In this paper, we study the discrete fractional Schrödinger equation \(\begin{aligned} (-\Delta )^\alpha u+h(x) u=f(x,u),\quad x\in {\mathbb {Z}}^d, \end{aligned}\) 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 the variational method.
