Localization of a multi-dimensional quantum walk with one defect

In this paper, we introduce a multi-dimensional generalization of Kitagawa’s split-step discrete-time quantum walk, study the spectrum of its evolution operator for the case of one-defect coins, and prove localization of the walk. Using a spectral mapping theorem, we can reduce the spectral analysis of the evolution operator to that of a discrete Schrödinger operator with variable coefficients, which is analyzed using the Feshbach map.