Feynman Integral in Quantum Walk, Barrier-top Scattering and Hadamard Walk

The aim of this article is to relate the discrete quantum walk on $\mathbb{Z}$ with the continuous Schr\"odinger operator on $\mathbb{R}$ in the scattering problem. Each point of $\mathbb{Z}$ is associated with a barrier of the potential, and the coin operator of the quantum walk is determined by the transfer matrix between bases of WKB solutions on the classically allowed regions of both sides of the barrier. This correspondence enables us to represent each entry of the scattering matrix of the Schr\"odinger operator as a countable sum of probability amplitudes associated with the paths of the quantum walker. In particular, the barrier-top scattering corresponds to the Hadamard walk in the semiclassical limit.