# Fractional Laplacian in Bounded Domains

### A. Zoia 1, A. Rosso 2, 3, M. Kardar 3

#### Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 76 (2007) 021116

The fractional Laplacian operator, $-(-\triangle)^{\frac{\alpha}{2}}$, appears in a wide class of physical systems, including Lévy 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 eigenvalues spectrum are also obtained.

• 1. Department of Nuclear Engineering,,
Polytechnic of Milan
• 2. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
CNRS : UMR8626 – Université Paris XI - Paris Sud
• 3. Department of Physics Massachusetts Institute of Technology,
Massachusetts Institute of Technology