Strong Convergence for Discrete Nonlinear Schr\"odinger equations in the Continuum Limit

We consider discrete nonlinear Schr\"odinger equations (DNLS) on the lattice $h\mathbb{Z}^d$ whose linear part is determined by the discrete Laplacian which accounts only for nearest neighbor interactions, or by its fractional power. We show that in the continuum limit $h\to 0$, solutions to DNLS converge strongly in $L^2$ to those to the corresponding continuum equations, but a precise rate of convergence is also calculated. In particular cases, this result improves weak convergence in Kirkpatrick, Lenzmann and Staffilani \cite{KLS}. Our proof is based on a suitable adjustment of dispersive PDE techniques to a discrete setting. Notably, we employ uniform-in-$h$ Strichartz estimates for discrete linear Schr\"odinger equations in \cite{HY}, which quantitatively measure dispersive phenomena on the lattice. Our approach could be adapted to a more general setting like \cite{KLS} as long as the desired Strichartz estimates are obtained.