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.
Funding Statement
The first author was supported by the JSPS KAKENHI Grant Numbers JP15K17568, JP17H02851, JP17H02853, 19K03579 and G19KK0066A. The second author was supported by JSPS KAKENHI Grant Number JP17K05311. The third author acknowledges financial support from JSPS the Grant-in-Aid of Scientific Research (C) Grant Number 19K03616 and Reserch Origin for Dressed Photon. The fourth author was supported by JSPS KAKENHI Grant Numbers JP26800054 and JP18K03327. The fifth author acknowledges JSPS the Grant-in-Aid for Scientific Research (C) 26400156 and 18K03354.
Citation
Masaya MAEDA. Hironobu SASAKI. Etsuo SEGAWA. Akito SUZUKI. Kanako SUZUKI. "Dispersive estimates for quantum walks on 1D lattice." J. Math. Soc. Japan 74 (1) 217 - 246, January, 2022. https://doi.org/10.2969/jmsj/85218521
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