Spectral analysis for a multi-dimensional split-step quantum walk with a defect

This paper studies the spectrum of a multi-dimensional split-step quantum walk with a defect that cannot be analyzed in previous papers by Fuda et al. (Quantum Inf Process 18:203–226, 2017; J Math Phys 59:082201, 2018). To this end, we have developed a new technique which allow us to use a spectral mapping theorem for the one-defect model. We also derive the time-averaged limit measure for one-dimensional case as an application of the spectral analysis.