Continuous limits of linear and nonlinear quantum walks

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 $δ\mathbb Z$ uniformly converges (in Sobolev space $H^s$) to the solution to a nonlinear Dirac equation (NLD) on a fixed time interval as $δ\to 0$. Here, to compare the walker defined on $δ\mathbb Z$ and the solution to the NLD defined on $\mathbb R$, we use Shannon interpolation.

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