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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 thegeneric case and $l^{1,2}$ perturbation for the exceptional case, where $U$ isthe evolution operator of a quantum walk and $P_c$ is the projection to thecontinuous spectrum. This is an analogous result for Schr\"odinger operatorsand discrete Schr\"odinger operators. The proof is based on the estimate ofoscillatory integrals expressed by Jost solutions.

Dispersive estimates for quantum walks on 1D lattice