Fractional Laplacian in bounded domains 论文

摘要

The fractional Laplacian operator $\ensuremath{-}{(\ensuremath{-}\ensuremath{\Delta})}^{\ensuremath{\alpha}∕2}$ appears in a wide class of physical systems, including L\'evy flights and stochastic interfaces. In this paper, we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval. The implementation of boundary conditions is justified by appealing to two physical models, namely, hopping particles and elastic springs. The eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions. Some analytical results concerning the structure of the eigenvalue spectrum are also obtained.