Dispersive estimates for discrete Klein-Gordon equations on one-dimensional lattice with quasi-periodic potentials

TL;DR

The paper addresses dispersive properties of the discrete Klein–Gordon equation on $\mathbb{Z}$ with small real-analytic quasi-periodic potentials. It develops a KAM-based reducibility framework for the associated Schrödinger cocycle, constructs a modified spectral transform, and reduces the dispersive analysis to oscillatory integrals on a Cantor spectrum, achieving an $\ell^{1}\to\ell^{\infty}$ decay of order $\langle t\rangle^{-1/3}$ up to logarithmic refinements. This dispersive estimate yields Strichartz estimates and enables small-data global well-posedness for the nonlinear equation with $p>7$, highlighting the robustness of dispersion under quasi-periodic perturbations. The results advance understanding of wave propagation in quasi-crystal-like lattices and provide tools for linear and nonlinear analysis in Cantor-spectrum settings.

Abstract

We prove $\ell^{1}\!\to\!\ell^{\infty}$ dispersive estimates for the discrete Klein--Gordon equation on $\mathbb Z$ with small real-analytic quasi-periodic potentials, showing that the time-decay rate persists as $(\tfrac13)^{-}$. As applications, we derive the corresponding Strichartz estimates and establish small-data global well-posedness for the associated nonlinear discrete Klein--Gordon equation.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1: Proposition 1 of Z16
• Proposition 2.2: Proposition 2 of Z16
• Proposition 2.3: Section 4.2 in Z16
• Lemma 3.1
• proof : Proof of Theorem \ref{['thm:disper']} assuming Lemma \ref{['lem:disper-log']}
• Lemma 3.2
• proof