Continuum limit for discrete NLS with memory effect

We consider a discrete nonlinear Schr\"odinger equation with long-range interactions and a memory effect on the infinite lattice $h\Z$ with mesh-size $h>0$. Such models are common in the study of charge and energy transport in biomolecules. Given that the distance between base pairs is small, we consider the continuum limit: a sharp approximation to the system as $h\rightarrow 0$. In this limit, we prove that solutions to this discrete equation converge strongly in $L^2$ to the solution to a continuous NLS-type equation with a memory effect, and we compute the precise rate of convergence. In order to obtain these results, we generalize some recent ideas proposed by Hong and Yang in $L^2$-based spaces to classical functional settings in dispersive PDEs involving the smoothing effect and maximal function estimates, as originally introduced in the pioneering works of Kenig, Ponce and Vega. We believe that our approach may therefore be adapted to tackle continuum limits of more general dispersive equations.