Dispersive estimates for quantum walks on 1D lattice

17 Aug 2018  ·  Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki, Kanako Suzuki ·

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. read more

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Mathematical Physics Analysis of PDEs Mathematical Physics Spectral Theory 35Q41, 81U30