Continuous limits of linear and nonlinear quantum walks

Abstract

In this paper, we consider the continuous limit of a nonlinear quantum walk (NLQW) that incorporates a linear quantum walk as a special case. In particular, we rigorously prove that the walker (solution) of the NLQW on a lattice $\delta \mathbb{Z}$ uniformly converges (in Sobolev space $H^s$) to the solution to a nonlinear Dirac equation (NLD) on a fixed time interval as $\delta \to 0$. Here, to compare the walker defined on $\delta \mathbb{Z}$ and the solution to the NLD defined on $\mathbb{R}$, we use Shannon interpolation.

19 pages, to appear in Reviews in Mathematical Physics