Solutions to discrete fractional Schödinger equations

In this paper, we study the discrete fractional Schrödinger equation $$(-\mathrm{\Delta }{)}^{\alpha }u+h(x)u=f(x,u),x\in {\mathbb{Z}}^d,$$ where $d\in {\mathbb{N}}^{\ast },\alpha \in (0,1)$ and the nonlocal operator $(-\mathrm{\Delta }{)}^{\alpha }$ 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.