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For given two unitary and self-adjoint operators on a Hilbert space, aspectral mapping theorem was proved in \cite{HiSeSu}. In this paper, as anapplication of the spectral mapping theorem, we investigate the spectrum of aone-dimensional split-step quantum walk. We give a criterion for when there isno eigenvalues around $\pm 1$ in terms of a discriminant operator. We alsoprovide a criterion for when eigenvalues $\pm 1$ exist in terms of birtheigenspaces. Moreover, we prove that eigenvectors from the birth eigenspacesdecay exponentially at spatial infinity and that the birth eigenspaces arerobust against perturbations.

Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations