Continuum limit of fourth-order Schrödinger equations on the lattice

In this paper, we consider the discrete fourth-order Schrödinger equation on the lattice $h{\mathbb{Z}}^2$. Uniform Strichartz estimates are established by analyzing frequency localized oscillatory integrals with the method of stationary phase and applying Littlewood-Paley inequalities. As an application, we obtain the precise rate of $L^2$ convergence from the solutions of discrete semilinear equations to those of the corresponding equations on the Euclidean plane ${\mathbb{R}}^2$ in the contimuum limit $h\to 0$.