Continuum limit of fourth-order Schr\"{o}dinger equations on the lattice

TL;DR

This paper establishes uniform Strichartz estimates for the discrete fourth-order Schrödinger equation on a lattice and determines the rate at which solutions converge to their continuum counterparts as the lattice spacing approaches zero.

Contribution

It provides the first rigorous analysis of the continuum limit for fourth-order Schrödinger equations on lattices, including precise convergence rates.

Findings

Established uniform Strichartz estimates for the lattice equation

Derived the rate of $L^2$ convergence to the continuum equation

Analyzed frequency localized oscillatory integrals using stationary phase

Abstract

In this paper, we consider the discrete fourth-order Schr\"{o}dinger 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 $h \rightarrow 0$.