Dispersive estimates for quantum walks on 1D lattice

TL;DR

This paper establishes dispersive decay estimates for quantum walks on a 1D lattice with position-dependent coins, extending techniques from Schr"odinger operator theory to quantum walk dynamics.

Contribution

It provides the first dispersive estimates for quantum walks with position-dependent coins, using oscillatory integral techniques similar to those in Schr"odinger operator analysis.

Findings

Dispersive estimate U^tP_c u_0_{l^} \, ext{decays as } (1+|t|)^{-1/3}

Results hold under specific perturbation conditions (l^{1,1} and l^{1,2})

Method relies on oscillatory integral estimates via Jost solutions

Abstract

We consider quantum walks with position dependent coin on 1D lattice $\mathbb{Z}$. The dispersive estimate $\|U^tP_c u_0\|_{l^\infty}\lesssim (1+|t|)^{-1/3} \|u_0\|_{l^1}$ is shown under $l^{1,1}$ perturbation for the generic case and $l^{1,2}$ perturbation for the exceptional case, where $U$ is the evolution operator of a quantum walk and $P_c$ is the projection to the continuous spectrum. This is an analogous result for Schr\"odinger operators and discrete Schr\"odinger operators. The proof is based on the estimate of oscillatory integrals expressed by Jost solutions.