  The fractional Laplacian operator, $-(-\triangle)^{\frac{\alpha}{2}}$,
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 eigenvalues spectrum are also obtained.
