Fractional Laplacian matrix on the finite periodic linear chain and its periodic Riesz fractional derivative continuum limit

International audience; The 1D discrete fractional Laplacian operator on a cyclically closed (periodic) linear chain with finite number $N$ of identical particles is introduced. We suggest a ''fractional elastic harmonic potential", and obtain the $N$-periodic fractional Laplacian operator in the form of a power law matrix function for the finite chain ($N$ arbitrary not necessarily large) in explicit form. In the limiting case $N\rightarrow \infty$ this fractional Laplacian matrix recovers the fractional Laplacian matrix of the infinite chain.The lattice model contains two free material constants, the particle mass $\mu$ and a frequency $\Omega_{\alpha}$.The ''periodic string continuum limit" of the fractional lattice model is analyzed where lattice constant $h\rightarrow 0$ and length $L=Nh$ of the chain (''string") is kept finite: Assuming finiteness of the total mass and total elastic energy of the chain in the continuum limit leads to asymptotic scaling behavior for $h\rightarrow 0$ of the two material constants, namely $\mu \sim h$ and $\Omega_{\alpha}^2 \sim h^{-\alpha}$. In this way we obtain the $L$-periodic fractional Laplacian (Riesz fractional derivative) kernel in explicit form. This $L$-periodic fractional Laplacian kernel recovers for $L\rightarrow\infty$ the well known 1D infinite space fractional Laplacian (Riesz fractional derivative) kernel. When the scaling exponent of the Laplacian takes integers, the fractional Laplacian kernel recovers, respectively, $L$-periodic and infinite space (localized) distributional representations of integer-order Laplacians.The results of this paper appear to be useful for the analysis of fractional finite domain problems for instance in anomalous diffusion (Levy flights), fractional Quantum Mechanics, and the development of fractional discrete calculus on finite lattices.