spectrum matches that of the infinite lattice operator and that interior spectral/dynamical results transfer between full- and half-lattice settings (Athmouni, 12 Oct 2025).

Generalizations to finite graphs employ the spectral functional calculus; for graphs with boundary conditions, Dirichlet, Neumann, and Robin extensions have been systematically discretized via finite element and quadrature-based methods (Cusimano et al., 2017). For negative fractional powers (fractional integrals), similar constructions yield discrete versions of Riesz potentials, preserving mapping properties and Hardy–Littlewood–Sobolev inequalities (Ciaurri et al., 2016, Li et al., 1 Mar 2026).

5. Analytic, Probabilistic, and Variational Structures

Discrete fractional Laplacians on graphs generate regular Dirichlet forms: $(V,E)$7 which serve as discrete energy functionals and underlie variational approaches to nonlocal equations. The fractional discrete Laplacian is the Dirichlet-to-Neumann map for a semidiscrete extension problem, mirroring the Caffarelli–Silvestre extension in the continuum (Ciaurri et al., 2016).

s-Harmonic functions satisfy a nonlocal mean-value property, which has a (fractional) random walk interpretation: the probability to "jump" is proportional to $(V,E)$8 for large $(V,E)$9 (Ciaurri et al., 2016). On graphs, fractional Sobolev spaces $-\Delta$0 are defined, yielding embeddings and variational principles for related nonlinear equations, such as the discrete fractional Schrödinger equation (Zhang et al., 2024).

For general weighted graphs, the structure of positive-critical Hardy weights for fractional Laplacians has been characterized, allowing for optimal Hardy inequalities and sharp control of ground-state behaviors, with explicit asymptotics on Cayley, curvature, and fractal graphs (Hake et al., 19 May 2026).

6. Spectral Theory, Scattering, and Propagation Estimates

On hypercubic lattices, the anisotropic fractional discrete Laplacian’s spectral theory, Mourre estimates, and threshold analysis are developed in detail:

• The spectrum is determined by the sum of powers of the discrete Laplacians along each axis, with thresholds at critical values of the group velocity.
• A strict Mourre estimate is established away from thresholds, yielding a Limiting Absorption Principle (LAP), absence of singular continuous spectrum, and finiteness of eigenvalues on spectral windows (Athmouni, 7 Sep 2025, Athmouni, 12 Oct 2025).
• Scattering theory for long-range perturbations involves stationary representations and Birman–Krein formulae for the scattering matrix, ensuring completeness and spectral-shift function representation (Athmouni, 7 Sep 2025).

Propagation and time-decay estimates (weighted and minimal velocity bounds) have been derived for fractional discrete Laplacians, matching sharp results from the continuum theory.

7. Numerical Discretization, Algorithms, and Applications

Multiple high-accuracy and structure-preserving discretizations for fractional Laplacians and their nonlinear variants have been constructed:

• Finite-difference schemes (spectral, regularized, tent, quadratic) yield discretizations with explicit positive weights, controlled accuracy, and convergent to the correct continuum limit (Huang et al., 2016, Huang et al., 2013).
• Factorization methods and Toeplitz structures enable efficient multidimensional matrix-vector operations via FFT (Wu et al., 2021, Minden et al., 2018). Monte Carlo and fast preconditioned solvers are effective for large-scale elliptic and parabolic problems.
• Discretizations on general bounded domains are resolved using heat-semigroup quadrature and finite element temporal-spatial discretization, with proven convergence for spectral and integral fractional operators under Dirichlet, Neumann, or Robin boundary conditions (Cusimano et al., 2017).

Operator and kernel regularity, mean-value formulas, and tail corrections are central to precise numerical treatment in practice. The stability, maximal principle, and convergence are well-understood for the schemes constructed using semigroup or quadrature-based frameworks (Chowdhury et al., 2024).

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References:

• (Michelitsch et al., 2014, Michelitsch et al., 2015, Ciaurri et al., 2015, Ciaurri et al., 2016, Huang et al., 2016, Cusimano et al., 2017, Minden et al., 2018, Jones et al., 2021, Wu et al., 2021, Chowdhury et al., 2024, Zhang et al., 2024, Zhang et al., 2024, Athmouni, 7 Sep 2025, Athmouni, 12 Oct 2025, Li et al., 1 Mar 2026, Hake et al., 19 May 2026).