Dispersive estimates for quantum walks on 1D lattice

Abstract

We consider quantum walks with position dependent coin on 1D lattice $\mathbb{Z}$. The dispersive estimate $\| U^{t} P_{c} u_0\|_{l^{\infty}} \lesssim (1+|t|)^{-1/3} \|u_0\|_{l^1}$ 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 Schrödinger operators and discrete Schrödinger operators. The proof is based on the estimate of oscillatory integrals expressed by Jost solutions.