  We consider a general class of discrete nonlinear Schroedinger equations
(DNLS) on the lattice $h \mathbb{Z}$ with mesh size $h>0$. In the continuum
limit when $h \to 0$, we prove that the limiting dynamics are given by a
nonlinear Schroedinger equation (NLS) on $\mathbb{R}$ with the fractional
Laplacian $(-\Delta)^\alpha$ as dispersive symbol. In particular, we obtain
that fractional powers $1/2 < \alpha < 1$ arise from long-range lattice
interactions when passing to the continuum limit, whereas NLS with the
non-fractional Laplacian $-\Delta$ describes the dispersion in the continuum
limit for short-range lattice interactions (e.g., nearest-neighbor
interactions).
  Our results rigorously justify certain NLS model equations with fractional
Laplacians proposed in the physics literature. Moreover, the arguments given in
our paper can be also applied to discuss the continuum limit for other lattice
systems with long-range interactions.
