Optimal dispersion for discrete periodic Schrödinger operators

TL;DR

This work establishes an optimal dispersive decay rate for time evolution under the periodic discrete Schrödinger operator on the line, showing $\|e^{-itH_V}\|_{\ell^1\to\ell^\infty} \lesssim \langle t\rangle^{-1/3}$ when $V$ is $p$-periodic. The authors achieve this via a detailed Floquet/Marchenko–Ostrovski analysis, leveraging the direct integral decomposition and precise control of band-function derivatives through the MO mapping and van der Corput-type estimates. The results extend to discrete nonlinear Schrödinger dynamics with small initial data and suitable nonlinearity, yielding $\ell^\infty$ decay at the same rate for $\sigma>5$, and improve prior periodic-disorder bounds by attaining the best-known free-rate decay for general period $p$. The methods illuminate how nondegeneracy of band-structure derivatives under periodicity drives dispersion, with potential implications for tight-binding models, optical lattices, and nonlinear lattice dynamics.

Abstract

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

Theorems & Definitions (12)

• Theorem 1.1
• Remark 1.2
• Corollary 1.3
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['thm:periodicdispersion']}