  In this paper, we study the discrete fractional Schr\"{o}dinger equation $$
(-\Delta)^\alpha u+h(x) u=f(x,u),\quad x\in \mathbb{Z}^d,$$
  where $d\in\mathbb{N}^*,\,\alpha \in(0, 1)$ and the nonlocal operator
$(-\Delta)^\alpha $ is defined by discrete Fourier transform, which differs
from the continuous case. Under suitable assumptions on $h$ and $f$, we prove
the existence and multiplicity of solutions to this equation by variational
method.

Analysis of PDEs