Optimal dispersion for discrete periodic Schr\"odinger operators

TL;DR

This paper establishes optimal dispersive decay estimates for periodic discrete Schr"odinger operators and extends these results to the nonlinear case with small initial data, enhancing understanding of wave dispersion in periodic quantum systems.

Contribution

It provides the first proof of optimal dispersive decay rates for periodic discrete Schr"odinger operators and applies these results to nonlinear equations.

Findings

Proved optimal dispersive decay estimates for periodic discrete Schr"odinger operators.

Extended dispersive estimates to the discrete nonlinear Schr"odinger equation with small initial data.

Demonstrated the applicability of linear results to nonlinear wave evolution in periodic settings.

Abstract

We prove a dispersive estimate for periodic discrete Schr\"odinger operators on the line with optimal rate of decay. Additionally, by standard methods, we deduce dispersive estimates for the discrete nonlinear Schr\"odinger equation with small initial data and suitable nonlinearity when the underlying Hamiltonian is periodic.