Scaling limit of the time averaged distribution for continuous time quantum walk and Szegedy's walk on the path

In this paper, we consider Szegedy’s walk, a type of discrete time quantum walk, and corresponding continuous time quantum walk related to the birth and death chain. We show that the scaling limit of time averaged distribution for the continuous time quantum walk induces that of Szegedy’s walk if there exists the spectral gap on so-called the corresponding Jacobi matrix .